[ { "id": "quadratic-formula", "name": "Quadratic Formula", "latex": "x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}", "category": "Algebra", "description": "Finds the roots of a quadratic equation ax² + bx + c = 0 when a is nonzero.", "variables": [ "a: coefficient of x² (a ≠ 0)", "b: coefficient of x", "c: constant term", "x: solution or root" ], "notes": [ "The discriminant b² − 4ac determines the number and type of real roots.", "Use the ± sign to calculate both possible roots." ], "examples": [ "For x² − 5x + 6 = 0, the roots are x = 2 and x = 3." ], "seo": { "title": "Quadratic Formula: Formula, Worked Example & Checks", "description": "Learn the quadratic formula with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence", "binomial-theorem" ], "assumptions": [ "The discriminant b² − 4ac determines the number and type of real roots.", "For the Quadratic Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible." ], "commonMistakes": [ "Do not collapse the ± in Quadratic Formula into one value; calculate each admissible branch and check it in the original problem.", "For the Quadratic Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Solve x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6. The discriminant is 1, so the two branches give x = 3 and x = 2.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the coefficients from standard form: a = 1, b = −5, and c = 6.", "Compute the discriminant: b² − 4ac = 25 − 24 = 1.", "Substitute into the formula: x = (5 ± √1) / 2.", "Evaluate both branches to obtain x = 3 and x = 2." ], "verification": "Substituting 3 gives 9 − 15 + 6 = 0, and substituting 2 gives 4 − 10 + 6 = 0, so both roots satisfy the original equation.", "domain": "Use for equations in the form ax² + bx + c = 0 with a ≠ 0; complex roots are allowed when the discriminant is negative.", "units": [ "a, b, c, and x use compatible problem-specific units" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Quadratic Formula, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=1, b=-5, c=6", "workedOutput": "x=2 and x=3", "notApplicableWhen": [ "Do not use Quadratic Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "slope-intercept-form", "name": "Slope-Intercept Form", "latex": "y=mx+b", "category": "Algebra", "description": "Expresses a straight line using its slope and vertical intercept.", "variables": [ "m: slope of the line", "b: y-intercept", "x: horizontal coordinate", "y: vertical coordinate" ], "notes": [ "A positive slope rises from left to right; a negative slope falls.", "The point (0, b) is where the line crosses the y-axis." ], "examples": [ "The line y = 2x + 3 has slope 2 and y-intercept 3." ], "seo": { "title": "Slope-Intercept Form: Formula, Worked Example & Checks", "description": "Learn the Slope-Intercept Form, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "distance-formula", "arithmetic-sequence", "geometric-sequence", "binomial-theorem" ], "assumptions": [ "A positive slope rises from left to right; a negative slope falls.", "For the Slope-Intercept Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Slope-Intercept Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Slope-Intercept Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For y = 2x + 3 and x = 4, the output is y = 2(4) + 3 = 11. The line crosses the y-axis at 3 and rises 2 units per 1 unit of run.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Read the slope and intercept from the equation: m = 2 and b = 3.", "Substitute x = 4 into y = mx + b.", "Multiply 2 × 4 = 8, then add the intercept 3.", "Report the point on the line as (4, 11)." ], "verification": "Using the intercept point (0, 3), the slope between (0, 3) and (4, 11) is (11 − 3)/(4 − 0) = 2, matching the equation.", "domain": "Represents nonvertical straight lines in a Cartesian plane; vertical lines cannot be written with a finite slope m.", "units": [ "m has y-units per x-unit", "b and y use the same units" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Slope-Intercept Form.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Slope-Intercept Form, check zero, negative, and extreme input values before relying on the result.", "When using Slope-Intercept Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form y=mx+b for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "m=2, b=3, x=4", "workedOutput": "y=11", "notApplicableWhen": [ "Do not use Slope-Intercept Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "distance-formula", "name": "Distance Formula", "latex": "d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}", "category": "Algebra", "description": "Calculates the straight-line distance between two points in a coordinate plane.", "variables": [ "(x₁, y₁): first point", "(x₂, y₂): second point", "d: distance between the points" ], "notes": [ "The formula follows from the Pythagorean theorem.", "The order of the two points does not change the result." ], "examples": [ "Points (1, 2) and (4, 6) are 5 units apart." ], "seo": { "title": "Distance Formula: Formula, Worked Example & Checks", "description": "Learn the distance formula with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "arithmetic-sequence", "geometric-sequence", "binomial-theorem" ], "assumptions": [ "The order of the two points does not change the result.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Distance Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Do not drop the radical boundary in Distance Formula; verify exactly which terms are inside the root and whether the chosen root is valid.", "For the Distance Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "The distance between (1, 2) and (4, 6) is √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = 5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Find the horizontal change: Δx = 4 − 1 = 3.", "Find the vertical change: Δy = 6 − 2 = 4.", "Square and add the changes: 3² + 4² = 25.", "Take the nonnegative square root to obtain a distance of 5." ], "verification": "The coordinate differences form a 3–4–5 right triangle, so the computed straight-line distance of 5 is consistent with the Pythagorean theorem.", "domain": "Applies to two points in a Euclidean Cartesian plane; the result is always nonnegative and independent of point order.", "units": [ "both coordinate axes must use compatible length units" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Distance Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Distance Formula, check zero, negative, and extreme input values before relying on the result.", "When using Distance Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "(x1,y1)=(1,2), (x2,y2)=(4,6)", "workedOutput": "d=5", "notApplicableWhen": [ "Do not use Distance Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "arithmetic-sequence", "name": "Arithmetic Sequence Formula", "latex": "a_n=a_1+(n-1)d", "category": "Algebra", "description": "Finds the nth term of a sequence with a constant difference.", "variables": [ "aₙ: nth term", "a₁: first term", "n: term number", "d: common difference" ], "notes": [ "The difference between consecutive terms must be constant.", "Term numbering begins at n = 1." ], "examples": [ "For 3, 7, 11, … the 10th term is 39." ], "seo": { "title": "Arithmetic Sequence Formula: Formula, Worked Example & Checks", "description": "Learn the Arithmetic Sequence Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "geometric-sequence", "binomial-theorem" ], "assumptions": [ "The difference between consecutive terms must be constant.", "For the Arithmetic Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Arithmetic Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Arithmetic Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For a sequence with first term 3 and common difference 4, the tenth term is a₁₀ = 3 + (10 − 1)4 = 39.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify a₁ = 3, d = 4, and n = 10.", "Compute n − 1 = 9 intervals from the first term.", "Multiply the common difference: 9 × 4 = 36.", "Add the first term to obtain a₁₀ = 39." ], "verification": "Listing the terms 3, 7, 11, … shows that adding 4 nine times reaches 39 at the tenth position.", "domain": "Applies to sequences with a constant additive difference between consecutive terms and integer term index n ≥ 1.", "units": [ "a₁, d, and aₙ use the same units", "n is dimensionless" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Sequence Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Arithmetic Sequence Formula, check zero, negative, and extreme input values before relying on the result.", "When using Arithmetic Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a_n=a_1+(n-1)d for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a1=3, d=4, n=10", "workedOutput": "a10=39", "notApplicableWhen": [ "Do not use Arithmetic Sequence Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "geometric-sequence", "name": "Geometric Sequence Formula", "latex": "a_n=a_1r^{n-1}", "category": "Algebra", "description": "Finds the nth term of a sequence with a constant ratio.", "variables": [ "aₙ: nth term", "a₁: first term", "r: common ratio", "n: term number" ], "notes": [ "Each term is obtained by multiplying the previous term by r.", "A negative ratio causes alternating signs." ], "examples": [ "For 2, 6, 18, … the 5th term is 162." ], "seo": { "title": "Geometric Sequence Formula: Formula, Worked Example & Checks", "description": "Learn the Geometric Sequence Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "binomial-theorem" ], "assumptions": [ "Each term is obtained by multiplying the previous term by r.", "For the Geometric Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Geometric Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Geometric Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For a sequence with first term 2 and common ratio 3, the fifth term is a₅ = 2·3⁴ = 162.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify a₁ = 2, r = 3, and n = 5.", "Compute the exponent n − 1 = 4.", "Evaluate the repeated factor: 3⁴ = 81.", "Multiply by the first term: 2 × 81 = 162." ], "verification": "The sequence 2, 6, 18, 54, 162 multiplies by 3 at every step and confirms the fifth term.", "domain": "Applies to sequences with a constant multiplicative ratio between consecutive nonzero terms and integer index n ≥ 1.", "units": [ "a₁ and aₙ use the same units", "r and n are dimensionless" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Geometric Sequence Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Geometric Sequence Formula, check zero, negative, and extreme input values before relying on the result.", "When using Geometric Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a_n=a_1r^{n-1} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a1=2, r=3, n=5", "workedOutput": "a5=162", "notApplicableWhen": [ "Do not use Geometric Sequence Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "binomial-theorem", "name": "Binomial Theorem", "latex": "(a+b)^n=\\sum_{k=0}^{n}\\binom{n}{k}a^{n-k}b^k", "category": "Algebra", "description": "Expands a nonnegative integer power of a binomial.", "variables": [ "a, b: binomial terms", "n: nonnegative integer exponent", "k: summation index", "C(n,k): binomial coefficient" ], "notes": [ "The coefficients are the entries in row n of Pascal’s triangle.", "The exponents in every term add to n." ], "examples": [ "(a + b)³ = a³ + 3a²b + 3ab² + b³." ], "seo": { "title": "Binomial Theorem: Formula, Worked Example & Checks", "description": "Learn the binomial theorem with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "The coefficients are the entries in row n of Pascal’s triangle.", "For the Binomial Theorem, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Binomial Theorem, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Do not omit the index or bounds in Binomial Theorem; changing either one changes which terms are included.", "For the Binomial Theorem, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Expand (2+3)³. The theorem gives 2³+3·2²·3+3·2·3²+3³.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Binomial Theorem.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Binomial Theorem only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Set n=3, a=2, and b=3.", "Evaluate the four binomial terms: 8, 36, 54, and 27.", "Add the terms to obtain 8+36+54+27=125." ], "verification": "Direct evaluation gives (2+3)³=5³=125, matching the expansion.", "boundaryCases": [ "For Binomial Theorem, check zero, negative, and extreme input values before relying on the result.", "When using Binomial Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form (a+b)^n=\\sum_{k=0}^{n}\\binom{n}{k}a^{n-k}b^k for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "(a+b)^3 with a=2 and b=3", "workedOutput": "125", "notApplicableWhen": [ "Do not use Binomial Theorem when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "pythagorean-theorem", "name": "Pythagorean Theorem", "latex": "a^2+b^2=c^2", "category": "Geometry", "description": "Relates the side lengths of a right triangle.", "variables": [ "a, b: perpendicular leg lengths", "c: hypotenuse length" ], "notes": [ "The theorem applies only to right triangles.", "The hypotenuse c is opposite the right angle and is the longest side." ], "examples": [ "A triangle with legs 3 and 4 has hypotenuse 5." ], "seo": { "title": "Pythagorean Theorem: Formula, Worked Example & Checks", "description": "Learn the pythagorean theorem with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "circle-area", "circle-circumference", "triangle-area", "sphere-volume", "midpoint-formula" ], "assumptions": [ "The theorem applies only to right triangles.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Pythagorean Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Pythagorean Theorem, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For a right triangle with legs 3 and 4, calculate the hypotenuse: c = √(3² + 4²) = √25 = 5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Confirm that the triangle is right-angled and that a and b are the perpendicular legs.", "Square the leg lengths: 3² = 9 and 4² = 16.", "Add the squared lengths: 9 + 16 = 25.", "Take the nonnegative square root to obtain c = 5." ], "verification": "The result forms the known 3–4–5 right triangle, and 5² = 25 equals 3² + 4² = 25.", "domain": "Applies only to Euclidean right triangles; c denotes the side opposite the right angle and must be the longest side.", "units": [ "a, b, and c must use the same length unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Pythagorean Theorem.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Pythagorean Theorem, check zero, negative, and extreme input values before relying on the result.", "When using Pythagorean Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a^2+b^2=c^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=3, b=4", "workedOutput": "c=5", "notApplicableWhen": [ "Do not use Pythagorean Theorem when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "circle-area", "name": "Area of a Circle", "latex": "A=\\pi r^2", "category": "Geometry", "description": "Calculates the area enclosed by a circle of radius r.", "variables": [ "A: area", "r: radius", "π: circle constant, approximately 3.14159" ], "notes": [ "The radius is half the diameter.", "Area is expressed in square units." ], "examples": [ "A circle with radius 3 has area 9π square units." ], "seo": { "title": "Area of a Circle: Formula, Worked Example & Checks", "description": "Learn the area of a circle with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-circumference", "triangle-area", "sphere-volume", "midpoint-formula" ], "assumptions": [ "The radius is half the diameter.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Area of a Circle, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Area of a Circle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For a circle with radius 3 cm, A = π(3 cm)² = 9π cm² ≈ 28.274 cm².", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the radius as r = 3 cm.", "Square the radius: r² = 9 cm².", "Multiply by π to obtain the exact area 9π cm².", "Round only at the end to obtain approximately 28.274 cm²." ], "verification": "Dividing the calculated area by r² gives π, and the area is positive with squared length units as required.", "domain": "Applies to a Euclidean circle with radius r ≥ 0; use the radius rather than the diameter unless you first divide the diameter by two.", "units": [ "r uses a length unit", "A uses the corresponding squared length unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of a Circle.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Area of a Circle, check zero, negative, and extreme input values before relying on the result.", "When using Area of a Circle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=\\pi r^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=3 cm", "workedOutput": "A=9π cm²≈28.274 cm²", "notApplicableWhen": [ "Do not use Area of a Circle when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "circle-circumference", "name": "Circumference of a Circle", "latex": "C=2\\pi r=\\pi d", "category": "Geometry", "description": "Calculates the distance around a circle from its radius or diameter.", "variables": [ "C: circumference", "r: radius", "d: diameter", "π: circle constant" ], "notes": [ "The diameter equals 2r.", "Circumference is measured in linear units." ], "examples": [ "A circle with radius 5 has circumference 10π units." ], "seo": { "title": "Circumference of a Circle: Formula, Worked Example & Checks", "description": "Learn the Circumference of a Circle, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "triangle-area", "sphere-volume", "midpoint-formula" ], "assumptions": [ "The diameter equals 2r.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Circumference of a Circle, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Circumference of a Circle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For a circle with radius 4 m, C = 2π(4 m) = 8π m ≈ 25.133 m.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the radius as r = 4 m.", "Multiply the radius by 2 to obtain the diameter 8 m.", "Multiply the diameter by π to obtain the exact circumference 8π m.", "Round at the final step to approximately 25.133 m." ], "verification": "The ratio C/(2r) equals π, and the result has a length unit rather than a squared unit.", "domain": "Applies to a Euclidean circle with radius r ≥ 0; the equivalent form C = πd may be used when the diameter is known.", "units": [ "r and C use the same length unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Circumference of a Circle.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Circumference of a Circle, check zero, negative, and extreme input values before relying on the result.", "When using Circumference of a Circle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form C=2\\pi r=\\pi d for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=4 m", "workedOutput": "C=8π m≈25.133 m", "notApplicableWhen": [ "Do not use Circumference of a Circle when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "triangle-area", "name": "Area of a Triangle", "latex": "A=\\frac{1}{2}bh", "category": "Geometry", "description": "Calculates triangle area from a base and its perpendicular height.", "variables": [ "A: area", "b: chosen base length", "h: perpendicular height to that base" ], "notes": [ "The height must be perpendicular to the base.", "Any side can serve as the base if the matching height is used." ], "examples": [ "A triangle with base 8 and height 5 has area 20 square units." ], "seo": { "title": "Area of a Triangle: Formula, Worked Example & Checks", "description": "Learn the area of a triangle with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "sphere-volume", "midpoint-formula" ], "assumptions": [ "The height must be perpendicular to the base.", "For the Area of a Triangle, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Area of a Triangle, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Area of a Triangle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For a triangle with base 10 cm and perpendicular height 6 cm, A = ½(10)(6) = 30 cm².", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Choose a side as the base, b = 10 cm.", "Use the perpendicular distance to that base as h = 6 cm.", "Multiply the base and height: 10 × 6 = 60 cm².", "Divide by two to obtain A = 30 cm²." ], "verification": "A rectangle with the same base and height has area 60 cm², so the triangle occupies exactly half that area.", "domain": "Applies to any Euclidean triangle when h is measured perpendicular to the selected base, including when the altitude falls outside an obtuse triangle.", "units": [ "b and h use compatible length units", "A uses squared length units" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of a Triangle.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Area of a Triangle, check zero, negative, and extreme input values before relying on the result.", "When using Area of a Triangle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=\\frac{1}{2}bh for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "b=10 cm, h=6 cm", "workedOutput": "A=30 cm²", "notApplicableWhen": [ "Do not use Area of a Triangle when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "sphere-volume", "name": "Volume of a Sphere", "latex": "V=\\frac{4}{3}\\pi r^3", "category": "Geometry", "description": "Calculates the volume enclosed by a sphere.", "variables": [ "V: volume", "r: sphere radius", "π: circle constant" ], "notes": [ "Volume is measured in cubic units.", "Doubling the radius multiplies the volume by eight." ], "examples": [ "A sphere with radius 3 has volume 36π cubic units." ], "seo": { "title": "Volume of a Sphere: Formula, Worked Example & Checks", "description": "Learn the volume of a sphere with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "midpoint-formula" ], "assumptions": [ "Volume is measured in cubic units.", "For the Volume of a Sphere, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Volume of a Sphere, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Volume of a Sphere, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For a sphere with radius 3 cm, V = (4/3)π(3³) = 36π cm³ ≈ 113.097 cm³.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the radius as r = 3 cm.", "Cube the radius: r³ = 27 cm³.", "Multiply 27 by 4/3 to obtain 36.", "Multiply by π for the exact volume 36π cm³, then approximate if needed." ], "verification": "Dividing the result by r³ gives 4π/3, and the answer uses cubic length units as a volume must.", "domain": "Applies to a Euclidean sphere with radius r ≥ 0; do not substitute the diameter directly for r.", "units": [ "r uses a length unit", "V uses the corresponding cubic length unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Volume of a Sphere.", "verifiedAgainst": "OpenStax Algebra and Trigonometry", "sourceIds": [ "openstax-algebra-trigonometry" ], "qualityTier": "core", "boundaryCases": [ "For Volume of a Sphere, check zero, negative, and extreme input values before relying on the result.", "When using Volume of a Sphere, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form V=\\frac{4}{3}\\pi r^3 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=3 cm", "workedOutput": "V=36π cm³≈113.097 cm³", "notApplicableWhen": [ "Do not use Volume of a Sphere when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "derivative-definition", "name": "Derivative Definition", "latex": "f'(x)=\\lim_{h\\to0}\\frac{f(x+h)-f(x)}{h}", "category": "Calculus", "description": "Defines the instantaneous rate of change of a function as a limit.", "variables": [ "f(x): original function", "h: change in the input", "f′(x): derivative at x" ], "notes": [ "The limit must exist for the derivative to exist.", "The quotient represents the slope of a secant line before h approaches zero." ], "examples": [ "Applying the definition to f(x) = x² gives f′(x) = 2x." ], "seo": { "title": "Derivative Definition: Formula, Worked Example & Checks", "description": "Learn the Derivative Definition, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts", "power-rule-derivative" ], "assumptions": [ "The limit must exist for the derivative to exist.", "For the Derivative Definition, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative Definition, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For f(x) = x², [f(x+h) − f(x)]/h = [(x+h)² − x²]/h = 2x + h. Taking h → 0 gives f′(x) = 2x.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Substitute f(x) = x² into the difference quotient.", "Expand (x + h)² and subtract x² to obtain 2xh + h².", "Factor h and cancel it for h ≠ 0, leaving 2x + h.", "Take the limit as h approaches 0 to obtain 2x." ], "verification": "The result agrees with the power rule, and numerical secant slopes near a chosen x approach 2x as h becomes smaller.", "domain": "The derivative exists at x only when the difference quotient approaches a finite common limit from all required directions.", "units": [ "derivative units are function-output units per x-unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Derivative Definition.", "verifiedAgainst": "OpenStax Calculus Volume 1", "sourceIds": [ "openstax-calculus-volume-1" ], "qualityTier": "core", "boundaryCases": [ "For Derivative Definition, check zero, negative, and extreme input values before relying on the result.", "When using Derivative Definition, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form f'(x)=\\lim_{h\\to0}\\frac{f(x+h)-f(x)}{h} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f(x)=x² at x=3", "workedOutput": "f′(3)=6", "notApplicableWhen": [ "Do not use Derivative Definition without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "product-rule", "name": "Product Rule", "latex": "(fg)'=f'g+fg'", "category": "Calculus", "description": "Differentiates the product of two differentiable functions.", "variables": [ "f, g: differentiable functions", "f′, g′: their derivatives" ], "notes": [ "Differentiate one factor at a time while keeping the other factor unchanged.", "Do not multiply the two derivatives together." ], "examples": [ "For x²sin x, the derivative is 2x sin x + x² cos x." ], "seo": { "title": "Product Rule: Formula, Worked Example & Checks", "description": "Learn the product rule with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "derivative-definition", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts", "power-rule-derivative" ], "assumptions": [ "Differentiate one factor at a time while keeping the other factor unchanged.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Before substituting values into Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For f(x) = x² sin x, f′(x) = (2x)sin x + x² cos x.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Separate the product into u(x) = x² and v(x) = sin x.", "Differentiate each factor: u′(x) = 2x and v′(x) = cos x.", "Apply (uv)′ = u′v + uv′ without dropping either term.", "Substitute the factors to obtain 2x sin x + x² cos x." ], "verification": "At x = 0, both the symbolic derivative and a local numerical slope are 0, providing a simple consistency check.", "domain": "Applies where both factor functions are differentiable; it is not equivalent to multiplying the two derivatives.", "units": [ "derivative units are product-output units per input unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Product Rule.", "verifiedAgainst": "OpenStax Calculus Volume 1", "sourceIds": [ "openstax-calculus-volume-1" ], "qualityTier": "core", "boundaryCases": [ "For Product Rule, check zero, negative, and extreme input values before relying on the result.", "When using Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form (fg)'=f'g+fg' for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f=x², g=sin x", "workedOutput": "(fg)′=2x sin x+x² cos x", "notApplicableWhen": [ "Do not use Product Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "chain-rule", "name": "Chain Rule", "latex": "\\frac{d}{dx}f(g(x))=f'(g(x))g'(x)", "category": "Calculus", "description": "Differentiates a composite function by combining outer and inner derivatives.", "variables": [ "f: outer function", "g: inner function", "x: independent variable" ], "notes": [ "Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.", "The rule can be applied repeatedly to nested compositions." ], "examples": [ "The derivative of sin(x²) is 2x cos(x²)." ], "seo": { "title": "Chain Rule: Formula, Worked Example & Checks", "description": "Learn the chain rule with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "fundamental-theorem-calculus", "integration-by-parts", "power-rule-derivative" ], "assumptions": [ "Differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative.", "For the Chain Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Chain Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For y = (3x + 1)⁴, let u = 3x + 1. Then dy/dx = 4u³·3 = 12(3x + 1)³.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the outer function u⁴ and the inner function u = 3x + 1.", "Differentiate the outer function with respect to u: 4u³.", "Differentiate the inner function with respect to x: du/dx = 3.", "Multiply the derivatives and replace u to obtain 12(3x + 1)³." ], "verification": "Expanding (3x + 1)⁴ first and differentiating term by term produces the same polynomial as expanding 12(3x + 1)³.", "domain": "Applies when a differentiable outer function is composed with a differentiable inner function at the point being evaluated.", "units": [ "derivative units are output units per input unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Chain Rule.", "verifiedAgainst": "OpenStax Calculus Volume 1", "sourceIds": [ "openstax-calculus-volume-1" ], "qualityTier": "core", "boundaryCases": [ "For Chain Rule, check zero, negative, and extreme input values before relying on the result.", "When using Chain Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\frac{d}{dx}f(g(x))=f'(g(x))g'(x) for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "y=(3x+1)^4", "workedOutput": "dy/dx=12(3x+1)^3", "notApplicableWhen": [ "Do not use Chain Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "fundamental-theorem-calculus", "name": "Fundamental Theorem of Calculus", "latex": "\\int_a^b f(x)\\,dx=F(b)-F(a),\\quad F'=f", "category": "Calculus", "description": "Connects definite integration with antiderivatives.", "variables": [ "f: integrand", "F: any antiderivative of f", "a, b: integration limits" ], "notes": [ "The function must satisfy the theorem’s continuity conditions on the interval.", "Evaluate the antiderivative at the upper limit minus the lower limit." ], "examples": [ "∫₀¹ 2x dx = [x²]₀¹ = 1." ], "seo": { "title": "Fundamental Theorem of Calculus: Formula, Worked Example & Checks", "description": "Learn the Fundamental Theorem of Calculus, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "integration-by-parts", "power-rule-derivative" ], "assumptions": [ "The function must satisfy the theorem’s continuity conditions on the interval.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Fundamental Theorem of Calculus, and include a constant of integration for an indefinite integral.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Evaluate ∫₀¹2x dx. An antiderivative is F(x)=x², so F(1)-F(0)=1.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Fundamental Theorem of Calculus.", "sourceIds": [ "openstax-calculus-volume-1" ], "verifiedAgainst": "OpenStax Calculus Volume 1", "domain": "Apply Fundamental Theorem of Calculus only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Choose F(x)=x² because F′(x)=2x.", "Evaluate the upper endpoint: F(1)=1.", "Evaluate the lower endpoint and subtract: 1-0=1." ], "verification": "A geometric check gives the area of a triangle with base 1 and height 2, which is also 1.", "boundaryCases": [ "For Fundamental Theorem of Calculus, check zero, negative, and extreme input values before relying on the result.", "When using Fundamental Theorem of Calculus, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\int_a^b f(x)\\,dx=F(b)-F(a),\\quad F'=f for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f(x)=2x on [0,1]", "workedOutput": "∫₀¹2x dx=1", "notApplicableWhen": [ "Do not use Fundamental Theorem of Calculus without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "integration-by-parts", "name": "Integration by Parts", "latex": "\\int u\\,dv=uv-\\int v\\,du", "category": "Calculus", "description": "Transforms an integral of a product using the product rule in reverse.", "variables": [ "u: selected function to differentiate", "dv: selected differential to integrate", "du: derivative of u", "v: antiderivative of dv" ], "notes": [ "Choose u so that differentiating it simplifies the expression.", "Remember the minus sign before the remaining integral." ], "examples": [ "∫x eˣ dx = xeˣ − eˣ + C." ], "seo": { "title": "Integration by Parts: Formula, Worked Example & Checks", "description": "Learn the Integration by Parts, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "power-rule-derivative" ], "assumptions": [ "Choose u so that differentiating it simplifies the expression.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Integration by Parts, and include a constant of integration for an indefinite integral.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Evaluate ∫x eˣ dx. Choose u = x and dv = eˣdx, giving du = dx and v = eˣ. The result is xeˣ − eˣ + C = eˣ(x − 1) + C.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Choose u = x because it simplifies when differentiated, and choose dv = eˣ dx.", "Compute du = dx and integrate dv to obtain v = eˣ.", "Apply ∫u dv = uv − ∫v du to get xeˣ − ∫eˣ dx.", "Integrate the remaining term and include the constant: eˣ(x − 1) + C." ], "verification": "Differentiating eˣ(x − 1) gives eˣ(x − 1) + eˣ = xeˣ, which recovers the original integrand.", "domain": "Applies to integrals where the integrand can be expressed as a product u·dv and both required derivatives and antiderivatives exist.", "units": [ "the antiderivative has integrand units multiplied by the input unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Integration by Parts.", "verifiedAgainst": "OpenStax Calculus Volume 1", "sourceIds": [ "openstax-calculus-volume-1" ], "qualityTier": "core", "boundaryCases": [ "For Integration by Parts, check zero, negative, and extreme input values before relying on the result.", "When using Integration by Parts, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\int u\\,dv=uv-\\int v\\,du for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "∫x e^x dx", "workedOutput": "e^x(x-1)+C", "notApplicableWhen": [ "Do not use Integration by Parts without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "sample-mean", "name": "Sample Mean", "latex": "\\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i", "category": "Statistics", "description": "Calculates the arithmetic average of n observed values.", "variables": [ "x̄: sample mean", "xᵢ: ith observation", "n: number of observations" ], "notes": [ "Every observation contributes equally.", "The sample mean is sensitive to extreme values." ], "examples": [ "The mean of 2, 4, and 9 is 5." ], "seo": { "title": "Sample Mean: Formula, Worked Example & Checks", "description": "Learn the sample mean with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "sample-variance", "z-score", "standard-error-mean", "population-variance", "covariance" ], "assumptions": [ "Every observation contributes equally.", "For the Sample Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Sample Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sample Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For the sample 2, 4, 6, 8, the mean is (2 + 4 + 6 + 8)/4 = 20/4 = 5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Add all sample observations to obtain a total of 20.", "Count the observations: n = 4.", "Divide the total by the sample size: 20/4.", "Report the sample mean as 5 in the original measurement unit." ], "verification": "The deviations from 5 are −3, −1, 1, and 3, which sum to zero as deviations from an arithmetic mean should.", "domain": "Applies to quantitative observations with sample size n > 0; sensitivity to extreme values should be considered for skewed data.", "units": [ "the sample mean has the same unit as the observations" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sample Mean.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Sample Mean, check zero, negative, and extreme input values before relying on the result.", "When using Sample Mean, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\bar{x}=\\frac{1}{n}\\sum_{i=1}^{n}x_i for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "data 2,4,6,8", "workedOutput": "x̄=5", "notApplicableWhen": [ "Do not use Sample Mean until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "sample-variance", "name": "Sample Variance", "latex": "s^2=\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2", "category": "Statistics", "description": "Measures sample spread using squared deviations and Bessel’s correction.", "variables": [ "s²: sample variance", "xᵢ: observation", "x̄: sample mean", "n: sample size" ], "notes": [ "The denominator n − 1 is used when estimating population variance from a sample.", "Variance is expressed in squared units." ], "examples": [ "For 1, 2, 3, the sample variance is 1." ], "seo": { "title": "Sample Variance: Formula, Worked Example & Checks", "description": "Learn the sample variance with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "sample-mean", "z-score", "standard-error-mean", "population-variance", "covariance" ], "assumptions": [ "The denominator n − 1 is used when estimating population variance from a sample.", "For the Sample Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sample Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sample Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Sample Variance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sample Variance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For 2, 4, 6, 8, the mean is 5. Squared deviations sum to 20, so s² = 20/(4 − 1) = 20/3 ≈ 6.667.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Calculate the sample mean: x̄ = 5.", "Compute deviations −3, −1, 1, and 3, then square them.", "Add the squared deviations: 9 + 1 + 1 + 9 = 20.", "Divide by n − 1 = 3 to obtain approximately 6.667." ], "verification": "The variance is nonnegative, and taking its square root gives a sample standard deviation of about 2.582 in the original unit.", "domain": "Uses the n − 1 denominator for a sample estimate when n > 1; population variance uses a different denominator.", "units": [ "variance uses the square of the observation unit" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sample Variance.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Sample Variance, check zero, negative, and extreme input values before relying on the result.", "When using Sample Variance, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form s^2=\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "sample 2,4,6", "workedOutput": "s²=4", "notApplicableWhen": [ "Do not use Sample Variance until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "z-score", "name": "Z-Score Formula", "latex": "z=\\frac{x-\\mu}{\\sigma}", "category": "Statistics", "description": "Expresses how many population standard deviations a value lies from the mean.", "variables": [ "z: standardized score", "x: observed value", "μ: population mean", "σ: population standard deviation" ], "notes": [ "A positive z-score lies above the mean; a negative one lies below.", "Use sample statistics only when the method specifically calls for them." ], "examples": [ "If x = 70, μ = 60, and σ = 5, then z = 2." ], "seo": { "title": "Z-Score Formula: Formula, Worked Example & Checks", "description": "Learn the z-score formula with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "standard-error-mean", "population-variance", "covariance" ], "assumptions": [ "A positive z-score lies above the mean; a negative one lies below.", "For the Z-Score Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Z-Score Formula, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Z-Score Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Z-Score Formula, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For x = 85, mean μ = 70, and standard deviation σ = 10, z = (85 − 70)/10 = 1.5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Subtract the mean from the observation: 85 − 70 = 15.", "Confirm the standard deviation is positive: σ = 10.", "Divide the deviation by the standard deviation: 15/10.", "Report z = 1.5, meaning the value is 1.5 standard deviations above the mean." ], "verification": "Rearranging x = μ + zσ gives 70 + 1.5(10) = 85, recovering the original observation.", "domain": "The standard deviation must be positive; interpretation as a normal-distribution percentile additionally requires an appropriate distribution model.", "units": [ "x, μ, and σ use the same unit", "z is dimensionless" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Z-Score Formula.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Z-Score Formula, check zero, negative, and extreme input values before relying on the result.", "When using Z-Score Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form z=\\frac{x-\\mu}{\\sigma} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "x=85, μ=70, σ=10", "workedOutput": "z=1.5", "notApplicableWhen": [ "Do not use Z-Score Formula until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "standard-error-mean", "name": "Standard Error of the Mean", "latex": "SE_{\\bar{x}}=\\frac{s}{\\sqrt{n}}", "category": "Statistics", "description": "Estimates the standard deviation of the sample mean’s sampling distribution.", "variables": [ "SE: standard error", "s: sample standard deviation", "n: sample size" ], "notes": [ "The standard error decreases as sample size increases.", "It is not the same as the standard deviation of individual observations." ], "examples": [ "If s = 12 and n = 36, the standard error is 2." ], "seo": { "title": "Standard Error of the Mean: Formula, Worked Example & Checks", "description": "Learn the Standard Error of the Mean, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "population-variance", "covariance" ], "assumptions": [ "The standard error decreases as sample size increases.", "For the Standard Error of the Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Standard Error of the Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Standard Error of the Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Standard Error of the Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For sample standard deviation s = 12 and sample size n = 36, SEM = 12/√36 = 12/6 = 2.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the sample standard deviation as s = 12.", "Identify the sample size as n = 36.", "Compute the square root of the sample size: √36 = 6.", "Divide s by √n to obtain a standard error of 2." ], "verification": "Quadrupling the sample size from 36 to 144 would halve the standard error from 2 to 1, consistent with the inverse square-root relationship.", "domain": "Assumes independent observations and a sampling setting where the sample mean has a meaningful standard error; n must be positive.", "units": [ "SEM has the same unit as the measured variable" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Standard Error of the Mean.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Standard Error of the Mean, check zero, negative, and extreme input values before relying on the result.", "When using Standard Error of the Mean, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form SE_{\\bar{x}}=\\frac{s}{\\sqrt{n}} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "s=12, n=36", "workedOutput": "SE=2", "notApplicableWhen": [ "Do not use Standard Error of the Mean until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "bayes-theorem", "name": "Bayes’ Theorem", "latex": "P(A\\mid B)=\\frac{P(B\\mid A)P(A)}{P(B)}", "category": "Probability", "description": "Updates the probability of an event after observing evidence.", "variables": [ "P(A|B): posterior probability", "P(B|A): likelihood", "P(A): prior probability", "P(B): evidence probability" ], "notes": [ "P(B) must be greater than zero.", "The denominator can be expanded with the law of total probability when needed." ], "examples": [ "Bayes’ theorem is commonly used to interpret medical test results." ], "seo": { "title": "Bayes’ Theorem: Formula, Worked Example & Checks", "description": "Learn the bayes’ theorem with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula", "probability-complement" ], "assumptions": [ "P(B) must be greater than zero.", "For the Bayes’ Theorem, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Bayes’ Theorem, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "With 1% prevalence, 90% sensitivity, and 5% false-positive rate, P(condition|positive) = 0.009/(0.009 + 0.0495) ≈ 0.1538, or 15.38%.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Compute the true-positive joint probability: 0.01 × 0.90 = 0.009.", "Compute the false-positive joint probability: 0.99 × 0.05 = 0.0495.", "Add both paths to get P(positive) = 0.0585.", "Divide 0.009 by 0.0585 to obtain approximately 0.1538." ], "verification": "In a hypothetical 10,000-person group, about 90 true positives and 495 false positives produce 90/585 = 15.38%, matching the formula.", "domain": "Requires defined prior and conditional probabilities with a nonzero probability for the observed evidence in the denominator.", "units": [ "all probabilities are dimensionless values from 0 to 1 or equivalent percentages" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Bayes’ Theorem.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Bayes’ Theorem, check zero, negative, and extreme input values before relying on the result.", "When using Bayes’ Theorem, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form P(A\\mid B)=\\frac{P(B\\mid A)P(A)}{P(B)} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "P(A)=0.01, P(B|A)=0.90, P(B)=0.05", "workedOutput": "P(A|B)=0.18", "notApplicableWhen": [ "Do not use Bayes’ Theorem until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "conditional-probability", "name": "Conditional Probability", "latex": "P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)}", "category": "Probability", "description": "Calculates the probability of A when B is known to have occurred.", "variables": [ "P(A|B): probability of A given B", "P(A ∩ B): probability both occur", "P(B): probability of the condition" ], "notes": [ "P(B) must be nonzero.", "Conditional probability changes the relevant sample space to outcomes in B." ], "examples": [ "If 20% of people meet both conditions and 50% meet B, then P(A|B) = 0.4." ], "seo": { "title": "Conditional Probability: Formula, Worked Example & Checks", "description": "Learn the Conditional Probability, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "binomial-probability", "permutation-formula", "combination-formula", "probability-complement" ], "assumptions": [ "P(B) must be nonzero.", "For the Conditional Probability, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Conditional Probability, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Among 18 students who study mathematics, 12 also study coding. P(coding|mathematics) = 12/18 = 2/3 ≈ 66.7%.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Restrict the sample space to the 18 students satisfying the condition.", "Count the 12 students who satisfy both mathematics and coding.", "Divide the intersection count by the condition count: 12/18.", "Simplify to 2/3 and convert to approximately 66.7%." ], "verification": "Multiplying P(coding|mathematics) by P(mathematics) recovers the joint probability P(coding ∩ mathematics) for the same population.", "domain": "The conditioning event must have nonzero probability; all counts or probabilities must refer to the same population and sampling frame.", "units": [ "probabilities are dimensionless", "counts use the same population base" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Conditional Probability.", "verifiedAgainst": "OpenStax Introductory Statistics", "sourceIds": [ "openstax-intro-statistics" ], "qualityTier": "core", "boundaryCases": [ "For Conditional Probability, check zero, negative, and extreme input values before relying on the result.", "When using Conditional Probability, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form P(A\\mid B)=\\frac{P(A\\cap B)}{P(B)} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "P(A∩B)=0.20, P(B)=0.50", "workedOutput": "P(A|B)=0.40", "notApplicableWhen": [ "Do not use Conditional Probability until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "binomial-probability", "name": "Binomial Probability Formula", "latex": "P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}", "category": "Probability", "description": "Finds the probability of exactly k successes in n independent Bernoulli trials.", "variables": [ "n: number of trials", "k: number of successes", "p: success probability per trial", "X: success count" ], "notes": [ "Trials must be independent and use the same success probability.", "k must be an integer from 0 through n." ], "examples": [ "For 3 fair coin flips, the probability of exactly 2 heads is 3/8." ], "seo": { "title": "Binomial Probability Formula: Formula, Worked Example & Checks", "description": "Learn the Binomial Probability Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "permutation-formula", "combination-formula", "probability-complement" ], "assumptions": [ "Trials must be independent and use the same success probability.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Binomial Probability Formula; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "For three fair coin flips, the probability of exactly two heads is C(3,2)(0.5)²(0.5)=3/8=0.375.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Binomial Probability Formula.", "sourceIds": [ "openstax-intro-statistics" ], "verifiedAgainst": "OpenStax Introductory Statistics", "domain": "Apply Binomial Probability Formula only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Set n=3, k=2, and p=0.5.", "Compute C(3,2)=3 and p²(1-p)=0.125.", "Multiply 3×0.125 to obtain 0.375." ], "verification": "The three equally likely outcomes HHT, HTH, and THH out of eight total outcomes confirm 3/8.", "boundaryCases": [ "For Binomial Probability Formula, check zero, negative, and extreme input values before relying on the result.", "When using Binomial Probability Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "n=3, k=2, p=0.5", "workedOutput": "P(X=2)=3/8=0.375", "notApplicableWhen": [ "Do not use Binomial Probability Formula until the sample, event, distribution, independence assumptions, and parameter convention match the problem." ] }, { "id": "dot-product", "name": "Vector Dot Product", "latex": "\\mathbf{a}\\cdot\\mathbf{b}=\\sum_{i=1}^{n}a_ib_i=\\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta", "category": "Linear Algebra", "description": "Combines two vectors into a scalar and relates their components to the angle between them.", "variables": [ "a, b: vectors", "aᵢ, bᵢ: vector components", "θ: angle between vectors" ], "notes": [ "Perpendicular nonzero vectors have dot product zero.", "The component and angle formulas describe the same operation." ], "examples": [ "(1, 2) · (3, 4) = 11." ], "seo": { "title": "Vector Dot Product: Formula, Worked Example & Checks", "description": "Learn the vector dot product with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two", "eigenvalue-equation" ], "assumptions": [ "Perpendicular nonzero vectors have dot product zero.", "For the Vector Dot Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit." ], "commonMistakes": [ "Do not omit the index or bounds in Vector Dot Product; changing either one changes which terms are included.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For vectors (1, 2, 3) and (4, −1, 2), the dot product is 1·4 + 2·(−1) + 3·2 = 4 − 2 + 6 = 8.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Confirm both vectors have the same number of components.", "Multiply corresponding components: 4, −2, and 6.", "Add the component products: 4 − 2 + 6.", "Report the scalar dot product as 8." ], "verification": "Using a·b = |a||b|cos θ gives the same scalar when the vector magnitudes and included angle are evaluated.", "domain": "Defined for equal-dimensional real vectors in a Euclidean inner-product space; the result is a scalar rather than a vector.", "units": [ "the result uses the product of the two vectors’ component units" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Vector Dot Product.", "verifiedAgainst": "OpenStax Calculus Volume 1; NIST Digital Library of Mathematical Functions", "sourceIds": [ "openstax-calculus-volume-1", "nist-dlmf" ], "qualityTier": "core", "boundaryCases": [ "For Vector Dot Product, check zero, negative, and extreme input values before relying on the result.", "When using Vector Dot Product, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\mathbf{a}\\cdot\\mathbf{b}=\\sum_{i=1}^{n}a_ib_i=\\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=(1,2,3), b=(4,5,6)", "workedOutput": "a·b=32", "notApplicableWhen": [ "Do not use Vector Dot Product when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "two-by-two-determinant", "name": "2×2 Matrix Determinant", "latex": "\\det\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}=ad-bc", "category": "Linear Algebra", "description": "Calculates the determinant of a two-by-two matrix.", "variables": [ "a, b, c, d: matrix entries", "det: scalar determinant" ], "notes": [ "A zero determinant means the matrix is singular and has no inverse.", "Keep the subtraction order ad − bc." ], "examples": [ "The determinant of [[1,2],[3,4]] is −2." ], "seo": { "title": "2×2 Matrix Determinant: Formula, Worked Example & Checks", "description": "Learn the 2×2 Matrix Determinant, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "dot-product", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two", "eigenvalue-equation" ], "assumptions": [ "A zero determinant means the matrix is singular and has no inverse.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using 2×2 Matrix Determinant; matrix operations are order-sensitive.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For the matrix [[3, 2], [1, 4]], det(A) = 3·4 − 2·1 = 12 − 2 = 10.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify a = 3, b = 2, c = 1, and d = 4.", "Multiply the main diagonal entries: ad = 3 × 4 = 12.", "Multiply the other diagonal entries: bc = 2 × 1 = 2.", "Subtract to obtain det(A) = 12 − 2 = 10." ], "verification": "Because the determinant is nonzero, the matrix is invertible; direct multiplication by its computed inverse returns the identity matrix.", "domain": "Applies to a 2 × 2 square matrix; a zero determinant indicates singularity and no two-sided inverse.", "units": [ "determinant units are the product of one entry from each row and column" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for 2×2 Matrix Determinant.", "verifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "sourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "qualityTier": "core", "boundaryCases": [ "For 2×2 Matrix Determinant, check zero, negative, and extreme input values before relying on the result.", "When using 2×2 Matrix Determinant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\det\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}=ad-bc for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "[[1,2],[3,4]]", "workedOutput": "det=-2", "notApplicableWhen": [ "Do not use 2×2 Matrix Determinant when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "compound-interest", "name": "Compound Interest Formula", "latex": "A=P\\left(1+\\frac{r}{n}\\right)^{nt}", "category": "Applied Math", "description": "Calculates a balance with interest compounded a fixed number of times per year.", "variables": [ "A: final amount", "P: principal", "r: annual rate as a decimal", "n: compounding periods per year", "t: years" ], "notes": [ "The stated rate must be converted from percent to decimal.", "This formula assumes a constant rate and no additional deposits or withdrawals." ], "examples": [ "$1,000 at 5% compounded monthly for one year grows to about $1,051.16." ], "seo": { "title": "Compound Interest Formula: Formula, Worked Example & Checks", "description": "Learn the Compound Interest Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "percentage-change", "logistic-function", "continuous-compound-interest", "exponential-population-growth" ], "assumptions": [ "The stated rate must be converted from percent to decimal.", "For the Compound Interest Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "commonMistakes": [ "When copying Compound Interest Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Compound Interest Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Invest $1,000 at a nominal annual rate of 5%, compounded monthly for 2 years: A = 1000(1 + 0.05/12)²⁴ ≈ $1,104.94.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Set P = 1000, r = 0.05, n = 12, and t = 2.", "Compute the periodic rate: r/n = 0.05/12.", "Compute the number of periods: nt = 12 × 2 = 24.", "Evaluate 1000(1 + 0.05/12)²⁴ and round currency at the end." ], "verification": "The balance exceeds the principal and is slightly above the annual-compounding result, which is consistent with positive monthly compounding at the same nominal rate.", "domain": "Assumes a fixed nominal annual rate, regular compounding, no deposits or withdrawals, and consistent time units for n and t.", "units": [ "P and A use the same currency", "r is a decimal rate", "t is measured in years when r is annual" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Compound Interest Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "sourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "qualityTier": "core", "boundaryCases": [ "For Compound Interest Formula, check zero, negative, and extreme input values before relying on the result.", "When using Compound Interest Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=P\\left(1+\\frac{r}{n}\\right)^{nt} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "P=1000, r=0.05, n=12, t=2", "workedOutput": "A≈1104.94", "notApplicableWhen": [ "Do not use Compound Interest Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "percentage-change", "name": "Percentage Change Formula", "latex": "\\%\\text{ change}=\\frac{\\text{new}-\\text{old}}{|\\text{old}|}\\times100\\%", "category": "Applied Math", "description": "Measures relative increase or decrease from an original nonzero value.", "variables": [ "old: original value", "new: later value", "% change: signed relative difference" ], "notes": [ "The original value must be nonzero.", "A positive result is an increase; a negative result is a decrease." ], "examples": [ "A change from 80 to 100 is a 25% increase." ], "seo": { "title": "Percentage Change Formula: Formula, Worked Example & Checks", "description": "Learn the Percentage Change Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "compound-interest", "logistic-function", "continuous-compound-interest", "exponential-population-growth" ], "assumptions": [ "The original value must be nonzero.", "For the Percentage Change Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "commonMistakes": [ "When copying Percentage Change Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Percentage Change Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A value rises from 80 to 100. Percentage change = (100 − 80)/80 × 100% = 25%.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "workedSteps": [ "Identify the original value as 80 and the new value as 100.", "Calculate the change: 100 − 80 = 20.", "Divide by the original value: 20/80 = 0.25.", "Multiply by 100% to report a 25% increase." ], "verification": "Applying a 25% increase to 80 gives 80 × 1.25 = 100, recovering the stated new value.", "domain": "The original value must be nonzero; the sign distinguishes an increase from a decrease, and context determines whether percentage change is meaningful.", "units": [ "old and new values must use the same unit", "the final result is a percentage" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Percentage Change Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions", "sourceIds": [ "openstax-algebra-trigonometry", "nist-dlmf" ], "qualityTier": "core", "boundaryCases": [ "For Percentage Change Formula, check zero, negative, and extreme input values before relying on the result.", "When using Percentage Change Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\%\\text{ change}=\\frac{\\text{new}-\\text{old}}{|\\text{old}|}\\times100\\% for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "old=80, new=100", "workedOutput": "25% increase", "notApplicableWhen": [ "Do not use Percentage Change Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "quadratic-discriminant", "name": "Quadratic Discriminant", "latex": "\\Delta=b^2-4ac", "category": "Algebra", "description": "Determines the number and type of roots of a quadratic equation.", "variables": [ "a, b, c: coefficients of ax²+bx+c", "Δ: discriminant" ], "notes": [ "Δ>0 gives two distinct real roots; Δ=0 gives one repeated real root." ], "examples": [ "For x²−4x+4, Δ=0." ], "seo": { "title": "Quadratic Discriminant: Formula, Worked Example & Checks", "description": "Learn the Quadratic Discriminant, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Δ>0 gives two distinct real roots; Δ=0 gives one repeated real root.", "For the Quadratic Discriminant, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Quadratic Discriminant, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Quadratic Discriminant, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For x²-4x+4=0, Δ=(-4)²-4(1)(4)=16-16=0.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Discriminant.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Quadratic Discriminant only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify a=1, b=-4, and c=4.", "Square b and compute 4ac.", "Subtract to obtain Δ=0, indicating one repeated real root." ], "verification": "Factoring gives (x-2)²=0, confirming the repeated root x=2.", "boundaryCases": [ "For Quadratic Discriminant, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Discriminant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\Delta=b^2-4ac for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=1, b=-4, c=4", "workedOutput": "Δ=0; one repeated real root", "notApplicableWhen": [ "Do not use Quadratic Discriminant when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "vertex-form", "name": "Quadratic Vertex Form", "latex": "y=a(x-h)^2+k", "category": "Algebra", "description": "Expresses a parabola using its vertex and vertical scale.", "variables": [ "(h,k): vertex", "a: vertical scale and opening direction" ], "notes": [ "a>0 opens upward; a<0 opens downward." ], "examples": [ "y=2(x−3)²+1 has vertex (3,1)." ], "seo": { "title": "Quadratic Vertex Form: Formula, Worked Example & Checks", "description": "Learn the Quadratic Vertex Form, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "a>0 opens upward; a<0 opens downward.", "For the Quadratic Vertex Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Quadratic Vertex Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Quadratic Vertex Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For y=2(x-3)²+1, the vertex is (3,1). At x=5, y=2(2²)+1=9.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Vertex Form.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Quadratic Vertex Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Read h=3 and k=1 from a(x-h)²+k.", "Use the positive coefficient a=2 to identify an upward-opening parabola.", "Substitute x=5 to obtain y=9." ], "verification": "Substituting x=3 gives the minimum y=1, confirming the stated vertex.", "boundaryCases": [ "For Quadratic Vertex Form, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Vertex Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form y=a(x-h)^2+k for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "y=2(x-3)²+1, evaluate x=5", "workedOutput": "y=9; vertex (3,1)", "notApplicableWhen": [ "Do not use Quadratic Vertex Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "point-slope-form", "name": "Point-Slope Form", "latex": "y-y_1=m(x-x_1)", "category": "Algebra", "description": "Defines a line from one point and its slope.", "variables": [ "m: slope", "(x₁,y₁): known point" ], "notes": [ "Convert to slope-intercept form when a y-intercept is needed." ], "examples": [ "Through (2,3) with slope 4: y−3=4(x−2)." ], "seo": { "title": "Point-Slope Form: Formula, Worked Example & Checks", "description": "Learn the point-slope form with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Convert to slope-intercept form when a y-intercept is needed.", "For the Point-Slope Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Point-Slope Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Point-Slope Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "A line through (2,3) with slope 4 is y-3=4(x-2), which simplifies to y=4x-5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Point-Slope Form.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Point-Slope Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Insert m=4, x₁=2, and y₁=3.", "Distribute 4 to obtain y-3=4x-8.", "Add 3 to both sides to obtain y=4x-5." ], "verification": "The simplified line gives y=3 at x=2 and has slope 4.", "boundaryCases": [ "For Point-Slope Form, check zero, negative, and extreme input values before relying on the result.", "When using Point-Slope Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form y-y_1=m(x-x_1) for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "point (2,3), slope 4", "workedOutput": "y-3=4(x-2); y=4x-5", "notApplicableWhen": [ "Do not use Point-Slope Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "midpoint-formula", "name": "Midpoint Formula", "latex": "M=\\left(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2}\\right)", "category": "Geometry", "description": "Finds the point halfway between two coordinate points.", "variables": [ "(x₁,y₁),(x₂,y₂): endpoints", "M: midpoint" ], "notes": [ "Average corresponding coordinates." ], "examples": [ "Between (0,2) and (4,6), M=(2,4)." ], "seo": { "title": "Midpoint Formula: Formula, Worked Example & Checks", "description": "Learn the midpoint formula with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Both endpoints use the same coordinate axes, scale, and units.", "The midpoint is obtained by averaging corresponding coordinates, not by averaging distances." ], "commonMistakes": [ "Do not pair x₁ with y₂ or otherwise mix coordinate positions.", "Keep the factor of 1/2 on both coordinate sums and preserve negative signs inside parentheses." ], "workedExample": "For endpoints (−2, 4) and (6, 10), the midpoint is ((−2+6)/2, (4+10)/2) = (2, 7).", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "sourceIds": [ "openstax-algebra-trigonometry" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Midpoint Formula.", "verifiedAgainst": "OpenStax Algebra and Trigonometry 2e", "domain": "Use the midpoint formula for two points expressed in the same Cartesian coordinate system and compatible units.", "workedSteps": [ "List the two endpoints in the same coordinate order and confirm that their axes and units match.", "Add the two x-coordinates and divide that sum by two to obtain the midpoint x-coordinate.", "Add the two y-coordinates and divide that sum by two to obtain the midpoint y-coordinate.", "Verify the result by checking that the displacement from the first endpoint to the midpoint equals the displacement from the midpoint to the second endpoint." ], "verification": "Subtract the first endpoint from the midpoint and the midpoint from the second endpoint; the two displacement vectors should be identical.", "boundaryCases": [ "If the endpoints are identical, the midpoint must equal that same point.", "If one coordinate pair is symmetric about zero, the corresponding midpoint coordinate should be zero." ], "alternativeForms": [ "In vector form, the midpoint can be written as M=(P_1+P_2)/2 when vector addition and scalar division are defined." ], "workedInput": "(1,2) and (5,8)", "workedOutput": "M=(3,5)", "notApplicableWhen": [ "Do not use Midpoint Formula when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "exponent-product-rule", "name": "Exponent Product Rule", "latex": "a^m a^n=a^{m+n}", "category": "Algebra", "description": "Combines powers with the same base by adding exponents.", "variables": [ "a: common base", "m,n: exponents" ], "notes": [ "The base must be the same." ], "examples": [ "x³x⁵=x⁸." ], "seo": { "title": "Exponent Product Rule: Formula, Worked Example & Checks", "description": "Learn the Exponent Product Rule, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "The base must be the same.", "For the Exponent Product Rule, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Exponent Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Exponent Product Rule, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Multiply 2³·2⁵. Because the base is the same, add the exponents: 2⁸=256.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Exponent Product Rule.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Exponent Product Rule only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Confirm that both powers have base 2.", "Add the exponents 3+5=8.", "Evaluate 2⁸=256." ], "verification": "Direct multiplication 8×32=256 confirms the result.", "boundaryCases": [ "For Exponent Product Rule, check zero, negative, and extreme input values before relying on the result.", "When using Exponent Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a^m a^n=a^{m+n} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "2³·2⁵", "workedOutput": "2⁸=256", "notApplicableWhen": [ "Do not use Exponent Product Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "exponent-quotient-rule", "name": "Exponent Quotient Rule", "latex": "\\frac{a^m}{a^n}=a^{m-n}", "category": "Algebra", "description": "Divides powers with the same nonzero base by subtracting exponents.", "variables": [ "a: nonzero base", "m,n: exponents" ], "notes": [ "a cannot be zero when negative exponents may result." ], "examples": [ "x⁷/x²=x⁵." ], "seo": { "title": "Exponent Quotient Rule: Formula, Worked Example & Checks", "description": "Learn the Exponent Quotient Rule, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "a cannot be zero when negative exponents may result.", "For the Exponent Quotient Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Exponent Quotient Rule, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Exponent Quotient Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Exponent Quotient Rule, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Divide 3⁷ by 3². Subtract the exponents to get 3⁵=243.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Exponent Quotient Rule.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Exponent Quotient Rule only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Confirm the common nonzero base 3.", "Subtract exponents: 7-2=5.", "Evaluate 3⁵=243." ], "verification": "Direct cancellation of two factors of 3 leaves five factors, confirming 243.", "boundaryCases": [ "For Exponent Quotient Rule, check zero, negative, and extreme input values before relying on the result.", "When using Exponent Quotient Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\frac{a^m}{a^n}=a^{m-n} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "3⁷/3²", "workedOutput": "3⁵=243", "notApplicableWhen": [ "Do not use Exponent Quotient Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "change-of-base", "name": "Logarithm Change of Base", "latex": "\\log_b x=\\frac{\\log_k x}{\\log_k b}", "category": "Algebra", "description": "Rewrites a logarithm using another convenient base.", "variables": [ "b: original base", "k: new base", "x: positive argument" ], "notes": [ "Bases must be positive and not equal to 1." ], "examples": [ "log₂8=ln8/ln2=3." ], "seo": { "title": "Logarithm Change of Base: Formula, Worked Example & Checks", "description": "Learn the Logarithm Change of Base, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Bases must be positive and not equal to 1.", "For the Logarithm Change of Base, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Logarithm Change of Base, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Logarithm Change of Base, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Logarithm Change of Base, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Compute log₂8 as ln8/ln2. Since ln8=ln(2³)=3ln2, the quotient is 3.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Logarithm Change of Base.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Logarithm Change of Base only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Choose k=e so the numerator is ln8 and the denominator is ln2.", "Use 8=2³ to rewrite ln8 as 3ln2.", "Cancel ln2 to obtain 3." ], "verification": "The defining relation 2³=8 confirms the logarithm.", "boundaryCases": [ "For Logarithm Change of Base, check zero, negative, and extreme input values before relying on the result.", "When using Logarithm Change of Base, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\log_b x=\\frac{\\log_k x}{\\log_k b} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "log₂8 using natural logarithms", "workedOutput": "ln8/ln2=3", "notApplicableWhen": [ "Do not use Logarithm Change of Base when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "permutation-formula", "name": "Permutation Formula", "latex": "P(n,r)=\\frac{n!}{(n-r)!}", "category": "Probability", "description": "Counts ordered selections of r items from n distinct items.", "variables": [ "n: total items", "r: selected positions" ], "notes": [ "Order matters and sampling is without replacement." ], "examples": [ "P(5,2)=20." ], "seo": { "title": "Permutation Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the permutation formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "combination-formula", "probability-complement" ], "assumptions": [ "Order matters and sampling is without replacement.", "For the Permutation Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Permutation Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "P(5,2)=20.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "combination-formula", "name": "Combination Formula", "latex": "\\binom{n}{r}=\\frac{n!}{r!(n-r)!}", "category": "Probability", "description": "Counts unordered selections of r items from n distinct items.", "variables": [ "n: total items", "r: selected items" ], "notes": [ "Order does not matter." ], "examples": [ "C(5,2)=10." ], "seo": { "title": "Combination Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the combination formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "probability-complement" ], "assumptions": [ "Order does not matter.", "For the Combination Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Combination Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "C(5,2)=10.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "rectangle-area", "name": "Rectangle Area", "latex": "A=lw", "category": "Geometry", "description": "Calculates the area of a rectangle.", "variables": [ "l: length", "w: width", "A: area" ], "notes": [ "Use consistent length units." ], "examples": [ "A 5 by 3 rectangle has area 15." ], "seo": { "title": "Rectangle Area: Formula, Worked Example & Checks", "description": "Learn the rectangle area with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Rectangle Area, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Rectangle Area, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "A rectangle 5 cm long and 3 cm wide has area A=5×3=15 cm².", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Rectangle Area.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Rectangle Area only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify l=5 cm and w=3 cm.", "Multiply the side lengths.", "Report the result in square centimetres." ], "verification": "Counting a 5-by-3 grid gives 15 unit squares.", "boundaryCases": [ "For Rectangle Area, check zero, negative, and extreme input values before relying on the result.", "When using Rectangle Area, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=lw for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "l=5 cm, w=3 cm", "workedOutput": "A=15 cm²", "notApplicableWhen": [ "Do not use Rectangle Area when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "rectangle-perimeter", "name": "Rectangle Perimeter", "latex": "P=2(l+w)", "category": "Geometry", "description": "Calculates the perimeter of a rectangle.", "variables": [ "l: length", "w: width", "P: perimeter" ], "notes": [ "Perimeter uses linear units." ], "examples": [ "For l=5,w=3, P=16." ], "seo": { "title": "Rectangle Perimeter: Formula, Worked Example & Checks", "description": "Learn the rectangle perimeter with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Perimeter uses linear units.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Rectangle Perimeter, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Rectangle Perimeter, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For l=5 cm and w=3 cm, P=2(5+3)=16 cm.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Rectangle Perimeter.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Rectangle Perimeter only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Add one length and one width: 5+3=8 cm.", "Double the sum to include opposite sides.", "Report 16 cm as a length." ], "verification": "Adding all four sides, 5+3+5+3, also gives 16.", "boundaryCases": [ "For Rectangle Perimeter, check zero, negative, and extreme input values before relying on the result.", "When using Rectangle Perimeter, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form P=2(l+w) for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "l=5 cm, w=3 cm", "workedOutput": "P=16 cm", "notApplicableWhen": [ "Do not use Rectangle Perimeter when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "parallelogram-area", "name": "Parallelogram Area", "latex": "A=bh", "category": "Geometry", "description": "Calculates area from a base and perpendicular height.", "variables": [ "b: base", "h: perpendicular height" ], "notes": [ "The slanted side is not the height unless perpendicular." ], "examples": [ "b=8,h=3 gives A=24." ], "seo": { "title": "Parallelogram Area Formula — Meaning, Variables & LaTeX", "description": "Reference the parallelogram area, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "The slanted side is not the height unless perpendicular.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Parallelogram Area, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Parallelogram Area, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "b=8,h=3 gives A=24.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "trapezoid-area", "name": "Trapezoid Area", "latex": "A=\\frac{1}{2}(b_1+b_2)h", "category": "Geometry", "description": "Calculates area from parallel bases and height.", "variables": [ "b₁,b₂: parallel side lengths", "h: perpendicular height" ], "notes": [ "Both bases must be parallel." ], "examples": [ "b₁=4,b₂=8,h=3 gives A=18." ], "seo": { "title": "Trapezoid Area Formula — Meaning, Variables & LaTeX", "description": "Reference the trapezoid area, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Both bases must be parallel.", "For the Trapezoid Area, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Trapezoid Area, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Trapezoid Area, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "b₁=4,b₂=8,h=3 gives A=18.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "heron-formula", "name": "Heron Formula", "latex": "A=\\sqrt{s(s-a)(s-b)(s-c)},\\quad s=\\frac{a+b+c}{2}", "category": "Geometry", "description": "Calculates triangle area from its three side lengths.", "variables": [ "a,b,c: side lengths", "s: semiperimeter" ], "notes": [ "The side lengths must satisfy the triangle inequality." ], "examples": [ "Sides 3,4,5 give area 6." ], "seo": { "title": "Heron Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the heron formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "The side lengths must satisfy the triangle inequality.", "For the Heron Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Heron Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Heron Formula, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "Sides 3,4,5 give area 6.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "cylinder-volume", "name": "Cylinder Volume", "latex": "V=\\pi r^2h", "category": "Geometry", "description": "Calculates the volume of a right circular cylinder.", "variables": [ "r: radius", "h: height" ], "notes": [ "Use cubic units." ], "examples": [ "r=2,h=5 gives V=20π." ], "seo": { "title": "Cylinder Volume Formula — Meaning, Variables & LaTeX", "description": "Reference the cylinder volume, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Cylinder Volume, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Cylinder Volume, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "r=2,h=5 gives V=20π.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "cone-volume", "name": "Cone Volume", "latex": "V=\\frac{1}{3}\\pi r^2h", "category": "Geometry", "description": "Calculates the volume of a right circular cone.", "variables": [ "r: base radius", "h: perpendicular height" ], "notes": [ "The volume is one third of the matching cylinder." ], "examples": [ "r=3,h=4 gives V=12π." ], "seo": { "title": "Cone Volume Formula — Meaning, Variables & LaTeX", "description": "Reference the cone volume, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "The volume is one third of the matching cylinder.", "For the Cone Volume, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Cone Volume, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Cone Volume, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "r=3,h=4 gives V=12π.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "sphere-surface-area", "name": "Sphere Surface Area", "latex": "A=4\\pi r^2", "category": "Geometry", "description": "Calculates the surface area of a sphere.", "variables": [ "r: radius" ], "notes": [ "Use square units." ], "examples": [ "r=2 gives A=16π." ], "seo": { "title": "Sphere Surface Area Formula — Meaning, Variables & LaTeX", "description": "Reference the sphere surface area, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Sphere Surface Area, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Sphere Surface Area, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "r=2 gives A=16π.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "power-rule-derivative", "name": "Derivative Power Rule", "latex": "\\frac{d}{dx}x^n=nx^{n-1}", "category": "Calculus", "description": "Differentiates a power of x.", "variables": [ "n: constant exponent", "x: variable" ], "notes": [ "For real exponents, domain restrictions may apply." ], "examples": [ "d(x⁵)/dx=5x⁴." ], "seo": { "title": "Power Rule for Derivatives: Formula & Example", "description": "Learn the Derivative Power Rule, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "The exponent and domain must make the original function and resulting derivative well defined.", "Constant multiples should be retained and differentiated using linearity." ], "commonMistakes": [ "Do not subtract one from the exponent without also multiplying by the original exponent.", "Do not use the elementary real-valued rule across points where a fractional or negative power is undefined." ], "workedExample": "For f(x)=3x^5, f′(x)=3·5x^4=15x^4.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "sourceIds": [ "openstax-calculus-volume-1" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Derivative Power Rule.", "verifiedAgainst": "OpenStax Calculus Volume 1", "domain": "Apply the power rule on intervals where x^n is differentiable under the stated exponent and number-system assumptions.", "workedSteps": [ "Identify the coefficient and exponent of each power term before differentiating.", "Multiply the coefficient by the original exponent for the derivative coefficient.", "Reduce the exponent by one while preserving the variable and any constant multiplier.", "Verify the derivative at a sample point with a difference quotient or a trusted symbolic calculation." ], "verification": "For a chosen value of x, compare the derivative with a small-step numerical difference quotient and confirm that the values converge as the step shrinks.", "boundaryCases": [ "For n=0, the derivative of the constant x^0=1 is zero even though the unsimplified pattern contains x^{-1}.", "For negative or fractional exponents, exclude points where the original real-valued power is not defined." ], "alternativeForms": [ "For a monomial ax^n, linearity gives d(ax^n)/dx=anx^{n-1}." ], "workedInput": "f(x)=4x³-2x²+7", "workedOutput": "f′(x)=12x²-4x", "notApplicableWhen": [ "Do not use Derivative Power Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements." ] }, { "id": "quotient-rule", "name": "Quotient Rule", "latex": "\\left(\\frac{f}{g}\\right)\\prime=\\frac{f\\prime g-fg\\prime}{g^2}", "category": "Calculus", "description": "Differentiates a quotient of two functions.", "variables": [ "f,g: differentiable functions", "g: nonzero denominator" ], "notes": [ "Keep the term order to preserve the sign." ], "examples": [ "Differentiate x/(x+1) to get 1/(x+1)²." ], "seo": { "title": "Quotient Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the quotient rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "Keep the term order to preserve the sign.", "For the Quotient Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Quotient Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Differentiate x/(x+1) to get 1/(x+1)².", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-sine", "name": "Derivative of Sine", "latex": "\\frac{d}{dx}\\sin x=\\cos x", "category": "Calculus", "description": "Gives the derivative of the sine function.", "variables": [ "x: angle in radians" ], "notes": [ "The standard identity assumes radians." ], "examples": [ "At x=0 the derivative is 1." ], "seo": { "title": "Derivative of Sine Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative of sine, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "The standard identity assumes radians.", "For the Derivative of Sine, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative of Sine, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "At x=0 the derivative is 1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-cosine", "name": "Derivative of Cosine", "latex": "\\frac{d}{dx}\\cos x=-\\sin x", "category": "Calculus", "description": "Gives the derivative of the cosine function.", "variables": [ "x: angle in radians" ], "notes": [ "Remember the negative sign." ], "examples": [ "At x=π/2 the derivative is −1." ], "seo": { "title": "Derivative of Cosine Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative of cosine, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "Remember the negative sign.", "For the Derivative of Cosine, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative of Cosine, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "At x=π/2 the derivative is −1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-exponential", "name": "Derivative of Exponential", "latex": "\\frac{d}{dx}e^x=e^x", "category": "Calculus", "description": "Shows that the natural exponential is its own derivative.", "variables": [ "e: Euler number", "x: variable" ], "notes": [ "For e^{u(x)}, multiply by u′(x)." ], "examples": [ "d(e^{3x})/dx=3e^{3x}." ], "seo": { "title": "Derivative of Exponential Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative of exponential, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For e^{u(x)}, multiply by u′(x).", "For the Derivative of Exponential, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative of Exponential, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "d(e^{3x})/dx=3e^{3x}.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-logarithm", "name": "Derivative of Natural Logarithm", "latex": "\\frac{d}{dx}\\ln x=\\frac{1}{x}", "category": "Calculus", "description": "Differentiates the natural logarithm on its real domain.", "variables": [ "x: positive real variable" ], "notes": [ "For ln|x| the same derivative holds where x≠0." ], "examples": [ "d ln(2x)/dx=1/x." ], "seo": { "title": "Derivative of Natural Logarithm Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative of natural logarithm, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For ln|x| the same derivative holds where x≠0.", "For the Derivative of Natural Logarithm, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative of Natural Logarithm, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "d ln(2x)/dx=1/x.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "integral-power-rule", "name": "Integral Power Rule", "latex": "\\int x^n\\,dx=\\frac{x^{n+1}}{n+1}+C", "category": "Calculus", "description": "Integrates a power of x when n is not −1.", "variables": [ "n: exponent not equal to −1", "C: integration constant" ], "notes": [ "The case n=−1 gives ln|x|+C." ], "examples": [ "∫x²dx=x³/3+C." ], "seo": { "title": "Integral Power Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the integral power rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "The case n=−1 gives ln|x|+C.", "For the Integral Power Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Integral Power Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "∫x²dx=x³/3+C.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "integral-exponential", "name": "Integral of Exponential", "latex": "\\int e^x\\,dx=e^x+C", "category": "Calculus", "description": "Integrates the natural exponential function.", "variables": [ "C: integration constant" ], "notes": [ "For e^{ax}, divide by a." ], "examples": [ "∫e^{2x}dx=e^{2x}/2+C." ], "seo": { "title": "Integral of Exponential Formula — Meaning, Variables & LaTeX", "description": "Reference the integral of exponential, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For e^{ax}, divide by a.", "Integration limits, the differential, and any constant of integration must be included where required.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Integral of Exponential, and include a constant of integration for an indefinite integral.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "∫e^{2x}dx=e^{2x}/2+C.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "taylor-series", "name": "Taylor Series", "latex": "f(x)=\\sum_{n=0}^{\\infty}\\frac{f^{(n)}(a)}{n!}(x-a)^n", "category": "Calculus", "description": "Represents a sufficiently smooth function as a power series around a point.", "variables": [ "a: expansion point", "f⁽ⁿ⁾: nth derivative" ], "notes": [ "Convergence must be checked for the function and interval." ], "examples": [ "For e^x at a=0, all derivatives equal 1." ], "seo": { "title": "Taylor Series Formula — Meaning, Variables & LaTeX", "description": "Reference the taylor series, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "Convergence must be checked for the function and interval.", "For the Taylor Series, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Taylor Series, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Taylor Series, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For e^x at a=0, all derivatives equal 1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "gradient", "name": "Gradient", "latex": "\\nabla f=\\left(\\frac{\\partial f}{\\partial x_1},\\ldots,\\frac{\\partial f}{\\partial x_n}\\right)", "category": "Calculus", "description": "Forms a vector of partial derivatives of a scalar field.", "variables": [ "f: scalar field", "xᵢ: coordinate variables" ], "notes": [ "The gradient points in the direction of steepest increase." ], "examples": [ "For f=x²+y², ∇f=(2x,2y)." ], "seo": { "title": "Gradient Formula — Meaning, Variables & LaTeX", "description": "Reference the gradient, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "The gradient points in the direction of steepest increase.", "For the Gradient, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Gradient, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For f=x²+y², ∇f=(2x,2y).", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "divergence", "name": "Divergence", "latex": "\\nabla\\cdot\\mathbf F=\\sum_i\\frac{\\partial F_i}{\\partial x_i}", "category": "Calculus", "description": "Measures local source or sink strength of a vector field.", "variables": [ "F: vector field", "Fᵢ: components" ], "notes": [ "Coordinate-specific forms differ outside Cartesian coordinates." ], "examples": [ "For F=(x,y), divergence is 2." ], "seo": { "title": "Divergence Formula — Meaning, Variables & LaTeX", "description": "Reference the divergence, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "Coordinate-specific forms differ outside Cartesian coordinates.", "For the Divergence, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Divergence, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined." ], "commonMistakes": [ "When copying Divergence, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For F=(x,y), divergence is 2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "curl", "name": "Curl", "latex": "\\nabla\\times\\mathbf F", "category": "Calculus", "description": "Measures local rotation of a three-dimensional vector field.", "variables": [ "F: vector field" ], "notes": [ "In two dimensions, curl is often represented by a scalar component." ], "examples": [ "For F=(−y,x,0), curl=(0,0,2)." ], "seo": { "title": "Curl Formula — Meaning, Variables & LaTeX", "description": "Reference the curl, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "In two dimensions, curl is often represented by a scalar component.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Before substituting values into Curl, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "For F=(−y,x,0), curl=(0,0,2).", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "population-variance", "name": "Population Variance", "latex": "\\sigma^2=\\frac{1}{N}\\sum_{i=1}^{N}(x_i-\\mu)^2", "category": "Statistics", "description": "Measures average squared deviation from the population mean.", "variables": [ "N: population size", "μ: population mean", "xᵢ: observations" ], "notes": [ "Use N, not N−1, for a complete population." ], "examples": [ "For 1,2,3, μ=2 and σ²=2/3." ], "seo": { "title": "Population Variance Formula — Meaning, Variables & LaTeX", "description": "Reference the population variance, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "covariance" ], "assumptions": [ "For the Population Variance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Population Variance, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Population Variance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Population Variance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Population Variance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For 1,2,3, μ=2 and σ²=2/3.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "covariance", "name": "Covariance", "latex": "\\operatorname{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]", "category": "Statistics", "description": "Measures how two variables vary together.", "variables": [ "X,Y: random variables", "E: expectation" ], "notes": [ "Magnitude depends on variable scales." ], "examples": [ "Positive covariance means larger values tend to occur together." ], "seo": { "title": "Covariance Formula — Meaning, Variables & LaTeX", "description": "Reference the covariance, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Magnitude depends on variable scales.", "For the Covariance, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Covariance; map every symbol to its definition first.", "For the Covariance, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Positive covariance means larger values tend to occur together.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "pearson-correlation", "name": "Pearson Correlation", "latex": "r=\\frac{\\sum (x_i-\\bar x)(y_i-\\bar y)}{\\sqrt{\\sum(x_i-\\bar x)^2\\sum(y_i-\\bar y)^2}}", "category": "Statistics", "description": "Measures linear association between two variables.", "variables": [ "r: correlation coefficient", "x̄,ȳ: sample means" ], "notes": [ "Correlation does not imply causation." ], "examples": [ "r=1 indicates perfect positive linear association." ], "seo": { "title": "Pearson Correlation Formula — Meaning, Variables & LaTeX", "description": "Reference the pearson correlation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Correlation does not imply causation.", "For the Pearson Correlation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Pearson Correlation, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear." ], "commonMistakes": [ "When copying Pearson Correlation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Pearson Correlation, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "r=1 indicates perfect positive linear association.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "weighted-mean", "name": "Weighted Mean", "latex": "\\bar x_w=\\frac{\\sum_i w_ix_i}{\\sum_i w_i}", "category": "Statistics", "description": "Calculates an average where observations have different weights.", "variables": [ "wᵢ: weights", "xᵢ: values" ], "notes": [ "Weights should be nonnegative for most applications." ], "examples": [ "Values 10,20 with weights 1,3 give 17.5." ], "seo": { "title": "Weighted Mean Formula — Meaning, Variables & LaTeX", "description": "Reference the weighted mean, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Weights should be nonnegative for most applications.", "For the Weighted Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Weighted Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Weighted Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Weighted Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Weighted Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Values 10,20 with weights 1,3 give 17.5.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "geometric-mean", "name": "Geometric Mean", "latex": "G=\\left(\\prod_{i=1}^{n}x_i\\right)^{1/n}", "category": "Statistics", "description": "Averages positive values multiplicatively.", "variables": [ "xᵢ: positive values", "n: count" ], "notes": [ "Useful for growth rates and ratios." ], "examples": [ "Geometric mean of 2 and 8 is 4." ], "seo": { "title": "Geometric Mean Formula — Meaning, Variables & LaTeX", "description": "Reference the geometric mean, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Useful for growth rates and ratios.", "For the Geometric Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Geometric Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Geometric Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not omit the index or bounds in Geometric Mean; changing either one changes which terms are included.", "For the Geometric Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Geometric mean of 2 and 8 is 4.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "harmonic-mean", "name": "Harmonic Mean", "latex": "H=\\frac{n}{\\sum_{i=1}^{n}1/x_i}", "category": "Statistics", "description": "Averages positive rates or ratios.", "variables": [ "xᵢ: positive values", "n: count" ], "notes": [ "Appropriate when equal distances are traveled at different speeds." ], "examples": [ "Harmonic mean of 2 and 3 is 2.4." ], "seo": { "title": "Harmonic Mean Formula — Meaning, Variables & LaTeX", "description": "Reference the harmonic mean, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Appropriate when equal distances are traveled at different speeds.", "For the Harmonic Mean, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Harmonic Mean, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Harmonic Mean, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Harmonic Mean, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Harmonic Mean, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Harmonic mean of 2 and 3 is 2.4.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "confidence-interval-mean", "name": "Mean Confidence Interval", "latex": "\\bar x\\pm z_{\\alpha/2}\\frac{\\sigma}{\\sqrt n}", "category": "Statistics", "description": "Estimates a population mean when population standard deviation is known.", "variables": [ "x̄: sample mean", "z: critical value", "σ: population standard deviation", "n: sample size" ], "notes": [ "Use a t critical value when σ is unknown and assumptions apply." ], "examples": [ "For 95%, z≈1.96." ], "seo": { "title": "Mean Confidence Interval: Formula & Example", "description": "Reference the mean confidence interval, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "For the Mean Confidence Interval, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible.", "For the Mean Confidence Interval, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not collapse the ± in Mean Confidence Interval into one value; calculate each admissible branch and check it in the original problem.", "For the Mean Confidence Interval, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For 95%, z≈1.96.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "chi-square-statistic", "name": "Chi-Square Statistic", "latex": "\\chi^2=\\sum\\frac{(O-E)^2}{E}", "category": "Statistics", "description": "Compares observed and expected categorical counts.", "variables": [ "O: observed count", "E: expected count" ], "notes": [ "Expected counts should be sufficiently large for the approximation." ], "examples": [ "A large statistic indicates poor agreement with expectations." ], "seo": { "title": "Chi-Square Statistic Formula — Meaning, Variables & LaTeX", "description": "Reference the chi-square statistic, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Expected counts should be sufficiently large for the approximation.", "For the Chi-Square Statistic, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Chi-Square Statistic, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Chi-Square Statistic, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Chi-Square Statistic, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Chi-Square Statistic, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "A large statistic indicates poor agreement with expectations.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "probability-complement", "name": "Probability Complement Rule", "latex": "P(A^c)=1-P(A)", "category": "Probability", "description": "Finds the probability that an event does not occur.", "variables": [ "A: event", "Aᶜ: complement" ], "notes": [ "The event and its complement partition the sample space." ], "examples": [ "If P(A)=0.3, P(Aᶜ)=0.7." ], "seo": { "title": "Probability Complement Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the probability complement rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "The event and its complement partition the sample space.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Probability Complement Rule; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "If P(A)=0.3, P(Aᶜ)=0.7.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "probability-addition", "name": "Probability Addition Rule", "latex": "P(A\\cup B)=P(A)+P(B)-P(A\\cap B)", "category": "Probability", "description": "Finds the probability of at least one of two events.", "variables": [ "A,B: events" ], "notes": [ "For mutually exclusive events, the intersection term is zero." ], "examples": [ "Use it to avoid double-counting overlap." ], "seo": { "title": "Probability Addition Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the probability addition rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "For mutually exclusive events, the intersection term is zero.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Probability Addition Rule; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Use it to avoid double-counting overlap.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "law-total-probability", "name": "Law of Total Probability", "latex": "P(A)=\\sum_i P(A\\mid B_i)P(B_i)", "category": "Probability", "description": "Combines conditional probabilities across a partition.", "variables": [ "Bᵢ: mutually exclusive exhaustive events" ], "notes": [ "Every Bᵢ should have positive probability when conditioning." ], "examples": [ "Useful before applying Bayes theorem." ], "seo": { "title": "Law of Total Probability: Formula & Example", "description": "Reference the law of total probability, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "Every Bᵢ should have positive probability when conditioning.", "For the Law of Total Probability, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not omit the index or bounds in Law of Total Probability; changing either one changes which terms are included.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Useful before applying Bayes theorem.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "poisson-pmf", "name": "Poisson Probability Mass Function", "latex": "P(X=k)=\\frac{e^{-\\lambda}\\lambda^k}{k!}", "category": "Probability", "description": "Models counts of independent events in a fixed interval.", "variables": [ "λ: expected count", "k: nonnegative integer count" ], "notes": [ "Assumes a constant average rate and independent occurrences." ], "examples": [ "For λ=2, P(X=0)=e^{-2}." ], "seo": { "title": "Poisson Probability Mass Function: Formula & Worked Example", "description": "Reference the poisson probability mass function, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "Assumes a constant average rate and independent occurrences.", "For the Poisson Probability Mass Function, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Poisson Probability Mass Function, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "For λ=2, P(X=0)=e^{-2}.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "normal-density", "name": "Normal Distribution Density", "latex": "f(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}", "category": "Probability", "description": "Defines the bell-shaped normal probability density.", "variables": [ "μ: mean", "σ: positive standard deviation" ], "notes": [ "Probabilities are areas under the density curve." ], "examples": [ "The density is symmetric around μ." ], "seo": { "title": "Normal Distribution Density Formula — Meaning, Variables & LaTeX", "description": "Reference the normal distribution density, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "Probabilities are areas under the density curve.", "For the Normal Distribution Density, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "When copying Normal Distribution Density, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "The density is symmetric around μ.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "geometric-pmf", "name": "Geometric Distribution PMF", "latex": "P(X=k)=(1-p)^{k-1}p", "category": "Probability", "description": "Models the trial number of the first success.", "variables": [ "p: success probability", "k: positive trial number" ], "notes": [ "This convention starts counting at k=1." ], "examples": [ "P(X=1)=p." ], "seo": { "title": "Geometric Distribution PMF Formula — Meaning, Variables & LaTeX", "description": "Reference the geometric distribution pmf, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "This convention starts counting at k=1.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Geometric Distribution PMF; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "P(X=1)=p.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "matrix-multiplication", "name": "Matrix Multiplication", "latex": "(AB)_{ij}=\\sum_k A_{ik}B_{kj}", "category": "Linear Algebra", "description": "Defines each entry of a matrix product.", "variables": [ "A,B: compatible matrices", "i,j,k: indices" ], "notes": [ "Inner dimensions must match." ], "examples": [ "A 2×3 matrix can multiply a 3×4 matrix." ], "seo": { "title": "Matrix Multiplication Formula — Meaning, Variables & LaTeX", "description": "Reference the matrix multiplication, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "three-by-three-determinant", "inverse-two-by-two", "eigenvalue-equation" ], "assumptions": [ "Inner dimensions must match.", "For the Matrix Multiplication, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not omit the index or bounds in Matrix Multiplication; changing either one changes which terms are included.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "A 2×3 matrix can multiply a 3×4 matrix.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "three-by-three-determinant", "name": "Three by Three Determinant", "latex": "\\det A=a(ei-fh)-b(di-fg)+c(dh-eg)", "category": "Linear Algebra", "description": "Expands the determinant of a 3×3 matrix along the first row.", "variables": [ "a through i: matrix entries" ], "notes": [ "Cofactor signs alternate +,−,+." ], "examples": [ "Use row operations for larger matrices." ], "seo": { "title": "Three by Three Determinant Formula — Meaning, Variables & LaTeX", "description": "Reference the three by three determinant, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "inverse-two-by-two", "eigenvalue-equation" ], "assumptions": [ "Cofactor signs alternate +,−,+.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using Three by Three Determinant; matrix operations are order-sensitive.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Use row operations for larger matrices.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "inverse-two-by-two", "name": "Two by Two Matrix Inverse", "latex": "A^{-1}=\\frac{1}{ad-bc}\\begin{bmatrix}d&-b\\\\-c&a\\end{bmatrix}", "category": "Linear Algebra", "description": "Finds the inverse of a nonsingular 2×2 matrix.", "variables": [ "a,b,c,d: matrix entries" ], "notes": [ "The determinant ad−bc must be nonzero." ], "examples": [ "Check by multiplying A A⁻¹=I." ], "seo": { "title": "Two by Two Matrix Inverse Formula — Meaning, Variables & LaTeX", "description": "Reference the two by two matrix inverse, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "eigenvalue-equation" ], "assumptions": [ "The determinant ad−bc must be nonzero.", "For the Two by Two Matrix Inverse, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "When copying Two by Two Matrix Inverse, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Check by multiplying A A⁻¹=I.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "eigenvalue-equation", "name": "Eigenvalue Equation", "latex": "A\\mathbf v=\\lambda\\mathbf v", "category": "Linear Algebra", "description": "Defines an eigenvector and its eigenvalue.", "variables": [ "A: square matrix", "v: nonzero eigenvector", "λ: eigenvalue" ], "notes": [ "Solve det(A−λI)=0 for eigenvalues." ], "examples": [ "A scales v without changing its direction." ], "seo": { "title": "Eigenvalue Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the eigenvalue equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "Solve det(A−λI)=0 for eigenvalues.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Before substituting values into Eigenvalue Equation, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "A scales v without changing its direction.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "vector-norm", "name": "Euclidean Vector Norm", "latex": "\\|\\mathbf v\\|_2=\\sqrt{\\sum_i v_i^2}", "category": "Linear Algebra", "description": "Calculates the Euclidean length of a vector.", "variables": [ "vᵢ: vector components" ], "notes": [ "The norm is nonnegative and zero only for the zero vector." ], "examples": [ "||(3,4)||=5." ], "seo": { "title": "Euclidean Vector Norm Formula — Meaning, Variables & LaTeX", "description": "Reference the euclidean vector norm, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "The norm is nonnegative and zero only for the zero vector.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Euclidean Vector Norm, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined." ], "commonMistakes": [ "Do not drop the radical boundary in Euclidean Vector Norm; verify exactly which terms are inside the root and whether the chosen root is valid.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "||(3,4)||=5.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "vector-projection", "name": "Vector Projection", "latex": "\\operatorname{proj}_{\\mathbf b}\\mathbf a=\\frac{\\mathbf a\\cdot\\mathbf b}{\\mathbf b\\cdot\\mathbf b}\\mathbf b", "category": "Linear Algebra", "description": "Projects one vector onto the direction of another.", "variables": [ "a: vector being projected", "b: nonzero direction vector" ], "notes": [ "b cannot be the zero vector." ], "examples": [ "The result is parallel to b." ], "seo": { "title": "Vector Projection: Formula & Worked Example", "description": "Reference the vector projection, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "b cannot be the zero vector.", "For the Vector Projection, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "When copying Vector Projection, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "The result is parallel to b.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "cross-product-formula", "name": "Cross Product Components", "latex": "\\mathbf a\\times\\mathbf b=\\begin{bmatrix}a_2b_3-a_3b_2\\\\a_3b_1-a_1b_3\\\\a_1b_2-a_2b_1\\end{bmatrix}", "category": "Linear Algebra", "description": "Calculates the three-dimensional vector cross product.", "variables": [ "a,b: 3D vectors", "result: the quantity produced after the stated inputs and conditions are applied" ], "notes": [ "The result is perpendicular to both vectors." ], "examples": [ "i×j=k." ], "seo": { "title": "Cross Product Components: Formula, Worked Example & Checks", "description": "Learn the Cross Product Components, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "The result is perpendicular to both vectors.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using Cross Product Components; matrix operations are order-sensitive.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For a=(1,0,0) and b=(0,1,0), a×b=(0,0,1).", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Cross Product Components.", "sourceIds": [ "nist-dlmf" ], "verifiedAgainst": "NIST Digital Library of Mathematical Functions", "domain": "Apply Cross Product Components only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute the first component 0·0-0·1=0.", "Compute the second component 0·0-1·0=0.", "Compute the third component 1·1-0·0=1." ], "verification": "The result is perpendicular to both input vectors and has unit length.", "boundaryCases": [ "For Cross Product Components, check zero, negative, and extreme input values before relying on the result.", "When using Cross Product Components, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\mathbf a\\times\\mathbf b=\\begin{bmatrix}a_2b_3-a_3b_2\\\\a_3b_1-a_1b_3\\\\a_1b_2-a_2b_1\\end{bmatrix} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=(1,0,0), b=(0,1,0)", "workedOutput": "a×b=(0,0,1)", "notApplicableWhen": [ "Do not use Cross Product Components when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "euclidean-algorithm", "name": "Euclidean Algorithm", "latex": "\\gcd(a,b)=\\gcd(b,a\\bmod b)", "category": "Number Theory", "description": "Computes a greatest common divisor by repeated remainders.", "variables": [ "a,b: integers", "mod: remainder operation" ], "notes": [ "The displayed recurrence is applied until the remainder is zero." ], "examples": [ "gcd(48,18)=6." ], "seo": { "title": "Euclidean Algorithm Formula — Meaning, Variables & LaTeX", "description": "Reference the euclidean algorithm, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-totient", "euler-totient-product", "chinese-remainder" ], "assumptions": [ "The displayed recurrence is applied until the remainder is zero.", "Apply the formula over the intended integer domain and check divisibility, coprimality, parity, or modulus conditions." ], "commonMistakes": [ "Before substituting values into Euclidean Algorithm, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Euclidean Algorithm with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "gcd(48,18)=6.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "euler-totient", "name": "Euler Totient Formula for Prime Powers", "latex": "\\varphi(p^k)=p^k-p^{k-1}", "category": "Number Theory", "description": "Counts positive integers up to a prime power that are coprime to it.", "variables": [ "p: prime", "k: positive integer" ], "notes": [ "For general n, use multiplicativity across coprime factors." ], "examples": [ "φ(8)=4." ], "seo": { "title": "Euler Totient Formula for Prime Powers: Formula & Worked Example", "description": "Reference the euler totient formula for prime powers, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euclidean-algorithm", "euler-totient-product", "chinese-remainder" ], "assumptions": [ "For general n, use multiplicativity across coprime factors.", "Apply the formula over the intended integer domain and check divisibility, coprimality, parity, or modulus conditions." ], "commonMistakes": [ "Before substituting values into Euler Totient Formula for Prime Powers, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Euler Totient Formula for Prime Powers with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "φ(8)=4.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "newton-second-law", "name": "Newton Second Law", "latex": "\\mathbf F_{net}=m\\mathbf a", "category": "Physics", "description": "Relates net force, mass, and acceleration.", "variables": [ "Fnet: net force", "m: mass", "a: acceleration" ], "notes": [ "Use consistent SI units for newtons, kilograms, and metres per second squared." ], "examples": [ "A 2 kg mass accelerating at 3 m/s² needs 6 N net force." ], "seo": { "title": "Newton Second Law Formula — Meaning, Variables & LaTeX", "description": "Reference the newton second law, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula", "ohms-law" ], "assumptions": [ "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Newton Second Law; perform a dimensional check before accepting the result.", "Verify the result of Newton Second Law with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A 2 kg mass accelerating at 3 m/s² needs 6 N net force.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "mass-energy-equivalence", "name": "Mass Energy Equivalence", "latex": "E=mc^2", "category": "Physics", "description": "Relates rest energy to mass.", "variables": [ "E: rest energy", "m: mass", "c: speed of light" ], "notes": [ "This is the rest-energy relation, not a complete statement of all relativistic energy." ], "examples": [ "One kilogram corresponds to about 9×10¹⁶ joules." ], "seo": { "title": "Mass Energy Equivalence Formula — Meaning, Variables & LaTeX", "description": "Reference the mass energy equivalence, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "kinetic-energy", "gravitational-potential-energy", "momentum-formula", "ohms-law" ], "assumptions": [ "This is the rest-energy relation, not a complete statement of all relativistic energy.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Mass Energy Equivalence; perform a dimensional check before accepting the result.", "Verify the result of Mass Energy Equivalence with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "One kilogram corresponds to about 9×10¹⁶ joules.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "kinetic-energy", "name": "Kinetic Energy", "latex": "K=\\frac{1}{2}mv^2", "category": "Physics", "description": "Calculates translational kinetic energy.", "variables": [ "m: mass", "v: speed" ], "notes": [ "Speed is squared, so kinetic energy is nonnegative." ], "examples": [ "m=2 kg,v=3 m/s gives K=9 J." ], "seo": { "title": "Kinetic Energy Formula — Meaning, Variables & LaTeX", "description": "Reference the kinetic energy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "gravitational-potential-energy", "momentum-formula", "ohms-law" ], "assumptions": [ "Speed is squared, so kinetic energy is nonnegative.", "For the Kinetic Energy, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Kinetic Energy, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Kinetic Energy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "m=2 kg,v=3 m/s gives K=9 J.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "gravitational-potential-energy", "name": "Gravitational Potential Energy", "latex": "U=mgh", "category": "Physics", "description": "Approximates gravitational potential energy near Earth surface.", "variables": [ "m: mass", "g: gravitational acceleration", "h: height change" ], "notes": [ "Choose a reference height consistently." ], "examples": [ "m=1 kg,h=2 m gives about 19.6 J." ], "seo": { "title": "Gravitational Potential Energy Formula — Meaning, Variables & LaTeX", "description": "Reference the gravitational potential energy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "momentum-formula", "ohms-law" ], "assumptions": [ "Choose a reference height consistently.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Gravitational Potential Energy; perform a dimensional check before accepting the result.", "Verify the result of Gravitational Potential Energy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "m=1 kg,h=2 m gives about 19.6 J.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "momentum-formula", "name": "Linear Momentum", "latex": "\\mathbf p=m\\mathbf v", "category": "Physics", "description": "Defines linear momentum as mass times velocity.", "variables": [ "p: momentum", "m: mass", "v: velocity" ], "notes": [ "Momentum is a vector and is conserved in isolated systems." ], "examples": [ "A 2 kg object at 4 m/s has momentum 8 kg·m/s." ], "seo": { "title": "Linear Momentum Formula — Meaning, Variables & LaTeX", "description": "Reference the linear momentum, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "ohms-law" ], "assumptions": [ "Momentum is a vector and is conserved in isolated systems.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Linear Momentum; perform a dimensional check before accepting the result.", "Verify the result of Linear Momentum with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A 2 kg object at 4 m/s has momentum 8 kg·m/s.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "ohms-law", "name": "Ohm Law", "latex": "V=IR", "category": "Physics", "description": "Relates voltage, current, and resistance for an ohmic element.", "variables": [ "V: voltage", "I: current", "R: resistance" ], "notes": [ "The relation may not hold for nonlinear components." ], "examples": [ "I=2 A,R=5 Ω gives V=10 V." ], "seo": { "title": "Ohm Law Formula — Meaning, Variables & LaTeX", "description": "Reference the ohm law, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "The relation may not hold for nonlinear components.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Ohm Law; perform a dimensional check before accepting the result.", "Verify the result of Ohm Law with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "I=2 A,R=5 Ω gives V=10 V.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "electric-power", "name": "Electric Power", "latex": "P=VI", "category": "Physics", "description": "Calculates electrical power from voltage and current.", "variables": [ "P: power", "V: voltage", "I: current" ], "notes": [ "For DC resistive circuits, equivalent forms include I²R and V²/R." ], "examples": [ "12 V at 2 A uses 24 W." ], "seo": { "title": "Electric Power Formula — Meaning, Variables & LaTeX", "description": "Reference the electric power, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For DC resistive circuits, equivalent forms include I²R and V²/R.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Electric Power; perform a dimensional check before accepting the result.", "Verify the result of Electric Power with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "12 V at 2 A uses 24 W.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "wave-speed", "name": "Wave Speed", "latex": "v=f\\lambda", "category": "Physics", "description": "Relates wave speed, frequency, and wavelength.", "variables": [ "v: wave speed", "f: frequency", "λ: wavelength" ], "notes": [ "Use consistent units, usually hertz and metres." ], "examples": [ "f=100 Hz,λ=3 m gives v=300 m/s." ], "seo": { "title": "Wave Speed Formula — Meaning, Variables & LaTeX", "description": "Reference the wave speed, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Wave Speed; perform a dimensional check before accepting the result.", "Verify the result of Wave Speed with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "f=100 Hz,λ=3 m gives v=300 m/s.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "ideal-gas-law", "name": "Ideal Gas Law", "latex": "PV=nRT", "category": "Physics", "description": "Relates pressure, volume, amount, and absolute temperature for an ideal gas.", "variables": [ "P: pressure", "V: volume", "n: amount", "R: gas constant", "T: absolute temperature" ], "notes": [ "Temperature must be absolute, typically kelvin." ], "examples": [ "At fixed n and T, pressure is inversely proportional to volume." ], "seo": { "title": "Ideal Gas Law Formula — Meaning, Variables & LaTeX", "description": "Reference the ideal gas law, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "Temperature must be absolute, typically kelvin.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Ideal Gas Law; perform a dimensional check before accepting the result.", "Verify the result of Ideal Gas Law with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "At fixed n and T, pressure is inversely proportional to volume.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "law-of-cosines", "name": "Law of Cosines", "latex": "c^2=a^2+b^2-2ab\\cos C", "category": "Geometry", "description": "Relates the sides of any triangle to the cosine of one included angle.", "variables": [ "a, b, c: side lengths", "C: angle opposite side c" ], "notes": [ "It reduces to the Pythagorean theorem when C is 90 degrees.", "Use matching opposite side-angle pairs." ], "examples": [ "For a = 3, b = 4, and C = 90°, c = 5." ], "seo": { "title": "Law of Cosines: Formula, Worked Example & Checks", "description": "Learn the law of cosines with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Side lengths are positive and form a valid triangle.", "Angle C is opposite side c, and all angle measurements use a consistent degree or radian mode." ], "commonMistakes": [ "Do not pair an angle with the wrong opposite side.", "Check calculator angle mode and preserve the negative sign before 2ab cos C." ], "workedExample": "For a=3, b=4, and C=90°, c²=9+16−24·0=25, so c=5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "sourceIds": [ "openstax-algebra-trigonometry" ], "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Law of Cosines.", "verifiedAgainst": "OpenStax Algebra and Trigonometry 2e", "domain": "Use the law of cosines for a Euclidean triangle when the paired side lengths and included or opposite angle are identified consistently.", "workedSteps": [ "Label each side opposite its corresponding angle and identify the known side-angle relationship.", "Substitute a, b, and C into c²=a²+b²−2ab cos C with the correct angle mode.", "Evaluate the squared length before taking the nonnegative square root for the physical side length.", "Verify that the three side lengths satisfy the triangle inequality and that the result reduces to the Pythagorean theorem when C=90°." ], "verification": "Check the triangle inequalities and, for a right angle, confirm that the cosine term vanishes and the equation becomes c²=a²+b².", "boundaryCases": [ "At C=0° or 180°, the configuration becomes degenerate and should be interpreted as a limiting case rather than an ordinary triangle.", "For C=90°, the law of cosines must reduce exactly to the Pythagorean theorem." ], "alternativeForms": [ "Solving for the angle gives C=arccos((a²+b²−c²)/(2ab)) when a and b are nonzero and the ratio lies in [−1,1]." ], "workedInput": "a=3, b=4, C=90°", "workedOutput": "c=5", "notApplicableWhen": [ "Do not use Law of Cosines when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "law-of-sines", "name": "Law of Sines", "latex": "\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}", "category": "Geometry", "description": "Relates each side of a triangle to the sine of its opposite angle.", "variables": [ "a, b, c: side lengths", "A, B, C: opposite angles" ], "notes": [ "The ambiguous SSA case can produce zero, one, or two triangles.", "Angles must use a consistent unit." ], "examples": [ "If A = 30°, a = 5, and B = 45°, then b = 5 sin45° / sin30°." ], "seo": { "title": "Law of Sines: Formula, Worked Example & Checks", "description": "Learn the law of sines with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "The ambiguous SSA case can produce zero, one, or two triangles.", "For the Law of Sines, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Law of Sines, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Law of Sines, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "With A=30°, a=5, and B=45°, b=5sin45°/sin30°=5√2≈7.071.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Law of Sines.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Law of Sines only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Write b/sin45°=5/sin30°.", "Solve for b by multiplying by sin45°.", "Use sin45°=√2/2 and sin30°=1/2." ], "verification": "The larger angle B=45° is opposite the larger side b≈7.071, which is consistent.", "boundaryCases": [ "For Law of Sines, check zero, negative, and extreme input values before relying on the result.", "When using Law of Sines, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "A=30°, a=5, B=45°", "workedOutput": "b=5sin45°/sin30°≈7.071", "notApplicableWhen": [ "Do not use Law of Sines when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "ellipse-area", "name": "Area of an Ellipse", "latex": "A=\\pi ab", "category": "Geometry", "description": "Calculates the area enclosed by an ellipse from its semi-major and semi-minor axes.", "variables": [ "A: area", "a: semi-major axis", "b: semi-minor axis" ], "notes": [ "The values a and b are half of the full axis lengths.", "A circle is the special case a = b." ], "examples": [ "An ellipse with a = 4 and b = 2 has area 8π square units." ], "seo": { "title": "Area of an Ellipse: Formula, Worked Example & Checks", "description": "Learn the area of an ellipse with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "The values a and b are half of the full axis lengths.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Area of an Ellipse, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Area of an Ellipse, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For semi-axes a=4 and b=2, A=πab=8π≈25.133 square units.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of an Ellipse.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Area of an Ellipse only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Confirm that 4 and 2 are semi-axis lengths, not full diameters.", "Multiply a·b=8.", "Multiply by π and attach square units." ], "verification": "When a=b=2, the formula reduces to the circle area 4π.", "boundaryCases": [ "For Area of an Ellipse, check zero, negative, and extreme input values before relying on the result.", "When using Area of an Ellipse, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=\\pi ab for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a=4, b=2", "workedOutput": "A=8π≈25.133 square units", "notApplicableWhen": [ "Do not use Area of an Ellipse when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "circular-arc-length", "name": "Circular Arc Length", "latex": "s=r\\theta", "category": "Geometry", "description": "Calculates an arc length from a circle radius and a central angle measured in radians.", "variables": [ "s: arc length", "r: radius", "θ: central angle in radians" ], "notes": [ "Convert degrees to radians before using this form.", "For a full circle, θ = 2π and s = 2πr." ], "examples": [ "With r = 3 and θ = π/2, the arc length is 3π/2." ], "seo": { "title": "Circular Arc Length: Formula & Example", "description": "Learn the circular arc length with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Convert degrees to radians before using this form.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Circular Arc Length, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Circular Arc Length, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For r=3 and θ=π/2 radians, s=rθ=3π/2≈4.712.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Circular Arc Length.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Circular Arc Length only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Convert the central angle to radians if necessary.", "Multiply radius 3 by π/2.", "Report the result in the same length unit as the radius." ], "verification": "A quarter of a circle with circumference 6π has length 6π/4=3π/2.", "boundaryCases": [ "For Circular Arc Length, check zero, negative, and extreme input values before relying on the result.", "When using Circular Arc Length, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form s=r\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=3, θ=π/2 radians", "workedOutput": "s=3π/2≈4.712", "notApplicableWhen": [ "Do not use Circular Arc Length when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "matrix-trace", "name": "Matrix Trace", "latex": "\\operatorname{tr}(A)=\\sum_{i=1}^{n}a_{ii}", "category": "Linear Algebra", "description": "Adds the entries on the main diagonal of a square matrix.", "variables": [ "A: n × n square matrix", "aᵢᵢ: diagonal entries", "tr(A): trace of A" ], "notes": [ "Trace is defined for square matrices.", "The trace is invariant under similarity transformations." ], "examples": [ "For diag(2, 3, 5), tr(A) = 10." ], "seo": { "title": "Matrix Trace Formula — Meaning, Variables & LaTeX", "description": "Reference the matrix trace, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "Trace is defined for square matrices.", "For the Matrix Trace, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not omit the index or bounds in Matrix Trace; changing either one changes which terms are included.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For diag(2, 3, 5), tr(A) = 10.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "cramers-rule-two-by-two", "name": "Cramer's Rule for a 2×2 System", "latex": "x=\\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\\quad y=\\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2}", "category": "Linear Algebra", "description": "Solves a nonsingular two-equation linear system using determinant ratios.", "variables": [ "a₁x + b₁y = d₁: first equation", "a₂x + b₂y = d₂: second equation", "a₁b₂ − b₁a₂: coefficient determinant" ], "notes": [ "The denominator must be nonzero.", "Elimination is often more efficient for large systems." ], "examples": [ "For x + y = 5 and x − y = 1, Cramer's rule gives x = 3 and y = 2." ], "seo": { "title": "Cramer's Rule for a 2×2 System Formula — Meaning, Variables & LaTeX", "description": "Reference the cramer's rule for a 2×2 system, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "The denominator must be nonzero.", "For the Cramer's Rule for a 2×2 System, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "When copying Cramer's Rule for a 2×2 System, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For x + y = 5 and x − y = 1, Cramer's rule gives x = 3 and y = 2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "logistic-function", "name": "Logistic Growth Function", "latex": "P(t)=\\frac{K}{1+Ae^{-rt}}", "category": "Applied Math", "description": "Models growth that begins nearly exponentially and then approaches a carrying capacity.", "variables": [ "P(t): population or quantity at time t", "K: carrying capacity", "A: constant set by the initial value", "r: positive growth rate" ], "notes": [ "The model assumes a fixed carrying capacity.", "Parameter estimates should be checked against observed data." ], "examples": [ "When K = 1000, A = 9, and r = 0.5, the curve approaches 1000 as t increases." ], "seo": { "title": "Logistic Growth Function Formula — Meaning, Variables & LaTeX", "description": "Reference the logistic growth function, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "compound-interest", "percentage-change", "continuous-compound-interest", "exponential-population-growth" ], "assumptions": [ "The model assumes a fixed carrying capacity.", "For the Logistic Growth Function, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "commonMistakes": [ "When copying Logistic Growth Function, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Logistic Growth Function with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "When K = 1000, A = 9, and r = 0.5, the curve approaches 1000 as t increases.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "simple-harmonic-motion", "name": "Simple Harmonic Motion", "latex": "x(t)=A\\cos(\\omega t+\\varphi)", "category": "Physics", "description": "Describes ideal periodic motion whose restoring force is proportional to displacement.", "variables": [ "x(t): displacement", "A: amplitude", "ω: angular frequency", "φ: phase constant" ], "notes": [ "The period is T = 2π/ω.", "Real systems may require damping and driving terms." ], "examples": [ "With A = 2, ω = π, and φ = 0, x(1) = −2." ], "seo": { "title": "Simple Harmonic Motion Formula — Meaning, Variables & LaTeX", "description": "Reference the simple harmonic motion, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "The period is T = 2π/ω.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Simple Harmonic Motion; perform a dimensional check before accepting the result.", "Verify the result of Simple Harmonic Motion with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "With A = 2, ω = π, and φ = 0, x(1) = −2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "coulombs-law", "name": "Coulomb's Law", "latex": "F=k\\frac{|q_1q_2|}{r^2}", "category": "Physics", "description": "Calculates the magnitude of the electrostatic force between two point charges.", "variables": [ "F: force magnitude", "k: Coulomb constant", "q₁, q₂: electric charges", "r: separation distance" ], "notes": [ "The force direction depends on the signs of the charges.", "The point-charge approximation must be appropriate." ], "examples": [ "Doubling the separation reduces the force to one quarter." ], "seo": { "title": "Coulomb's Law Formula — Meaning, Variables & LaTeX", "description": "Reference the coulomb's law, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "The force direction depends on the signs of the charges.", "For the Coulomb's Law, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Coulomb's Law, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Coulomb's Law with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Doubling the separation reduces the force to one quarter.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "pressure-formula", "name": "Pressure Formula", "latex": "P=\\frac{F}{A}", "category": "Physics", "description": "Calculates average pressure as perpendicular force divided by the area receiving that force.", "variables": [ "P: pressure", "F: perpendicular force", "A: area" ], "notes": [ "The SI unit is the pascal, equal to one newton per square metre.", "Use the normal component of force." ], "examples": [ "A 100 N force spread over 0.5 m² produces 200 Pa." ], "seo": { "title": "Pressure Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the pressure formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "The SI unit is the pascal, equal to one newton per square metre.", "For the Pressure Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Pressure Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Pressure Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A 100 N force spread over 0.5 m² produces 200 Pa.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "density-formula", "name": "Density Formula", "latex": "\\rho=\\frac{m}{V}", "category": "Physics", "description": "Calculates mass per unit volume for a material or object.", "variables": [ "ρ: density", "m: mass", "V: volume" ], "notes": [ "Use consistent units such as kg and m³.", "Density may change with temperature and pressure." ], "examples": [ "A 2 kg object occupying 0.001 m³ has density 2000 kg/m³." ], "seo": { "title": "Density Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the density formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "Density may change with temperature and pressure.", "For the Density Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Density Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Density Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A 2 kg object occupying 0.001 m³ has density 2000 kg/m³.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "molarity-formula", "name": "Molarity Formula", "latex": "M=\\frac{n}{V}", "category": "Chemistry", "description": "Calculates solution concentration as moles of solute per litre of solution.", "variables": [ "M: molarity in mol/L", "n: amount of solute in moles", "V: solution volume in litres" ], "notes": [ "Use total solution volume, not solvent volume alone.", "Convert millilitres to litres before calculation." ], "examples": [ "0.25 mol dissolved to make 0.5 L of solution gives 0.5 M." ], "seo": { "title": "Molarity Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the molarity formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [], "assumptions": [ "Convert millilitres to litres before calculation.", "For the Molarity Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Balance quantities and units, and state the temperature, pressure, concentration, or idealization assumptions used." ], "commonMistakes": [ "When copying Molarity Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Molarity Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "0.25 mol dissolved to make 0.5 L of solution gives 0.5 M.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "sum-first-n-integers", "name": "Sum of the First n Positive Integers", "latex": "\\sum_{k=1}^{n}k=\\frac{n(n+1)}{2}", "category": "Discrete Math", "description": "Calculates the total of the integers from one through n.", "variables": [ "n: positive integer upper limit", "k: summation index" ], "notes": [ "The identity can be proved by pairing terms from opposite ends.", "For n = 0 the empty sum is zero." ], "examples": [ "For n = 10, the sum is 10×11÷2 = 55." ], "seo": { "title": "Sum of the First n Positive Integers: Formula & Worked Example", "description": "Reference the sum of the first n positive integers, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sum-first-n-squares", "sum-first-n-cubes", "binet-fibonacci" ], "assumptions": [ "The identity can be proved by pairing terms from opposite ends.", "For the Sum of the First n Positive Integers, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First n Positive Integers, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression." ], "commonMistakes": [ "When copying Sum of the First n Positive Integers, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Sum of the First n Positive Integers with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For n = 10, the sum is 10×11÷2 = 55.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "sum-first-n-squares", "name": "Sum of the First n Squares", "latex": "\\sum_{k=1}^{n}k^2=\\frac{n(n+1)(2n+1)}{6}", "category": "Discrete Math", "description": "Calculates the sum of the squared integers from one through n.", "variables": [ "n: positive integer upper limit", "k: summation index" ], "notes": [ "The result is a cubic polynomial in n.", "Use exact integer arithmetic before decimal conversion." ], "examples": [ "For n = 3, 1² + 2² + 3² = 14." ], "seo": { "title": "Sum of the First n Squares Formula — Meaning, Variables & LaTeX", "description": "Reference the sum of the first n squares, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sum-first-n-integers", "sum-first-n-cubes", "binet-fibonacci" ], "assumptions": [ "The result is a cubic polynomial in n.", "For the Sum of the First n Squares, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First n Squares, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression." ], "commonMistakes": [ "When copying Sum of the First n Squares, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Sum of the First n Squares with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For n = 3, 1² + 2² + 3² = 14.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "sum-first-n-cubes", "name": "Sum of the First n Cubes", "latex": "\\sum_{k=1}^{n}k^3=\\left(\\frac{n(n+1)}{2}\\right)^2", "category": "Discrete Math", "description": "Calculates the sum of the cubes from one through n.", "variables": [ "n: positive integer upper limit", "k: summation index" ], "notes": [ "The result is the square of the triangular number.", "The identity is useful in induction and finite-difference arguments." ], "examples": [ "For n = 3, 1³ + 2³ + 3³ = 36." ], "seo": { "title": "Sum of the First n Cubes Formula — Meaning, Variables & LaTeX", "description": "Reference the sum of the first n cubes, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sum-first-n-integers", "sum-first-n-squares", "binet-fibonacci" ], "assumptions": [ "The result is the square of the triangular number.", "For the Sum of the First n Cubes, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First n Cubes, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression." ], "commonMistakes": [ "When copying Sum of the First n Cubes, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Sum of the First n Cubes with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For n = 3, 1³ + 2³ + 3³ = 36.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "inclusion-exclusion-two-sets", "name": "Inclusion–Exclusion for Two Sets", "latex": "|A\\cup B|=|A|+|B|-|A\\cap B|", "category": "Combinatorics", "description": "Counts elements in a union without counting the intersection twice.", "variables": [ "A, B: finite sets", "|A|: number of elements in A" ], "notes": [ "The formula assumes ordinary finite cardinalities.", "For probabilities, replace cardinalities with probability measures." ], "examples": [ "If |A|=12, |B|=9, and |A∩B|=4, then |A∪B|=17." ], "seo": { "title": "Inclusion–Exclusion for Two Sets: Formula & Worked Example", "description": "Reference the inclusion–exclusion for two sets, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "pigeonhole-bound", "catalan-number" ], "assumptions": [ "For probabilities, replace cardinalities with probability measures.", "Define whether selections are ordered, repeated, labeled, or constrained before applying the counting formula." ], "commonMistakes": [ "Before substituting values into Inclusion–Exclusion for Two Sets, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Inclusion–Exclusion for Two Sets with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "If |A|=12, |B|=9, and |A∩B|=4, then |A∪B|=17.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "pigeonhole-bound", "name": "Pigeonhole Principle Bound", "latex": "\\max_i n_i\\ge\\left\\lceil\\frac{N}{m}\\right\\rceil", "category": "Combinatorics", "description": "Guarantees that at least one of m containers holds at least the ceiling of N divided by m objects.", "variables": [ "N: total number of objects", "m: number of containers", "n_i: occupancy of container i" ], "notes": [ "The principle guarantees existence but does not identify the container.", "Containers and assignments must be clearly defined." ], "examples": [ "Placing 13 objects into 4 boxes guarantees one box contains at least 4 objects." ], "seo": { "title": "Pigeonhole Principle Bound Formula — Meaning, Variables & LaTeX", "description": "Reference the pigeonhole principle bound, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "inclusion-exclusion-two-sets", "catalan-number" ], "assumptions": [ "The principle guarantees existence but does not identify the container.", "For the Pigeonhole Principle Bound, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Define whether selections are ordered, repeated, labeled, or constrained before applying the counting formula." ], "commonMistakes": [ "When copying Pigeonhole Principle Bound, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Pigeonhole Principle Bound with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Placing 13 objects into 4 boxes guarantees one box contains at least 4 objects.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "catalan-number", "name": "Catalan Number", "latex": "C_n=\\frac{1}{n+1}\\binom{2n}{n}", "category": "Combinatorics", "description": "Counts many recursively structured objects such as balanced parentheses and polygon triangulations.", "variables": [ "n: nonnegative integer index", "C_n: nth Catalan number" ], "notes": [ "Equivalent forms include C_n=(2n)!/(n!(n+1)!).", "Different counting problems require a proof of the Catalan correspondence." ], "examples": [ "C₄ = 14, so a convex hexagon has 14 triangulations." ], "seo": { "title": "Catalan Number Formula — Meaning, Variables & LaTeX", "description": "Reference the catalan number, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "inclusion-exclusion-two-sets", "pigeonhole-bound" ], "assumptions": [ "Equivalent forms include C_n=(2n)!/(n!(n+1)!).", "For the Catalan Number, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Define whether selections are ordered, repeated, labeled, or constrained before applying the counting formula." ], "commonMistakes": [ "When copying Catalan Number, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Catalan Number with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "C₄ = 14, so a convex hexagon has 14 triangulations.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "binet-fibonacci", "name": "Binet Formula for Fibonacci Numbers", "latex": "F_n=\\frac{\\varphi^n-(1-\\varphi)^n}{\\sqrt5}", "category": "Discrete Math", "description": "Expresses the nth Fibonacci number using powers of the golden ratio.", "variables": [ "F_n: nth Fibonacci number", "n: nonnegative integer", "φ: (1+√5)/2" ], "notes": [ "Floating-point evaluation should be rounded carefully for large n.", "Indexing conventions may start with F₀=0 or F₁=1." ], "examples": [ "Using n=10 gives F₁₀=55." ], "seo": { "title": "Binet Formula for Fibonacci Numbers: Formula & Worked Example", "description": "Reference the binet formula for fibonacci numbers, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sum-first-n-integers", "sum-first-n-squares", "sum-first-n-cubes" ], "assumptions": [ "Floating-point evaluation should be rounded carefully for large n.", "For the Binet Formula for Fibonacci Numbers, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression." ], "commonMistakes": [ "When copying Binet Formula for Fibonacci Numbers, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Binet Formula for Fibonacci Numbers with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Using n=10 gives F₁₀=55.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "de-morgan-sets", "name": "De Morgan Law for Sets", "latex": "(A\\cup B)^c=A^c\\cap B^c", "category": "Set Theory", "description": "Relates the complement of a union to the intersection of complements.", "variables": [ "A, B: subsets of a universal set", "c: set complement" ], "notes": [ "The universal set must be understood.", "A second law exchanges union and intersection." ], "examples": [ "If an element is outside A∪B, it is outside both A and B." ], "seo": { "title": "De Morgan Law for Sets Formula — Meaning, Variables & LaTeX", "description": "Reference the de morgan law for sets, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [], "assumptions": [ "The universal set must be understood.", "Define the universe and the meanings of inclusion, equality, complement, and cardinality used in the expression." ], "commonMistakes": [ "Before substituting values into De Morgan Law for Sets, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of De Morgan Law for Sets with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "If an element is outside A∪B, it is outside both A and B.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "de-morgan-logic", "name": "De Morgan Law for Logic", "latex": "\\neg(p\\land q)\\equiv(\\neg p)\\lor(\\neg q)", "category": "Logic", "description": "Rewrites the negation of a conjunction as a disjunction of negations.", "variables": [ "p, q: propositions", "¬: logical negation", "∧, ∨: conjunction and disjunction" ], "notes": [ "The dual law negates a disjunction into a conjunction.", "The equivalence holds for classical Boolean logic." ], "examples": [ "“Not both conditions hold” is equivalent to “at least one condition fails.”" ], "seo": { "title": "De Morgan Law for Logic Formula — Meaning, Variables & LaTeX", "description": "Reference the de morgan law for logic, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [], "assumptions": [ "The dual law negates a disjunction into a conjunction.", "State the logical system, operator precedence, and whether equivalence is semantic, syntactic, or truth-functional." ], "commonMistakes": [ "Before substituting values into De Morgan Law for Logic, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of De Morgan Law for Logic with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "“Not both conditions hold” is equivalent to “at least one condition fails.”", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "complex-modulus", "name": "Complex Number Modulus", "latex": "|z|=\\sqrt{a^2+b^2},\\quad z=a+bi", "category": "Algebra", "description": "Calculates the distance of a complex number from the origin in the complex plane.", "variables": [ "z: complex number", "a: real part", "b: imaginary part" ], "notes": [ "The modulus is always nonnegative.", "It obeys |zw|=|z||w|." ], "examples": [ "For z=3+4i, |z|=5." ], "seo": { "title": "Complex Number Modulus: Formula, Worked Example & Checks", "description": "Learn the Complex Number Modulus, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "The modulus is always nonnegative.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Complex Number Modulus, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Do not drop the radical boundary in Complex Number Modulus; verify exactly which terms are inside the root and whether the chosen root is valid.", "For the Complex Number Modulus, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For z=3+4i, |z|=√(3²+4²)=5.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Complex Number Modulus.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Complex Number Modulus only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify the real part a=3 and imaginary coefficient b=4.", "Compute a²+b²=9+16=25.", "Take the nonnegative square root to obtain 5." ], "verification": "Multiplying z by its conjugate gives 25, so |z|²=25.", "boundaryCases": [ "For Complex Number Modulus, check zero, negative, and extreme input values before relying on the result.", "When using Complex Number Modulus, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form |z|=\\sqrt{a^2+b^2},\\quad z=a+bi for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "z=3+4i", "workedOutput": "|z|=5", "notApplicableWhen": [ "Do not use Complex Number Modulus when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "euler-identity", "name": "Euler Identity", "latex": "e^{i\\pi}+1=0", "category": "Algebra", "description": "Connects the constants e, i, π, one, and zero in a single complex exponential identity.", "variables": [ "e: Euler number", "i: imaginary unit", "π: circle constant" ], "notes": [ "It follows from Euler’s formula e^{ix}=cos x+i sin x.", "The exponent uses radians." ], "examples": [ "Setting x=π gives cos π+i sin π=-1." ], "seo": { "title": "Euler Identity: Formula, Worked Example & Checks", "description": "Learn the euler identity with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "It follows from Euler’s formula e^{ix}=cos x+i sin x.", "For the Euler Identity, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Euler Identity, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Euler Identity, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Euler’s formula at x=π gives e^(iπ)=cosπ+i sinπ=-1, hence e^(iπ)+1=0.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Euler Identity.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Euler Identity only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Substitute x=π into e^(ix)=cosx+i sinx.", "Use cosπ=-1 and sinπ=0.", "Add 1 to both sides to obtain zero." ], "verification": "The real part is -1 and the imaginary part is 0, matching the unit-circle point at angle π.", "boundaryCases": [ "For Euler Identity, check zero, negative, and extreme input values before relying on the result.", "When using Euler Identity, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form e^{i\\pi}+1=0 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "x=π in e^(ix)=cos x+i sin x", "workedOutput": "e^(iπ)+1=0", "notApplicableWhen": [ "Do not use Euler Identity when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "complex-polar-form", "name": "Complex Number Polar Form", "latex": "z=r(\\cos\\theta+i\\sin\\theta)=re^{i\\theta}", "category": "Algebra", "description": "Expresses a complex number by magnitude and argument.", "variables": [ "r: nonnegative modulus", "θ: argument in radians", "i: imaginary unit" ], "notes": [ "The argument is defined modulo 2π.", "At z=0 the argument is not unique." ], "examples": [ "The number 1+i has r=√2 and θ=π/4." ], "seo": { "title": "Complex Number Polar Form: Formula, Worked Example & Checks", "description": "Learn the Complex Number Polar Form, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "The argument is defined modulo 2π.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "For the Complex Number Polar Form, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Complex Number Polar Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Complex Number Polar Form, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For z=1+i, r=√(1²+1²)=√2 and θ=atan2(1,1)=π/4.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Complex Number Polar Form.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Complex Number Polar Form only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute the modulus r=√2.", "Determine the first-quadrant argument θ=π/4.", "Write z=√2(cosπ/4+i sinπ/4)=√2e^(iπ/4)." ], "verification": "Converting back gives real and imaginary parts equal to 1.", "boundaryCases": [ "For Complex Number Polar Form, check zero, negative, and extreme input values before relying on the result.", "When using Complex Number Polar Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form z=r(\\cos\\theta+i\\sin\\theta)=re^{i\\theta} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "z=1+i", "workedOutput": "r=√2, θ=π/4", "notApplicableWhen": [ "Do not use Complex Number Polar Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "logarithm-product-rule", "name": "Logarithm Product Rule", "latex": "\\log_b(xy)=\\log_b x+\\log_b y", "category": "Algebra", "description": "Converts a logarithm of a product into a sum of logarithms.", "variables": [ "b: positive base not equal to one", "x, y: positive arguments" ], "notes": [ "The real-valued rule requires positive x and y.", "Do not apply it to log(x+y)." ], "examples": [ "log₁₀(100×10)=2+1=3." ], "seo": { "title": "Logarithm Product Rule: Formula, Worked Example & Checks", "description": "Learn the Logarithm Product Rule, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "The real-valued rule requires positive x and y.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Logarithm Product Rule, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Logarithm Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Logarithm Product Rule, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "log₁₀(100×10)=log₁₀100+log₁₀10=2+1=3.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Logarithm Product Rule.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Logarithm Product Rule only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Confirm that both factors are positive.", "Apply the product rule.", "Evaluate the two simple logarithms and add." ], "verification": "The product is 1000 and log₁₀1000=3.", "boundaryCases": [ "For Logarithm Product Rule, check zero, negative, and extreme input values before relying on the result.", "When using Logarithm Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\log_b(xy)=\\log_b x+\\log_b y for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "log₁₀(100×10)", "workedOutput": "3=2+1", "notApplicableWhen": [ "Do not use Logarithm Product Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "sine-double-angle", "name": "Sine Double-Angle Identity", "latex": "\\sin(2\\theta)=2\\sin\\theta\\cos\\theta", "category": "Trigonometry", "description": "Rewrites the sine of a doubled angle as a product.", "variables": [ "θ: angle measured consistently" ], "notes": [ "The identity holds for degrees or radians when the same unit is used throughout.", "It follows from the sine addition formula." ], "examples": [ "For θ=30°, sin60°=2 sin30° cos30°=√3/2." ], "seo": { "title": "Sine Double-Angle Identity: Formula & Example", "description": "Reference the sine double-angle identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "cosine-double-angle", "pythagorean-trig-identity", "double-angle-sine", "double-angle-cosine", "half-angle-sine" ], "assumptions": [ "The identity holds for degrees or radians when the same unit is used throughout.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "Do not mix degrees and radians in Sine Double-Angle Identity; use one angle convention throughout the calculation.", "Verify the result of Sine Double-Angle Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For θ=30°, sin60°=2 sin30° cos30°=√3/2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "cosine-double-angle", "name": "Cosine Double-Angle Identity", "latex": "\\cos(2\\theta)=\\cos^2\\theta-\\sin^2\\theta", "category": "Trigonometry", "description": "Rewrites the cosine of a doubled angle using squared sine and cosine.", "variables": [ "θ: angle" ], "notes": [ "Equivalent forms are 2cos²θ−1 and 1−2sin²θ.", "Choose the form that matches the available information." ], "examples": [ "For θ=45°, cos90°=1/2−1/2=0." ], "seo": { "title": "Cosine Double-Angle Identity: Formula & Example", "description": "Reference the cosine double-angle identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sine-double-angle", "pythagorean-trig-identity", "double-angle-sine", "double-angle-cosine", "half-angle-sine" ], "assumptions": [ "Equivalent forms are 2cos²θ−1 and 1−2sin²θ.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "Do not mix degrees and radians in Cosine Double-Angle Identity; use one angle convention throughout the calculation.", "Verify the result of Cosine Double-Angle Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For θ=45°, cos90°=1/2−1/2=0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "pythagorean-trig-identity", "name": "Pythagorean Trigonometric Identity", "latex": "\\sin^2\\theta+\\cos^2\\theta=1", "category": "Trigonometry", "description": "Relates sine and cosine for the same angle.", "variables": [ "θ: angle" ], "notes": [ "The identity comes from the unit circle.", "Rearrange it to replace sin²θ or cos²θ." ], "examples": [ "If sinθ=3/5 in the first quadrant, cosθ=4/5." ], "seo": { "title": "Pythagorean Trigonometric Identity: Variables & Worked Example", "description": "Reference the pythagorean trigonometric identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sine-double-angle", "cosine-double-angle", "double-angle-sine", "double-angle-cosine", "half-angle-sine" ], "assumptions": [ "The identity comes from the unit circle.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "Do not mix degrees and radians in Pythagorean Trigonometric Identity; use one angle convention throughout the calculation.", "Verify the result of Pythagorean Trigonometric Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "If sinθ=3/5 in the first quadrant, cosθ=4/5.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "sector-area", "name": "Area of a Circular Sector", "latex": "A=\\frac12 r^2\\theta", "category": "Geometry", "description": "Calculates sector area when the central angle is measured in radians.", "variables": [ "A: sector area", "r: circle radius", "θ: central angle in radians" ], "notes": [ "For degrees, use A=(θ/360°)πr².", "The radius and area units must be compatible." ], "examples": [ "With r=3 and θ=π/2, A=9π/4." ], "seo": { "title": "Area of a Circular Sector: Formula, Worked Example & Checks", "description": "Learn the Area of a Circular Sector, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "For degrees, use A=(θ/360°)πr².", "For the Area of a Circular Sector, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "When copying Area of a Circular Sector, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Area of a Circular Sector, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For r=3 and θ=π/2 radians, A=½·9·π/2=9π/4≈7.069.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Area of a Circular Sector.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Area of a Circular Sector only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Use radians for θ.", "Square the radius to obtain 9.", "Multiply by θ/2." ], "verification": "A 90° sector is one quarter of the circle area 9π, which is 9π/4.", "boundaryCases": [ "For Area of a Circular Sector, check zero, negative, and extreme input values before relying on the result.", "When using Area of a Circular Sector, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form A=\\frac12 r^2\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=3, θ=π/2 radians", "workedOutput": "A=9π/4≈7.069", "notApplicableWhen": [ "Do not use Area of a Circular Sector when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "arc-length-radians", "name": "Arc Length from a Radian Angle", "latex": "s=r\\theta", "category": "Geometry", "description": "Calculates the length of a circular arc from radius and angle in radians.", "variables": [ "s: arc length", "r: radius", "θ: central angle in radians" ], "notes": [ "Convert degrees to radians before using the formula.", "The result has the same length unit as r." ], "examples": [ "For r=5 and θ=π/3, s=5π/3." ], "seo": { "title": "Arc Length from a Radian Angle — Formula, Variables, Steps & Example", "description": "Use the arc length from a radian angle with defined variables, applicability checks, worked steps, verification guidance, common mistakes, and copyable LaTeX." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Convert degrees to radians before using the formula.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Arc Length from a Radian Angle, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Arc Length from a Radian Angle, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "Worked setup for Arc Length from a Radian Angle: For r=5 and θ=π/3, s=5π/3. The Arc Length from a Radian Angle calculation should preserve its original grouping and record the final unit or interpretation.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arc Length from a Radian Angle.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Arc Length from a Radian Angle only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify every input required by Arc Length from a Radian Angle and confirm that each value matches the variable definition and domain.", "Substitute the values into the complete expression s=r\\theta without dropping parentheses, signs, powers, or branches.", "Evaluate Arc Length from a Radian Angle in a stable order, preserve sufficient precision, and write the result together with its unit or mathematical interpretation.", "Verify the result independently by substitution, dimensional analysis, a limiting case, or an equivalent form appropriate to Geometry." ], "verification": "Check Arc Length from a Radian Angle by substituting the result back into the defining relation when possible, confirming units or dimensions, and testing a simple limiting or known case.", "boundaryCases": [ "For Arc Length from a Radian Angle, check zero, negative, and extreme input values before relying on the result.", "When using Arc Length from a Radian Angle, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form s=r\\theta for copying; rearrange only after preserving equivalence and domain restrictions." ] }, { "id": "rectangular-prism-volume", "name": "Rectangular Prism Volume", "latex": "V=lwh", "category": "Geometry", "description": "Calculates volume from length, width, and height.", "variables": [ "V: volume", "l: length", "w: width", "h: height" ], "notes": [ "All dimensions must use compatible units.", "The formula applies to right rectangular prisms." ], "examples": [ "A 2×3×4 prism has volume 24 cubic units." ], "seo": { "title": "Rectangular Prism Volume: Formula, Worked Example & Checks", "description": "Learn the Rectangular Prism Volume, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "All dimensions must use compatible units.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Rectangular Prism Volume, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Rectangular Prism Volume, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "A prism with dimensions 2×3×4 has volume V=24 cubic units.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Rectangular Prism Volume.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Rectangular Prism Volume only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Use mutually perpendicular length, width, and height.", "Multiply 2×3=6.", "Multiply 6×4=24 and use cubic units." ], "verification": "The prism contains 4 layers of 6 unit cubes.", "boundaryCases": [ "For Rectangular Prism Volume, check zero, negative, and extreme input values before relying on the result.", "When using Rectangular Prism Volume, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form V=lwh for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "l=2, w=3, h=4", "workedOutput": "V=24 cubic units", "notApplicableWhen": [ "Do not use Rectangular Prism Volume when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "cylinder-surface-area", "name": "Total Surface Area of a Cylinder", "latex": "S=2\\pi r^2+2\\pi rh", "category": "Geometry", "description": "Adds the areas of two circular bases and the curved lateral surface.", "variables": [ "S: total surface area", "r: radius", "h: height" ], "notes": [ "For lateral area only, omit the 2πr² term.", "Use consistent length units." ], "examples": [ "For r=2 and h=5, S=28π." ], "seo": { "title": "Total Surface Area of a Cylinder: Formula, Worked Example & Checks", "description": "Learn the Total Surface Area of a Cylinder, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "For lateral area only, omit the 2πr² term.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Total Surface Area of a Cylinder, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Total Surface Area of a Cylinder, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For r=2 and h=5, S=2π(2²)+2π(2)(5)=8π+20π=28π≈87.965.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Total Surface Area of a Cylinder.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Total Surface Area of a Cylinder only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute the two circular ends: 2πr²=8π.", "Compute the lateral area: 2πrh=20π.", "Add the components to obtain 28π." ], "verification": "The net consists of two circles and a rectangle of width 2πr and height h, giving the same total.", "boundaryCases": [ "For Total Surface Area of a Cylinder, check zero, negative, and extreme input values before relying on the result.", "When using Total Surface Area of a Cylinder, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form S=2\\pi r^2+2\\pi rh for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "r=2, h=5", "workedOutput": "S=28π≈87.965 square units", "notApplicableWhen": [ "Do not use Total Surface Area of a Cylinder when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "continuous-compound-interest", "name": "Continuous Compound Interest", "latex": "A=Pe^{rt}", "category": "Applied Math", "description": "Models continuously compounded growth of a principal amount.", "variables": [ "A: final amount", "P: initial principal", "r: annual rate as a decimal", "t: time in years" ], "notes": [ "The rate and time units must match.", "Real financial products may include fees and discrete posting rules." ], "examples": [ "At P=1000, r=0.05, t=2, A≈1105.17." ], "seo": { "title": "Continuous Compound Interest Formula — Meaning, Variables & LaTeX", "description": "Reference the continuous compound interest, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "compound-interest", "percentage-change", "logistic-function", "exponential-population-growth" ], "assumptions": [ "The rate and time units must match.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "commonMistakes": [ "Before substituting values into Continuous Compound Interest, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Continuous Compound Interest with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "At P=1000, r=0.05, t=2, A≈1105.17.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "exponential-population-growth", "name": "Exponential Population Growth", "latex": "N(t)=N_0e^{rt}", "category": "Applied Math", "description": "Models unrestricted growth or decay at a constant proportional rate.", "variables": [ "N(t): quantity at time t", "N₀: initial quantity", "r: growth rate", "t: elapsed time" ], "notes": [ "Positive r produces growth and negative r produces decay.", "Resource limits require a logistic rather than exponential model." ], "examples": [ "With N₀=100 and r=0.1, N(5)≈164.87." ], "seo": { "title": "Exponential Population Growth Formula — Meaning, Variables & LaTeX", "description": "Reference the exponential population growth, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "compound-interest", "percentage-change", "logistic-function", "continuous-compound-interest" ], "assumptions": [ "Positive r produces growth and negative r produces decay.", "Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions." ], "commonMistakes": [ "Before substituting values into Exponential Population Growth, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Exponential Population Growth with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "With N₀=100 and r=0.1, N(5)≈164.87.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "shannon-entropy", "name": "Shannon Entropy", "latex": "H(X)=-\\sum_i p_i\\log_2 p_i", "category": "Statistics", "description": "Measures uncertainty in a discrete probability distribution in bits.", "variables": [ "H(X): entropy", "p_i: probability of outcome i" ], "notes": [ "Terms with p_i=0 contribute zero by continuity.", "Changing the logarithm base changes the unit." ], "examples": [ "A fair binary variable has entropy 1 bit." ], "seo": { "title": "Shannon Entropy Formula — Meaning, Variables & LaTeX", "description": "Reference the shannon entropy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Terms with p_i=0 contribute zero by continuity.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Shannon Entropy, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Shannon Entropy, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not omit the index or bounds in Shannon Entropy; changing either one changes which terms are included.", "For the Shannon Entropy, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "A fair binary variable has entropy 1 bit.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "root-mean-square", "name": "Root Mean Square", "latex": "x_{\\mathrm{rms}}=\\sqrt{\\frac1n\\sum_{i=1}^{n}x_i^2}", "category": "Statistics", "description": "Measures the quadratic mean of a set of values.", "variables": [ "x_i: observations", "n: number of observations" ], "notes": [ "RMS is always nonnegative.", "For alternating signals it relates to equivalent power." ], "examples": [ "The RMS of 3 and 4 is √12.5≈3.536." ], "seo": { "title": "Root Mean Square Formula — Meaning, Variables & LaTeX", "description": "Reference the root mean square, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "RMS is always nonnegative.", "For the Root Mean Square, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Root Mean Square, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear." ], "commonMistakes": [ "When copying Root Mean Square, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Root Mean Square, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "The RMS of 3 and 4 is √12.5≈3.536.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "coefficient-variation", "name": "Coefficient of Variation", "latex": "CV=\\frac{s}{\\bar{x}}\\times100\\%", "category": "Statistics", "description": "Expresses sample standard deviation relative to the sample mean.", "variables": [ "CV: coefficient of variation", "s: sample standard deviation", "x̄: sample mean" ], "notes": [ "It is unstable when the mean is near zero.", "Use only when a ratio scale and meaningful zero are appropriate." ], "examples": [ "If x̄=50 and s=5, CV=10%." ], "seo": { "title": "Coefficient of Variation Formula — Meaning, Variables & LaTeX", "description": "Reference the coefficient of variation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "It is unstable when the mean is near zero.", "For the Coefficient of Variation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Coefficient of Variation, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Coefficient of Variation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Coefficient of Variation, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "If x̄=50 and s=5, CV=10%.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "handshake-lemma", "name": "Handshake Lemma", "latex": "\\sum_{v\\in V}\\deg(v)=2|E|", "category": "Graph Theory", "description": "Relates the sum of all vertex degrees in a finite undirected graph to twice its edge count.", "variables": [ "V: vertex set", "E: edge set", "deg(v): degree of vertex v" ], "notes": [ "Every edge contributes one to the degree of each endpoint.", "The number of odd-degree vertices is therefore even." ], "examples": [ "For degrees 3, 2, 2, 1, the sum is 8, so the graph has 4 edges." ], "seo": { "title": "Handshake Lemma Formula — Meaning, Variables & LaTeX", "description": "Reference the handshake lemma, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "complete-graph-edges", "tree-edge-count", "euler-planar-formula", "average-graph-degree", "adjacency-matrix-degree" ], "assumptions": [ "Every edge contributes one to the degree of each endpoint.", "For the Handshake Lemma, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "Do not omit the index or bounds in Handshake Lemma; changing either one changes which terms are included.", "Verify the result of Handshake Lemma with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For degrees 3, 2, 2, 1, the sum is 8, so the graph has 4 edges.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "complete-graph-edges", "name": "Complete Graph Edge Count", "latex": "|E(K_n)|=\\frac{n(n-1)}{2}", "category": "Graph Theory", "description": "Counts the edges in a simple complete graph on n vertices.", "variables": [ "K_n: complete graph on n vertices", "n: number of vertices" ], "notes": [ "Every unordered pair of distinct vertices determines one edge.", "Loops and repeated edges are excluded." ], "examples": [ "K_5 has 5·4/2=10 edges." ], "seo": { "title": "Complete Graph Edge Count Formula — Meaning, Variables & LaTeX", "description": "Reference the complete graph edge count, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "handshake-lemma", "tree-edge-count", "euler-planar-formula", "average-graph-degree", "adjacency-matrix-degree" ], "assumptions": [ "Every unordered pair of distinct vertices determines one edge.", "For the Complete Graph Edge Count, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "When copying Complete Graph Edge Count, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Complete Graph Edge Count with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "K_5 has 5·4/2=10 edges.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "tree-edge-count", "name": "Tree Edge Count", "latex": "|E|=|V|-1", "category": "Graph Theory", "description": "States the edge count of a finite connected acyclic graph.", "variables": [ "V: vertex set", "E: edge set" ], "notes": [ "The formula characterizes trees among connected simple graphs.", "A forest with c components has |E|=|V|-c." ], "examples": [ "A tree with 12 vertices has 11 edges." ], "seo": { "title": "Tree Edge Count Formula — Meaning, Variables & LaTeX", "description": "Reference the tree edge count, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "handshake-lemma", "complete-graph-edges", "euler-planar-formula", "average-graph-degree", "adjacency-matrix-degree" ], "assumptions": [ "A forest with c components has |E|=|V|-c.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "Before substituting values into Tree Edge Count, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Tree Edge Count with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A tree with 12 vertices has 11 edges.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "euler-planar-formula", "name": "Euler Formula for Planar Graphs", "latex": "|V|-|E|+|F|=2", "category": "Graph Theory", "description": "Relates vertices, edges, and faces in a connected planar embedding.", "variables": [ "V: vertices", "E: edges", "F: faces including the outer face" ], "notes": [ "The graph must be connected and planar in the chosen embedding.", "For c components, the right side becomes 1+c." ], "examples": [ "A cube graph has 8−12+6=2." ], "seo": { "title": "Euler Formula for Planar Graphs Formula — Meaning, Variables & LaTeX", "description": "Reference the euler formula for planar graphs, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "handshake-lemma", "complete-graph-edges", "tree-edge-count", "average-graph-degree", "adjacency-matrix-degree" ], "assumptions": [ "The graph must be connected and planar in the chosen embedding.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "Before substituting values into Euler Formula for Planar Graphs, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Euler Formula for Planar Graphs with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A cube graph has 8−12+6=2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "average-graph-degree", "name": "Average Graph Degree", "latex": "\\bar d=\\frac{2|E|}{|V|}", "category": "Graph Theory", "description": "Computes the mean degree of a finite undirected graph.", "variables": [ "d̄: average degree", "E: edge set", "V: vertex set" ], "notes": [ "Derived directly from the handshake lemma.", "Isolated vertices are included with degree zero." ], "examples": [ "A graph with 7 vertices and 9 edges has average degree 18/7." ], "seo": { "title": "Average Graph Degree Formula — Meaning, Variables & LaTeX", "description": "Reference the average graph degree, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "handshake-lemma", "complete-graph-edges", "tree-edge-count", "euler-planar-formula", "adjacency-matrix-degree" ], "assumptions": [ "Derived directly from the handshake lemma.", "For the Average Graph Degree, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "When copying Average Graph Degree, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Average Graph Degree with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A graph with 7 vertices and 9 edges has average degree 18/7.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "adjacency-matrix-degree", "name": "Degree from Adjacency Matrix", "latex": "\\deg(v_i)=\\sum_j A_{ij}", "category": "Graph Theory", "description": "Computes a vertex degree by summing its row in a simple undirected graph adjacency matrix.", "variables": [ "A: adjacency matrix", "A_ij: adjacency entry", "v_i: selected vertex" ], "notes": [ "For directed graphs, row and column sums usually represent out-degree and in-degree.", "Weights change the interpretation to weighted degree." ], "examples": [ "If row i is [0,1,1,0], then deg(v_i)=2." ], "seo": { "title": "Degree from Adjacency Matrix Formula — Meaning, Variables & LaTeX", "description": "Reference the degree from adjacency matrix, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "handshake-lemma", "complete-graph-edges", "tree-edge-count", "euler-planar-formula", "average-graph-degree" ], "assumptions": [ "For directed graphs, row and column sums usually represent out-degree and in-degree.", "For the Degree from Adjacency Matrix, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "State whether the graph is directed, weighted, simple, connected, or finite whenever the formula depends on those properties." ], "commonMistakes": [ "Do not omit the index or bounds in Degree from Adjacency Matrix; changing either one changes which terms are included.", "Verify the result of Degree from Adjacency Matrix with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "If row i is [0,1,1,0], then deg(v_i)=2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "first-order-linear-ode", "name": "First-Order Linear ODE Solution", "latex": "y=e^{-\\int P(x)dx}\\left(\\int Q(x)e^{\\int P(x)dx}dx+C\\right)", "category": "Differential Equations", "description": "Gives the integrating-factor solution of y′+P(x)y=Q(x).", "variables": [ "P(x): coefficient function", "Q(x): forcing function", "C: integration constant" ], "notes": [ "Continuity assumptions are needed on the interval of interest.", "Apply initial data after finding the general solution." ], "examples": [ "For y′+y=1, the solution is y=1+Ce^{-x}." ], "seo": { "title": "First-Order Linear ODE Solution Formula — Meaning, Variables & LaTeX", "description": "Reference the first-order linear ode solution, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation", "wave-equation" ], "assumptions": [ "Continuity assumptions are needed on the interval of interest.", "Integration limits, the differential, and any constant of integration must be included where required.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Keep the integration bounds and differential attached to First-Order Linear ODE Solution, and include a constant of integration for an indefinite integral.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "For y′+y=1, the solution is y=1+Ce^{-x}.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "exponential-decay-ode", "name": "Exponential Decay Model", "latex": "y(t)=y_0e^{-kt}", "category": "Differential Equations", "description": "Solves a proportional decay law y′=−ky with positive rate k.", "variables": [ "y_0: initial value", "k: decay constant", "t: elapsed time" ], "notes": [ "The model assumes the instantaneous decay rate is proportional to the current amount.", "Half-life is ln(2)/k." ], "examples": [ "With y0=100 and k=0.1, y(10)=100e^{-1}." ], "seo": { "title": "Exponential Decay Model Formula — Meaning, Variables & LaTeX", "description": "Reference the exponential decay model, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation", "wave-equation" ], "assumptions": [ "The model assumes the instantaneous decay rate is proportional to the current amount.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Before substituting values into Exponential Decay Model, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "With y0=100 and k=0.1, y(10)=100e^{-1}.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "logistic-differential-equation", "name": "Logistic Differential Equation", "latex": "\\frac{dP}{dt}=rP\\left(1-\\frac{P}{K}\\right)", "category": "Differential Equations", "description": "Models growth that slows as a population approaches carrying capacity.", "variables": [ "P: population", "r: intrinsic growth rate", "K: carrying capacity", "t: time" ], "notes": [ "P=0 and P=K are equilibrium solutions.", "The model assumes a constant carrying capacity and homogeneous population." ], "examples": [ "When P=K/2, the growth rate is rK/4." ], "seo": { "title": "Logistic Differential Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the logistic differential equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "harmonic-oscillator-ode", "heat-equation", "wave-equation" ], "assumptions": [ "P=0 and P=K are equilibrium solutions.", "For the Logistic Differential Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "When copying Logistic Differential Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "When P=K/2, the growth rate is rK/4.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "harmonic-oscillator-ode", "name": "Simple Harmonic Oscillator Equation", "latex": "m\\ddot x+kx=0", "category": "Differential Equations", "description": "Describes an ideal mass-spring oscillator without damping or forcing.", "variables": [ "m: mass", "k: spring constant", "x: displacement" ], "notes": [ "Angular frequency is sqrt(k/m).", "Real systems may require damping and forcing terms." ], "examples": [ "For m=1 and k=9, the angular frequency is 3 rad/s." ], "seo": { "title": "Simple Harmonic Oscillator Equation: Variables & Worked Example", "description": "Reference the simple harmonic oscillator equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "heat-equation", "wave-equation" ], "assumptions": [ "Angular frequency is sqrt(k/m).", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Before substituting values into Simple Harmonic Oscillator Equation, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "For m=1 and k=9, the angular frequency is 3 rad/s.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "heat-equation", "name": "Heat Equation", "latex": "\\frac{\\partial u}{\\partial t}=\\alpha\\nabla^2u", "category": "Differential Equations", "description": "Models diffusion of heat or another conserved quantity through a medium.", "variables": [ "u: temperature or concentration field", "α: diffusivity", "t: time", "∇²: Laplacian" ], "notes": [ "Boundary and initial conditions are required for a unique solution.", "The constant-coefficient form assumes homogeneous material properties." ], "examples": [ "In one dimension, u_t=αu_xx." ], "seo": { "title": "Heat Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the heat equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "wave-equation" ], "assumptions": [ "Boundary and initial conditions are required for a unique solution.", "For the Heat Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "When copying Heat Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "In one dimension, u_t=αu_xx.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "wave-equation", "name": "Wave Equation", "latex": "\\frac{\\partial^2u}{\\partial t^2}=c^2\\nabla^2u", "category": "Differential Equations", "description": "Models ideal wave propagation with constant speed c.", "variables": [ "u: displacement field", "c: wave speed", "t: time", "∇²: Laplacian" ], "notes": [ "Initial displacement and velocity are normally required.", "Damping and dispersion need additional terms." ], "examples": [ "A vibrating string satisfies u_tt=c²u_xx." ], "seo": { "title": "Wave Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the wave equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation" ], "assumptions": [ "Initial displacement and velocity are normally required.", "For the Wave Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "When copying Wave Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "A vibrating string satisfies u_tt=c²u_xx.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "laplace-equation", "name": "Laplace Equation", "latex": "\\nabla^2u=0", "category": "Differential Equations", "description": "Defines harmonic functions and steady-state potential fields.", "variables": [ "u: scalar field", "∇²: Laplacian operator" ], "notes": [ "Boundary conditions determine the solution.", "It is the steady-state form of the heat equation without sources." ], "examples": [ "Electrostatic potential in a charge-free region satisfies ∇²u=0." ], "seo": { "title": "Laplace Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the laplace equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation" ], "assumptions": [ "Boundary conditions determine the solution.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Before substituting values into Laplace Equation, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "Electrostatic potential in a charge-free region satisfies ∇²u=0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "euler-method", "name": "Euler Method", "latex": "y_{n+1}=y_n+h f(t_n,y_n)", "category": "Numerical Methods", "description": "Advances an ordinary differential equation solution using the current slope.", "variables": [ "h: step size", "f: derivative function", "y_n: current approximation", "t_n: current time" ], "notes": [ "The method is first-order accurate.", "Smaller steps usually reduce truncation error but increase work." ], "examples": [ "For y′=y, h=0.1, and y0=1, y1=1.1." ], "seo": { "title": "Euler Method Formula — Meaning, Variables & LaTeX", "description": "Reference the euler method, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule", "central-difference-first" ], "assumptions": [ "The method is first-order accurate.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "Before substituting values into Euler Method, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "For y′=y, h=0.1, and y0=1, y1=1.1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "runge-kutta-four", "name": "Classical Runge–Kutta Method", "latex": "y_{n+1}=y_n+\\frac{h}{6}(k_1+2k_2+2k_3+k_4)", "category": "Numerical Methods", "description": "Combines four slope estimates to produce a fourth-order ODE step.", "variables": [ "h: step size", "k1…k4: staged slope estimates", "y_n: current approximation" ], "notes": [ "The displayed formula assumes the standard RK4 stage definitions.", "Adaptive methods estimate error and vary h." ], "examples": [ "RK4 is commonly used when a robust fixed-step solver is sufficient." ], "seo": { "title": "Classical Runge–Kutta Method Formula — Meaning, Variables & LaTeX", "description": "Reference the classical runge–kutta method, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "newton-raphson", "trapezoidal-rule", "simpson-rule", "central-difference-first" ], "assumptions": [ "The displayed formula assumes the standard RK4 stage definitions.", "For the Classical Runge–Kutta Method, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Classical Runge–Kutta Method, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "RK4 is commonly used when a robust fixed-step solver is sufficient.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "newton-raphson", "name": "Newton–Raphson Iteration", "latex": "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}", "category": "Numerical Methods", "description": "Iteratively approximates a root using the tangent line at the current estimate.", "variables": [ "x_n: current estimate", "f: target function", "f′: derivative" ], "notes": [ "Convergence depends on the initial guess and a nonzero derivative.", "Safeguards are needed near flat derivatives or discontinuities." ], "examples": [ "For f(x)=x²−2, the iteration converges to sqrt(2)." ], "seo": { "title": "Newton–Raphson Iteration Formula — Meaning, Variables & LaTeX", "description": "Reference the newton–raphson iteration, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "runge-kutta-four", "trapezoidal-rule", "simpson-rule", "central-difference-first" ], "assumptions": [ "Convergence depends on the initial guess and a nonzero derivative.", "For the Newton–Raphson Iteration, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Newton–Raphson Iteration, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "For f(x)=x²−2, the iteration converges to sqrt(2).", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "trapezoidal-rule", "name": "Trapezoidal Rule", "latex": "\\int_a^b f(x)dx\\approx\\frac{h}{2}\\left[f(x_0)+2\\sum_{i=1}^{n-1}f(x_i)+f(x_n)\\right]", "category": "Numerical Methods", "description": "Approximates a definite integral with piecewise linear segments.", "variables": [ "h=(b−a)/n: step width", "x_i: grid point", "n: number of subintervals" ], "notes": [ "The composite rule assumes equal spacing.", "Error decreases quadratically for sufficiently smooth functions." ], "examples": [ "With one interval, the estimate is the trapezoid area (b−a)(f(a)+f(b))/2." ], "seo": { "title": "Trapezoidal Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the trapezoidal rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "simpson-rule", "central-difference-first" ], "assumptions": [ "The composite rule assumes equal spacing.", "For the Trapezoidal Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Trapezoidal Rule, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Integration limits, the differential, and any constant of integration must be included where required." ], "commonMistakes": [ "When copying Trapezoidal Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "With one interval, the estimate is the trapezoid area (b−a)(f(a)+f(b))/2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "simpson-rule", "name": "Simpson’s Rule", "latex": "\\int_a^b f(x)dx\\approx\\frac{h}{3}\\left[f(x_0)+4\\sum_{i\\ odd}f(x_i)+2\\sum_{i\\ even}f(x_i)+f(x_n)\\right]", "category": "Numerical Methods", "description": "Approximates a definite integral with piecewise quadratic interpolation.", "variables": [ "h=(b−a)/n: step width", "n: even number of subintervals", "x_i: grid points" ], "notes": [ "The composite rule requires even n.", "It is exact for polynomials up to degree three." ], "examples": [ "For n=2, use weights 1,4,1 across the three sampled points." ], "seo": { "title": "Simpson’s Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the simpson’s rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "central-difference-first" ], "assumptions": [ "The composite rule requires even n.", "For the Simpson’s Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Simpson’s Rule, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Integration limits, the differential, and any constant of integration must be included where required." ], "commonMistakes": [ "When copying Simpson’s Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "For n=2, use weights 1,4,1 across the three sampled points.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "central-difference-first", "name": "Central Difference First Derivative", "latex": "f'(x)\\approx\\frac{f(x+h)-f(x-h)}{2h}", "category": "Numerical Methods", "description": "Approximates a first derivative using symmetric samples around x.", "variables": [ "h: small step", "f: sampled function", "x: evaluation point" ], "notes": [ "The truncation error is second order for smooth functions.", "Roundoff error can dominate when h is excessively small." ], "examples": [ "Use samples at x−h and x+h rather than a one-sided difference." ], "seo": { "title": "Central Difference First Derivative: Formula & Worked Example", "description": "Reference the central difference first derivative, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule" ], "assumptions": [ "The truncation error is second order for smooth functions.", "For the Central Difference First Derivative, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Central Difference First Derivative, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "Use samples at x−h and x+h rather than a one-sided difference.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "central-difference-second", "name": "Central Difference Second Derivative", "latex": "f''(x)\\approx\\frac{f(x+h)-2f(x)+f(x-h)}{h^2}", "category": "Numerical Methods", "description": "Approximates a second derivative using three equally spaced samples.", "variables": [ "h: grid spacing", "f: sampled function", "x: center point" ], "notes": [ "The formula is second-order accurate for smooth functions.", "It is a standard building block for finite-difference PDE solvers." ], "examples": [ "The stencil coefficients are 1, −2, 1 divided by h²." ], "seo": { "title": "Central Difference Second Derivative: Formula & Worked Example", "description": "Reference the central difference second derivative, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule" ], "assumptions": [ "It is a standard building block for finite-difference PDE solvers.", "For the Central Difference Second Derivative, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Central Difference Second Derivative, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "The stencil coefficients are 1, −2, 1 divided by h².", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "least-squares-normal-equation", "name": "Least-Squares Normal Equation", "latex": "\\hat\\beta=(X^TX)^{-1}X^Ty", "category": "Statistics", "description": "Computes ordinary least-squares coefficients when XᵀX is invertible.", "variables": [ "X: design matrix", "y: response vector", "β̂: estimated coefficients" ], "notes": [ "Numerical software typically uses QR or SVD rather than forming the inverse.", "Collinearity can make XᵀX singular or poorly conditioned." ], "examples": [ "For a full-rank design matrix, β̂ minimizes ||Xβ−y||²." ], "seo": { "title": "Least-Squares Normal Equation Formula — Meaning, Variables & LaTeX", "description": "Reference the least-squares normal equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Numerical software typically uses QR or SVD rather than forming the inverse.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "For the Least-Squares Normal Equation, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using Least-Squares Normal Equation; matrix operations are order-sensitive.", "For the Least-Squares Normal Equation, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "For a full-rank design matrix, β̂ minimizes ||Xβ−y||².", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "gradient-descent", "name": "Gradient Descent Update", "latex": "x_{k+1}=x_k-\\eta\\nabla f(x_k)", "category": "Optimization", "description": "Updates parameters in the direction opposite the objective gradient.", "variables": [ "x_k: current iterate", "η: step size", "∇f: gradient" ], "notes": [ "Step-size selection strongly affects convergence.", "Constraints require projection or another constrained method." ], "examples": [ "For f(x)=x², the update is x_{k+1}=(1−2η)x_k." ], "seo": { "title": "Gradient Descent Update: Formula & Example", "description": "Reference the gradient descent update, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "newton-optimization", "lagrange-multiplier-condition", "kkt-stationarity", "jensen-inequality", "linear-program-standard" ], "assumptions": [ "Step-size selection strongly affects convergence.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Gradient Descent Update, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "For f(x)=x², the update is x_{k+1}=(1−2η)x_k.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "newton-optimization", "name": "Newton Optimization Step", "latex": "x_{k+1}=x_k-[\\nabla^2f(x_k)]^{-1}\\nabla f(x_k)", "category": "Optimization", "description": "Uses gradient and Hessian information to form a local quadratic step.", "variables": [ "∇f: gradient", "∇²f: Hessian", "x_k: current iterate" ], "notes": [ "The Hessian must be invertible or regularized.", "A line search or trust region can improve robustness." ], "examples": [ "For a positive-definite quadratic, Newton’s method reaches the minimizer in one exact step." ], "seo": { "title": "Newton Optimization Step Formula — Meaning, Variables & LaTeX", "description": "Reference the newton optimization step, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "lagrange-multiplier-condition", "kkt-stationarity", "jensen-inequality", "linear-program-standard" ], "assumptions": [ "The Hessian must be invertible or regularized.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Do not reorder factors or mix incompatible dimensions when using Newton Optimization Step; matrix operations are order-sensitive.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "For a positive-definite quadratic, Newton’s method reaches the minimizer in one exact step.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "lagrange-multiplier-condition", "name": "Lagrange Multiplier Condition", "latex": "\\nabla f(x)=\\lambda\\nabla g(x)", "category": "Optimization", "description": "States the first-order condition for an equality-constrained extremum with one regular constraint.", "variables": [ "f: objective", "g: equality constraint", "λ: Lagrange multiplier" ], "notes": [ "Constraint qualification is required.", "Multiple constraints use a sum of multiplier-weighted gradients." ], "examples": [ "Optimize f subject to g=0 by solving ∇f=λ∇g together with g=0." ], "seo": { "title": "Lagrange Multiplier Condition Formula — Meaning, Variables & LaTeX", "description": "Reference the lagrange multiplier condition, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "newton-optimization", "kkt-stationarity", "jensen-inequality", "linear-program-standard" ], "assumptions": [ "Constraint qualification is required.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Lagrange Multiplier Condition, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "Optimize f subject to g=0 by solving ∇f=λ∇g together with g=0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "kkt-stationarity", "name": "KKT Stationarity", "latex": "\\nabla f(x)+\\sum_i\\lambda_i\\nabla g_i(x)+\\sum_j\\nu_j\\nabla h_j(x)=0", "category": "Optimization", "description": "Gives the stationarity part of the Karush–Kuhn–Tucker conditions.", "variables": [ "f: objective", "g_i: inequality constraints", "h_j: equality constraints", "λ_i,ν_j: multipliers" ], "notes": [ "Full KKT conditions also include feasibility and complementary slackness.", "For convex problems under regularity, KKT conditions can be sufficient." ], "examples": [ "Combine stationarity with λ_i≥0 and λ_i g_i(x)=0." ], "seo": { "title": "KKT Stationarity Formula — Meaning, Variables & LaTeX", "description": "Reference the kkt stationarity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "newton-optimization", "lagrange-multiplier-condition", "jensen-inequality", "linear-program-standard" ], "assumptions": [ "Full KKT conditions also include feasibility and complementary slackness.", "For the KKT Stationarity, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Do not omit the index or bounds in KKT Stationarity; changing either one changes which terms are included.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "Combine stationarity with λ_i≥0 and λ_i g_i(x)=0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "jensen-inequality", "name": "Jensen’s Inequality", "latex": "f(\\mathbb E[X])\\leq\\mathbb E[f(X)]", "category": "Optimization", "description": "Relates a convex function of an expectation to the expectation of the convex function.", "variables": [ "f: convex function", "X: random variable", "E: expectation" ], "notes": [ "The inequality reverses for concave f.", "Integrability and domain conditions must hold." ], "examples": [ "For f(x)=x², (E[X])²≤E[X²]." ], "seo": { "title": "Jensen’s Inequality Formula — Meaning, Variables & LaTeX", "description": "Reference the jensen’s inequality, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "newton-optimization", "lagrange-multiplier-condition", "kkt-stationarity", "linear-program-standard" ], "assumptions": [ "The inequality reverses for concave f.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Jensen’s Inequality, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "For f(x)=x², (E[X])²≤E[X²].", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "linear-program-standard", "name": "Linear Program Standard Form", "latex": "\\min_x c^Tx\\quad\\text{s.t.}\\quad Ax=b,\\;x\\geq0", "category": "Optimization", "description": "Expresses a linear objective with equality constraints and nonnegative variables.", "variables": [ "c: objective coefficients", "A: constraint matrix", "b: right-hand side", "x: decision vector" ], "notes": [ "Other linear programs can be transformed into this form.", "Feasibility and boundedness must be checked." ], "examples": [ "Slack variables convert many inequalities into equalities." ], "seo": { "title": "Linear Program Standard Form Formula — Meaning, Variables & LaTeX", "description": "Reference the linear program standard form, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "newton-optimization", "lagrange-multiplier-condition", "kkt-stationarity", "jensen-inequality" ], "assumptions": [ "Other linear programs can be transformed into this form.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Linear Program Standard Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "Slack variables convert many inequalities into equalities.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "quadratic-form", "name": "Quadratic Form", "latex": "q(x)=x^TAx", "category": "Linear Algebra", "description": "Defines a scalar quadratic expression associated with a square matrix.", "variables": [ "x: column vector", "A: square matrix", "q(x): scalar value" ], "notes": [ "Only the symmetric part of A affects q(x).", "Positive definiteness is central in optimization and stability." ], "examples": [ "For A=I, q(x)=||x||²." ], "seo": { "title": "Quadratic Form Formula — Meaning, Variables & LaTeX", "description": "Reference the quadratic form, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "Only the symmetric part of A affects q(x).", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Before substituting values into Quadratic Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For A=I, q(x)=||x||².", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "simple-interest", "name": "Simple Interest", "latex": "A=P(1+rt)", "category": "Financial Mathematics", "description": "Computes accumulated value when interest is applied only to the original principal.", "variables": [ "P: principal", "r: rate per period", "t: number of periods", "A: accumulated amount" ], "notes": [ "Rate and time units must match.", "Simple interest differs from compounding." ], "examples": [ "P=1000, r=5%, t=2 gives A=1100." ], "seo": { "title": "Simple Interest Formula — Meaning, Variables & LaTeX", "description": "Reference the simple interest, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "present-value", "future-value-annuity", "present-value-annuity", "loan-payment", "net-present-value" ], "assumptions": [ "Rate and time units must match.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "Do not mix annual, monthly, nominal, and effective rates in Simple Interest; convert the rate and number of periods to the same time basis.", "Verify the result of Simple Interest with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "P=1000, r=5%, t=2 gives A=1100.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "present-value", "name": "Present Value", "latex": "PV=\\frac{FV}{(1+r)^n}", "category": "Financial Mathematics", "description": "Discounts a future amount back n compound periods.", "variables": [ "PV: present value", "FV: future value", "r: rate per period", "n: periods" ], "notes": [ "The discount rate must match the period length.", "Risk and inflation assumptions determine the chosen rate." ], "examples": [ "FV=1210 at 10% for two periods has PV=1000." ], "seo": { "title": "Present Value Formula — Meaning, Variables & LaTeX", "description": "Reference the present value, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "future-value-annuity", "present-value-annuity", "loan-payment", "net-present-value" ], "assumptions": [ "The discount rate must match the period length.", "For the Present Value, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Present Value, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Present Value with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "FV=1210 at 10% for two periods has PV=1000.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "future-value-annuity", "name": "Future Value of Ordinary Annuity", "latex": "FV=PMT\\frac{(1+r)^n-1}{r}", "category": "Financial Mathematics", "description": "Computes the future value of equal end-of-period payments.", "variables": [ "PMT: payment per period", "r: rate per period", "n: number of payments" ], "notes": [ "For payments at the beginning, multiply by 1+r.", "The formula assumes a constant rate and payment." ], "examples": [ "Ten annual payments of 100 at 5% accumulate to 100((1.05)^10−1)/0.05." ], "seo": { "title": "Future Value of Ordinary Annuity: Formula & Worked Example", "description": "Reference the future value of ordinary annuity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "present-value-annuity", "loan-payment", "net-present-value" ], "assumptions": [ "For payments at the beginning, multiply by 1+r.", "For the Future Value of Ordinary Annuity, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Future Value of Ordinary Annuity, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Future Value of Ordinary Annuity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Ten annual payments of 100 at 5% accumulate to 100((1.05)^10−1)/0.05.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "present-value-annuity", "name": "Present Value of Ordinary Annuity", "latex": "PV=PMT\\frac{1-(1+r)^{-n}}{r}", "category": "Financial Mathematics", "description": "Discounts equal end-of-period payments to a present value.", "variables": [ "PMT: payment", "r: rate per period", "n: payments" ], "notes": [ "Beginning-of-period annuities require an additional factor 1+r.", "Taxes and fees are not included unless added to cash flows." ], "examples": [ "A five-payment stream is discounted using the matching per-period rate." ], "seo": { "title": "Present Value of Ordinary Annuity: Formula & Worked Example", "description": "Reference the present value of ordinary annuity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "loan-payment", "net-present-value" ], "assumptions": [ "Beginning-of-period annuities require an additional factor 1+r.", "For the Present Value of Ordinary Annuity, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Present Value of Ordinary Annuity, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Present Value of Ordinary Annuity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A five-payment stream is discounted using the matching per-period rate.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "loan-payment", "name": "Level Loan Payment", "latex": "PMT=P\\frac{r(1+r)^n}{(1+r)^n-1}", "category": "Financial Mathematics", "description": "Calculates the fixed payment for a fully amortizing loan.", "variables": [ "P: principal", "r: periodic rate", "n: number of payments", "PMT: payment" ], "notes": [ "The formula excludes fees and changing rates.", "Use the payment frequency to convert the nominal rate correctly." ], "examples": [ "A monthly mortgage uses monthly r and total monthly payment count n." ], "seo": { "title": "Level Loan Payment Formula — Meaning, Variables & LaTeX", "description": "Reference the level loan payment, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "net-present-value" ], "assumptions": [ "For the Level Loan Payment, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Level Loan Payment, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Level Loan Payment with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A monthly mortgage uses monthly r and total monthly payment count n.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "net-present-value", "name": "Net Present Value", "latex": "NPV=\\sum_{t=0}^{n}\\frac{CF_t}{(1+r)^t}", "category": "Financial Mathematics", "description": "Sums discounted cash flows including the initial investment.", "variables": [ "CF_t: cash flow at time t", "r: discount rate", "n: final period" ], "notes": [ "Cash-flow signs must be consistent.", "NPV depends on the selected discount rate." ], "examples": [ "An initial outflow is usually CF0<0." ], "seo": { "title": "Net Present Value Formula — Meaning, Variables & LaTeX", "description": "Reference the net present value, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "Cash-flow signs must be consistent.", "For the Net Present Value, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Net Present Value, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Net Present Value, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Net Present Value with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "An initial outflow is usually CF0<0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "internal-rate-return", "name": "Internal Rate of Return Equation", "latex": "0=\\sum_{t=0}^{n}\\frac{CF_t}{(1+IRR)^t}", "category": "Financial Mathematics", "description": "Defines an internal rate of return as a discount rate that makes NPV zero.", "variables": [ "CF_t: cash flow", "IRR: internal rate of return", "t: time index" ], "notes": [ "Multiple sign changes can create multiple IRRs.", "Numerical root finding is usually required." ], "examples": [ "Solve the NPV equation for IRR rather than rearranging algebraically." ], "seo": { "title": "Internal Rate of Return Equation: Variables & Worked Example", "description": "Reference the internal rate of return equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "Multiple sign changes can create multiple IRRs.", "For the Internal Rate of Return Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Internal Rate of Return Equation, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Internal Rate of Return Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Internal Rate of Return Equation with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Solve the NPV equation for IRR rather than rearranging algebraically.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "effective-annual-rate", "name": "Effective Annual Rate", "latex": "EAR=\\left(1+\\frac{r}{m}\\right)^m-1", "category": "Financial Mathematics", "description": "Converts a nominal annual rate compounded m times into an effective annual rate.", "variables": [ "r: nominal annual rate", "m: compounding periods per year", "EAR: effective annual rate" ], "notes": [ "The nominal rate convention must be known.", "Continuous compounding uses e^r−1." ], "examples": [ "A 12% nominal rate compounded monthly has EAR=(1.01)^12−1." ], "seo": { "title": "Effective Annual Rate Formula — Meaning, Variables & LaTeX", "description": "Reference the effective annual rate, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "The nominal rate convention must be known.", "For the Effective Annual Rate, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Effective Annual Rate, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Effective Annual Rate with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A 12% nominal rate compounded monthly has EAR=(1.01)^12−1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "compound-annual-growth-rate", "name": "Compound Annual Growth Rate", "latex": "CAGR=\\left(\\frac{V_f}{V_i}\\right)^{1/n}-1", "category": "Financial Mathematics", "description": "Finds the constant annual growth rate connecting an initial and final value.", "variables": [ "V_i: initial value", "V_f: final value", "n: years" ], "notes": [ "CAGR hides volatility between endpoints.", "Both values should be positive for the standard real-valued formula." ], "examples": [ "Growth from 100 to 121 in two years gives 10% CAGR." ], "seo": { "title": "Compound Annual Growth Rate Formula — Meaning, Variables & LaTeX", "description": "Reference the compound annual growth rate, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "CAGR hides volatility between endpoints.", "For the Compound Annual Growth Rate, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Compound Annual Growth Rate, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Compound Annual Growth Rate with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Growth from 100 to 121 in two years gives 10% CAGR.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "bond-price", "name": "Coupon Bond Price", "latex": "P=\\sum_{t=1}^{n}\\frac{C}{(1+y)^t}+\\frac{F}{(1+y)^n}", "category": "Financial Mathematics", "description": "Discounts coupon payments and face value using yield per period.", "variables": [ "P: bond price", "C: coupon per period", "F: face value", "y: yield per period", "n: periods" ], "notes": [ "Coupon frequency and yield units must match.", "Market conventions may include accrued interest and day-count rules." ], "examples": [ "A zero-coupon bond sets C=0." ], "seo": { "title": "Coupon Bond Price Formula — Meaning, Variables & LaTeX", "description": "Reference the coupon bond price, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "Coupon frequency and yield units must match.", "For the Coupon Bond Price, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Coupon Bond Price, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "When copying Coupon Bond Price, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Coupon Bond Price with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A zero-coupon bond sets C=0.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "t-statistic-one-sample", "name": "One-Sample t Statistic", "latex": "t=\\frac{\\bar x-\\mu_0}{s/\\sqrt n}", "category": "Statistics", "description": "Standardizes a sample mean when population standard deviation is unknown.", "variables": [ "x̄: sample mean", "μ0: hypothesized mean", "s: sample standard deviation", "n: sample size" ], "notes": [ "The classic test assumes independent observations and approximate normality.", "Degrees of freedom are n−1." ], "examples": [ "Compare the computed t with a t distribution having n−1 degrees of freedom." ], "seo": { "title": "One-Sample t Statistic Formula — Meaning, Variables & LaTeX", "description": "Reference the one-sample t statistic, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "The classic test assumes independent observations and approximate normality.", "For the One-Sample t Statistic, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the One-Sample t Statistic, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying One-Sample t Statistic, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the One-Sample t Statistic, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Compare the computed t with a t distribution having n−1 degrees of freedom.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "f-statistic", "name": "F Statistic for Two Variances", "latex": "F=\\frac{s_1^2}{s_2^2}", "category": "Statistics", "description": "Compares two sample variances under an F-distribution model.", "variables": [ "s1²,s2²: sample variances" ], "notes": [ "Place the chosen numerator and denominator consistently.", "The test is sensitive to non-normal data." ], "examples": [ "If s1²=12 and s2²=3, F=4." ], "seo": { "title": "F Statistic for Two Variances Formula — Meaning, Variables & LaTeX", "description": "Reference the f statistic for two variances, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Place the chosen numerator and denominator consistently.", "For the F Statistic for Two Variances, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the F Statistic for Two Variances, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying F Statistic for Two Variances, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the F Statistic for Two Variances, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "If s1²=12 and s2²=3, F=4.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "regression-slope", "name": "Simple Regression Slope", "latex": "b_1=\\frac{\\sum(x_i-\\bar x)(y_i-\\bar y)}{\\sum(x_i-\\bar x)^2}", "category": "Statistics", "description": "Computes the least-squares slope in simple linear regression.", "variables": [ "b1: slope estimate", "x_i,y_i: observations", "x̄,ȳ: sample means" ], "notes": [ "The denominator must be nonzero.", "A slope describes association, not necessarily causation." ], "examples": [ "After finding b1, use b0=ȳ−b1x̄." ], "seo": { "title": "Simple Regression Slope Formula — Meaning, Variables & LaTeX", "description": "Reference the simple regression slope, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "The denominator must be nonzero.", "For the Simple Regression Slope, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Simple Regression Slope, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Simple Regression Slope, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Simple Regression Slope, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Simple Regression Slope, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "After finding b1, use b0=ȳ−b1x̄.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "regression-intercept", "name": "Simple Regression Intercept", "latex": "b_0=\\bar y-b_1\\bar x", "category": "Statistics", "description": "Computes the intercept of a fitted simple regression line.", "variables": [ "b0: intercept", "b1: slope", "x̄,ȳ: sample means" ], "notes": [ "The intercept may have little practical meaning when x=0 is outside the data range.", "Use the same slope and means from the fitted sample." ], "examples": [ "The fitted line is ŷ=b0+b1x." ], "seo": { "title": "Simple Regression Intercept Formula — Meaning, Variables & LaTeX", "description": "Reference the simple regression intercept, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "The intercept may have little practical meaning when x=0 is outside the data range.", "For the Simple Regression Intercept, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Simple Regression Intercept; map every symbol to its definition first.", "For the Simple Regression Intercept, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "The fitted line is ŷ=b0+b1x.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "coefficient-determination", "name": "Coefficient of Determination", "latex": "R^2=1-\\frac{SS_{res}}{SS_{tot}}", "category": "Statistics", "description": "Measures the fraction of response variation explained by a regression model relative to a constant-mean baseline.", "variables": [ "SS_res: residual sum of squares", "SS_tot: total sum of squares", "R²: coefficient of determination" ], "notes": [ "R² does not establish causation or model adequacy.", "Out-of-sample R² can be negative." ], "examples": [ "Perfect in-sample predictions give SS_res=0 and R²=1." ], "seo": { "title": "Coefficient of Determination Formula — Meaning, Variables & LaTeX", "description": "Reference the coefficient of determination, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "R² does not establish causation or model adequacy.", "For the Coefficient of Determination, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Coefficient of Determination, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Coefficient of Determination, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Coefficient of Determination, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Perfect in-sample predictions give SS_res=0 and R²=1.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "mean-squared-error", "name": "Mean Squared Error", "latex": "MSE=\\frac{1}{n}\\sum_{i=1}^{n}(y_i-\\hat y_i)^2", "category": "Statistics", "description": "Averages squared prediction errors.", "variables": [ "y_i: observed value", "ŷ_i: prediction", "n: observation count" ], "notes": [ "Squaring gives large errors greater weight.", "Training and test MSE answer different questions." ], "examples": [ "Errors 1, −1, and 2 give MSE=(1+1+4)/3=2." ], "seo": { "title": "Mean Squared Error Formula — Meaning, Variables & LaTeX", "description": "Reference the mean squared error, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Squaring gives large errors greater weight.", "For the Mean Squared Error, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Mean Squared Error, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Mean Squared Error, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Mean Squared Error, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Mean Squared Error, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Errors 1, −1, and 2 give MSE=(1+1+4)/3=2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "root-mean-squared-error", "name": "Root Mean Squared Error", "latex": "RMSE=\\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}(y_i-\\hat y_i)^2}", "category": "Statistics", "description": "Returns the square root of mean squared error in the response variable’s units.", "variables": [ "y_i: observed value", "ŷ_i: prediction", "n: count" ], "notes": [ "RMSE is sensitive to large errors.", "Compare models on the same response scale and dataset." ], "examples": [ "If MSE=9, RMSE=3." ], "seo": { "title": "Root Mean Squared Error Formula — Meaning, Variables & LaTeX", "description": "Reference the root mean squared error, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "RMSE is sensitive to large errors.", "For the Root Mean Squared Error, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "For the Root Mean Squared Error, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear." ], "commonMistakes": [ "When copying Root Mean Squared Error, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Root Mean Squared Error, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "If MSE=9, RMSE=3.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "mean-absolute-error", "name": "Mean Absolute Error", "latex": "MAE=\\frac{1}{n}\\sum_{i=1}^{n}|y_i-\\hat y_i|", "category": "Statistics", "description": "Averages absolute prediction errors.", "variables": [ "y_i: observed value", "ŷ_i: prediction", "n: count" ], "notes": [ "MAE is less sensitive to extreme errors than MSE.", "It remains in the original response units." ], "examples": [ "Absolute errors 1,1,2 give MAE=4/3." ], "seo": { "title": "Mean Absolute Error Formula — Meaning, Variables & LaTeX", "description": "Reference the mean absolute error, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "MAE is less sensitive to extreme errors than MSE.", "For the Mean Absolute Error, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Mean Absolute Error, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Mean Absolute Error, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Mean Absolute Error, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Mean Absolute Error, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Absolute errors 1,1,2 give MAE=4/3.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "logistic-regression", "name": "Logistic Regression Probability", "latex": "p(x)=\\frac{1}{1+e^{-(\\beta_0+\\beta^Tx)}}", "category": "Statistics", "description": "Maps a linear predictor to a probability between zero and one.", "variables": [ "β0: intercept", "β: coefficient vector", "x: feature vector", "p: modeled probability" ], "notes": [ "Coefficients act linearly on log-odds.", "Classification requires a separate decision threshold." ], "examples": [ "At linear predictor zero, p=0.5." ], "seo": { "title": "Logistic Regression Probability Formula — Meaning, Variables & LaTeX", "description": "Reference the logistic regression probability, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Coefficients act linearly on log-odds.", "For the Logistic Regression Probability, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Logistic Regression Probability, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Logistic Regression Probability, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Logistic Regression Probability, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "At linear predictor zero, p=0.5.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "softmax-function", "name": "Softmax Function", "latex": "p_i=\\frac{e^{z_i}}{\\sum_j e^{z_j}}", "category": "Statistics", "description": "Converts a vector of real scores into positive values that sum to one.", "variables": [ "z_i: score for class i", "p_i: normalized probability-like value" ], "notes": [ "Subtract max(z) before exponentiating for numerical stability.", "Softmax output depends on all scores simultaneously." ], "examples": [ "Equal scores produce equal probabilities." ], "seo": { "title": "Softmax Function Formula — Meaning, Variables & LaTeX", "description": "Reference the softmax function, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Subtract max(z) before exponentiating for numerical stability.", "For the Softmax Function, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Softmax Function, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Softmax Function, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Softmax Function, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Softmax Function, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Equal scores produce equal probabilities.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "inclusion-exclusion-two", "name": "Two-Set Inclusion–Exclusion Formula", "latex": "|A\\cup B|=|A|+|B|-|A\\cap B|", "category": "Discrete Mathematics", "description": "Counts a union while correcting for elements counted in both sets.", "variables": [ "A, B: finite sets", "|A|: cardinality of A" ], "notes": [ "The intersection term removes double counting." ], "examples": [ "If |A|=12, |B|=9, and |A∩B|=4, then |A∪B|=17." ], "seo": { "title": "Two-Set Inclusion–Exclusion Formula: Variables & Worked Example", "description": "Reference the two-set inclusion–exclusion formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [], "assumptions": [ "The intersection term removes double counting.", "Confirm the finite-set, indexing, ordering, and counting conventions assumed by the expression." ], "commonMistakes": [ "Before substituting values into Two-Set Inclusion–Exclusion Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Two-Set Inclusion–Exclusion Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "If |A|=12, |B|=9, and |A∩B|=4, then |A∪B|=17.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "cauchy-schwarz", "name": "Cauchy–Schwarz Inequality", "latex": "|\\langle x,y\\rangle|^2\\le \\langle x,x\\rangle\\langle y,y\\rangle", "category": "Linear Algebra", "description": "Bounds the absolute inner product by the product of vector norms.", "variables": [ "x, y: vectors in an inner-product space", "⟨x,y⟩: inner product" ], "notes": [ "Equality holds exactly when the vectors are linearly dependent." ], "examples": [ "For x=(1,2) and y=(3,4), 11² ≤ 5·25." ], "seo": { "title": "Cauchy–Schwarz Inequality Formula — Meaning, Variables & LaTeX", "description": "Reference the cauchy–schwarz inequality, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "Equality holds exactly when the vectors are linearly dependent.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Before substituting values into Cauchy–Schwarz Inequality, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For x=(1,2) and y=(3,4), 11² ≤ 5·25.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "triangle-inequality-norm", "name": "Triangle Inequality for Norms", "latex": "\\|x+y\\|\\le \\|x\\|+\\|y\\|", "category": "Linear Algebra", "description": "States that the direct displacement is no longer than a path through an intermediate point.", "variables": [ "x, y: vectors", "||·||: a norm" ], "notes": [ "The statement is part of the definition of a norm." ], "examples": [ "For Euclidean vectors, it matches the geometric triangle inequality." ], "seo": { "title": "Triangle Inequality for Norms Formula — Meaning, Variables & LaTeX", "description": "Reference the triangle inequality for norms, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "The statement is part of the definition of a norm.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Before substituting values into Triangle Inequality for Norms, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "For Euclidean vectors, it matches the geometric triangle inequality.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "lagrange-multipliers", "name": "Single-Constraint Lagrange Condition", "latex": "\\nabla f(x)=\\lambda\\nabla g(x)", "category": "Optimization", "description": "Gives a first-order condition for a constrained extremum with one equality constraint.", "variables": [ "f: objective", "g(x)=c: constraint", "λ: Lagrange multiplier" ], "notes": [ "Constraint qualifications and second-order checks are still required." ], "examples": [ "Optimize area subject to a fixed perimeter by solving the gradient equations." ], "seo": { "title": "Single-Constraint Lagrange Condition: Formula & Worked Example", "description": "Reference the single-constraint lagrange condition, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "gradient-descent", "newton-optimization", "lagrange-multiplier-condition", "kkt-stationarity", "jensen-inequality" ], "assumptions": [ "Constraint qualifications and second-order checks are still required.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Single-Constraint Lagrange Condition, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "Optimize area subject to a fixed perimeter by solving the gradient equations.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "fourier-transform", "name": "Fourier Transform", "latex": "\\hat f(\\omega)=\\int_{-\\infty}^{\\infty}f(t)e^{-i\\omega t}\\,dt", "category": "Analysis", "description": "Transforms a time- or space-domain function into frequency components.", "variables": [ "f(t): input function", "ω: angular frequency", "i: imaginary unit" ], "notes": [ "Normalization conventions differ by field." ], "examples": [ "A Gaussian transforms to another Gaussian under common conventions." ], "seo": { "title": "Fourier Transform: Formula, Worked Example & Checks", "description": "Learn the fourier transform with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "inverse-fourier-transform", "multivariable-taylor" ], "assumptions": [ "Normalization conventions differ by field.", "Integration limits, the differential, and any constant of integration must be included where required.", "Check the domain and the convergence, continuity, differentiability, or measurability conditions used by the result." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Fourier Transform, and include a constant of integration for an indefinite integral.", "Verify the result of Fourier Transform with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Using F(ω)=∫f(t)e^(-iωt)dt, the transform of f(t)=e^(-t²/2) is √(2π)e^(-ω²/2).", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Fourier Transform.", "sourceIds": [ "nist-dlmf" ], "verifiedAgainst": "NIST Digital Library of Mathematical Functions", "domain": "Apply Fourier Transform only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Complete the square in -t²/2-iωt.", "Shift the Gaussian integral; the oscillatory term contributes e^(-ω²/2).", "Use ∫e^(-u²/2)du=√(2π)." ], "verification": "At ω=0, the transform equals the area under the Gaussian, √(2π), as the formula predicts.", "boundaryCases": [ "For Fourier Transform, check zero, negative, and extreme input values before relying on the result.", "When using Fourier Transform, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\hat f(\\omega)=\\int_{-\\infty}^{\\infty}f(t)e^{-i\\omega t}\\,dt for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f(t)=e^(-t²/2), using F(ω)=∫f(t)e^(-iωt)dt", "workedOutput": "F(ω)=√(2π)e^(-ω²/2)", "notApplicableWhen": [ "Do not use Fourier Transform when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "inverse-fourier-transform", "name": "Inverse Fourier Transform", "latex": "f(t)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\\hat f(\\omega)e^{i\\omega t}\\,d\\omega", "category": "Analysis", "description": "Reconstructs a function from its angular-frequency representation.", "variables": [ "f̂(ω): Fourier transform", "t: original-domain variable" ], "notes": [ "The 2π factor depends on the transform convention." ], "examples": [ "Apply the inverse transform to recover a signal after frequency-domain filtering." ], "seo": { "title": "Inverse Fourier Transform Formula — Meaning, Variables & LaTeX", "description": "Reference the inverse fourier transform, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "fourier-transform", "multivariable-taylor" ], "assumptions": [ "The 2π factor depends on the transform convention.", "For the Inverse Fourier Transform, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Integration limits, the differential, and any constant of integration must be included where required.", "Check the domain and the convergence, continuity, differentiability, or measurability conditions used by the result." ], "commonMistakes": [ "When copying Inverse Fourier Transform, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Inverse Fourier Transform with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the inverse transform to recover a signal after frequency-domain filtering.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "laplace-transform", "name": "Laplace Transform", "latex": "F(s)=\\int_0^{\\infty}f(t)e^{-st}\\,dt", "category": "Differential Equations", "description": "Maps a time-domain function to a complex-frequency function useful for initial-value problems.", "variables": [ "f(t): time-domain function", "s: complex frequency" ], "notes": [ "Convergence requires a suitable region in the complex plane." ], "examples": [ "The transform of 1 is 1/s for Re(s)>0." ], "seo": { "title": "Laplace Transform: Formula, Worked Example & Checks", "description": "Learn the laplace transform with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation" ], "assumptions": [ "Convergence requires a suitable region in the complex plane.", "Integration limits, the differential, and any constant of integration must be included where required.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Laplace Transform, and include a constant of integration for an indefinite integral.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "For f(t)=1 and Re(s)>0, F(s)=∫₀∞e^(-st)dt=[-e^(-st)/s]₀∞=1/s.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Laplace Transform.", "sourceIds": [ "nist-dlmf" ], "verifiedAgainst": "NIST Digital Library of Mathematical Functions", "domain": "Apply Laplace Transform only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Require Re(s)>0 so e^(-st) tends to zero.", "Integrate e^(-st) to -e^(-st)/s.", "Evaluate the endpoints to obtain 1/s." ], "verification": "Differentiating the antiderivative and checking the endpoint limit confirms the result.", "boundaryCases": [ "For Laplace Transform, check zero, negative, and extreme input values before relying on the result.", "When using Laplace Transform, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form F(s)=\\int_0^{\\infty}f(t)e^{-st}\\,dt for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f(t)=1, Re(s)>0", "workedOutput": "F(s)=1/s", "notApplicableWhen": [ "Do not use Laplace Transform when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "cross-entropy", "name": "Cross-Entropy", "latex": "H(p,q)=-\\sum_x p(x)\\log q(x)", "category": "Information Theory", "description": "Measures the expected coding cost when q is used for data distributed as p.", "variables": [ "p: target distribution", "q: model distribution" ], "notes": [ "q(x) must be positive wherever p(x)>0." ], "examples": [ "Classification loss often averages cross-entropy across examples." ], "seo": { "title": "Cross-Entropy Formula — Meaning, Variables & LaTeX", "description": "Reference the cross-entropy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "kl-divergence", "mutual-information" ], "assumptions": [ "q(x) must be positive wherever p(x)>0.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Cross-Entropy, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "State the logarithm base, probability model, and whether the reported unit is bits, nats, or another convention." ], "commonMistakes": [ "Do not omit the index or bounds in Cross-Entropy; changing either one changes which terms are included.", "Do not compare entropy or information values computed with different logarithm bases without converting the units." ], "workedExample": "Classification loss often averages cross-entropy across examples.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "kl-divergence", "name": "Kullback–Leibler Divergence", "latex": "D_{\\mathrm{KL}}(p\\|q)=\\sum_x p(x)\\log\\frac{p(x)}{q(x)}", "category": "Information Theory", "description": "Measures directed discrepancy from a reference distribution q to a target distribution p.", "variables": [ "p, q: probability distributions" ], "notes": [ "KL divergence is not symmetric and is not a metric." ], "examples": [ "D_KL(p||q)=0 only when p and q agree almost everywhere." ], "seo": { "title": "Kullback–Leibler Divergence Formula — Meaning, Variables & LaTeX", "description": "Reference the kullback–leibler divergence, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "cross-entropy", "mutual-information" ], "assumptions": [ "KL divergence is not symmetric and is not a metric.", "For the Kullback–Leibler Divergence, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Kullback–Leibler Divergence, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear." ], "commonMistakes": [ "When copying Kullback–Leibler Divergence, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not compare entropy or information values computed with different logarithm bases without converting the units." ], "workedExample": "D_KL(p||q)=0 only when p and q agree almost everywhere.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "mutual-information", "name": "Mutual Information", "latex": "I(X;Y)=\\sum_{x,y}p(x,y)\\log\\frac{p(x,y)}{p(x)p(y)}", "category": "Information Theory", "description": "Measures how much knowing one random variable reduces uncertainty about another.", "variables": [ "p(x,y): joint distribution", "p(x), p(y): marginals" ], "notes": [ "The value is nonnegative and zero for independent variables." ], "examples": [ "Used in feature selection and communication theory." ], "seo": { "title": "Mutual Information Formula — Meaning, Variables & LaTeX", "description": "Reference the mutual information, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "cross-entropy", "kl-divergence" ], "assumptions": [ "The value is nonnegative and zero for independent variables.", "For the Mutual Information, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Mutual Information, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear." ], "commonMistakes": [ "When copying Mutual Information, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not compare entropy or information values computed with different logarithm bases without converting the units." ], "workedExample": "Used in feature selection and communication theory.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "time-dependent-schrodinger", "name": "Time-Dependent Schrödinger Equation", "latex": "i\\hbar\\frac{\\partial}{\\partial t}|\\psi(t)\\rangle=\\hat H|\\psi(t)\\rangle", "category": "Quantum Mechanics", "description": "Describes the time evolution of a quantum state under a Hamiltonian.", "variables": [ "ħ: reduced Planck constant", "|ψ(t)⟩: state vector", "Ĥ: Hamiltonian operator" ], "notes": [ "The Hamiltonian and boundary conditions define the physical system." ], "examples": [ "For a time-independent energy eigenstate, evolution adds a phase factor." ], "seo": { "title": "Time-Dependent Schrödinger Equation: Formula & Example", "description": "Reference the time-dependent schrödinger equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "heisenberg-uncertainty", "commutator", "planck-relation", "uncertainty-principle", "schrodinger-equation" ], "assumptions": [ "The Hamiltonian and boundary conditions define the physical system.", "For the Time-Dependent Schrödinger Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "When copying Time-Dependent Schrödinger Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Time-Dependent Schrödinger Equation with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For a time-independent energy eigenstate, evolution adds a phase factor.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "heisenberg-uncertainty", "name": "Heisenberg Uncertainty Relation", "latex": "\\sigma_x\\sigma_p\\ge\\frac{\\hbar}{2}", "category": "Quantum Mechanics", "description": "Bounds the product of position and momentum standard deviations.", "variables": [ "σx: position standard deviation", "σp: momentum standard deviation" ], "notes": [ "The relation concerns statistical spread, not instrument disturbance alone." ], "examples": [ "Gaussian minimum-uncertainty states can attain equality." ], "seo": { "title": "Heisenberg Uncertainty Relation Formula — Meaning, Variables & LaTeX", "description": "Reference the heisenberg uncertainty relation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "time-dependent-schrodinger", "commutator", "planck-relation", "uncertainty-principle", "schrodinger-equation" ], "assumptions": [ "The relation concerns statistical spread, not instrument disturbance alone.", "For the Heisenberg Uncertainty Relation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "When copying Heisenberg Uncertainty Relation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Heisenberg Uncertainty Relation with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Gaussian minimum-uncertainty states can attain equality.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "commutator", "name": "Operator Commutator", "latex": "[A,B]=AB-BA", "category": "Quantum Mechanics", "description": "Measures the failure of two operators to commute.", "variables": [ "A, B: linear operators" ], "notes": [ "Commutators help determine simultaneous measurability and symmetry structure." ], "examples": [ "For position and momentum, [x,p]=iħ." ], "seo": { "title": "Operator Commutator Formula — Meaning, Variables & LaTeX", "description": "Reference the operator commutator, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "time-dependent-schrodinger", "heisenberg-uncertainty", "planck-relation", "uncertainty-principle", "schrodinger-equation" ], "assumptions": [ "Commutators help determine simultaneous measurability and symmetry structure.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Operator Commutator; perform a dimensional check before accepting the result.", "Verify the result of Operator Commutator with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For position and momentum, [x,p]=iħ.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "einstein-summation", "name": "Einstein Summation Convention", "latex": "a_i b^i=\\sum_i a_i b^i", "category": "Tensor Calculus", "description": "Suppresses an explicit summation sign when an index appears once up and once down.", "variables": [ "i: repeated index", "a_i: covector component", "b^i: vector component" ], "notes": [ "Free indexes must match on both sides of an equation." ], "examples": [ "The contraction a_i b^i is a scalar." ], "seo": { "title": "Einstein Summation Convention Formula — Meaning, Variables & LaTeX", "description": "Reference the einstein summation convention, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "covariant-derivative-vector" ], "assumptions": [ "Free indexes must match on both sides of an equation.", "For the Einstein Summation Convention, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "State the index convention, metric signature, coordinate basis, and differentiability assumptions." ], "commonMistakes": [ "Do not omit the index or bounds in Einstein Summation Convention; changing either one changes which terms are included.", "Verify the result of Einstein Summation Convention with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "The contraction a_i b^i is a scalar.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "covariant-derivative-vector", "name": "Covariant Derivative of a Vector", "latex": "\\nabla_\\mu V^\\nu=\\partial_\\mu V^\\nu+\\Gamma^\\nu_{\\mu\\lambda}V^\\lambda", "category": "Tensor Calculus", "description": "Differentiates vector components while accounting for basis variation.", "variables": [ "Γ: connection coefficients", "Vν: vector components" ], "notes": [ "Sign conventions differ for covectors and curvature tensors." ], "examples": [ "In Cartesian coordinates with zero connection, it reduces to a partial derivative." ], "seo": { "title": "Covariant Derivative of a Vector: Formula & Worked Example", "description": "Reference the covariant derivative of a vector, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "einstein-summation" ], "assumptions": [ "Sign conventions differ for covectors and curvature tensors.", "State the index convention, metric signature, coordinate basis, and differentiability assumptions." ], "commonMistakes": [ "Before substituting values into Covariant Derivative of a Vector, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Covariant Derivative of a Vector with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "In Cartesian coordinates with zero connection, it reduces to a partial derivative.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "divergence-theorem", "name": "Divergence Theorem", "latex": "\\iiint_V \\nabla\\cdot\\mathbf F\\,dV=\\iint_{\\partial V}\\mathbf F\\cdot\\mathbf n\\,dS", "category": "Vector Calculus", "description": "Relates volume divergence to outward flux across the boundary.", "variables": [ "V: volume", "∂V: boundary surface", "n: outward unit normal" ], "notes": [ "The field and region need suitable smoothness." ], "examples": [ "Apply it to convert a closed-surface flux integral to a volume integral." ], "seo": { "title": "Divergence Theorem Formula — Meaning, Variables & LaTeX", "description": "Reference the divergence theorem, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "stokes-theorem", "greens-theorem" ], "assumptions": [ "The field and region need suitable smoothness.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Use a consistent orientation and coordinate system, and confirm the smoothness and boundary assumptions required by the theorem." ], "commonMistakes": [ "Before substituting values into Divergence Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Divergence Theorem with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply it to convert a closed-surface flux integral to a volume integral.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "stokes-theorem", "name": "Stokes’ Theorem", "latex": "\\iint_S(\\nabla\\times\\mathbf F)\\cdot\\mathbf n\\,dS=\\oint_{\\partial S}\\mathbf F\\cdot d\\mathbf r", "category": "Vector Calculus", "description": "Relates surface curl to circulation around the oriented boundary.", "variables": [ "S: oriented surface", "∂S: boundary curve" ], "notes": [ "Boundary orientation follows the right-hand rule." ], "examples": [ "For a planar surface, it specializes to Green’s theorem." ], "seo": { "title": "Stokes’ Theorem Formula — Meaning, Variables & LaTeX", "description": "Reference the stokes’ theorem, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "divergence-theorem", "greens-theorem" ], "assumptions": [ "Boundary orientation follows the right-hand rule.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Use a consistent orientation and coordinate system, and confirm the smoothness and boundary assumptions required by the theorem." ], "commonMistakes": [ "Before substituting values into Stokes’ Theorem, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Stokes’ Theorem with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "For a planar surface, it specializes to Green’s theorem.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "greens-theorem", "name": "Green’s Theorem", "latex": "\\oint_C(P\\,dx+Q\\,dy)=\\iint_D\\left(\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\\right)dA", "category": "Vector Calculus", "description": "Relates a planar line integral to a double integral over its enclosed region.", "variables": [ "C: positively oriented boundary", "D: enclosed region" ], "notes": [ "Assume P and Q have continuous partial derivatives on a suitable region." ], "examples": [ "Use it to compute area with a line integral." ], "seo": { "title": "Green’s Theorem Formula — Meaning, Variables & LaTeX", "description": "Reference the green’s theorem, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "divergence-theorem", "stokes-theorem" ], "assumptions": [ "Assume P and Q have continuous partial derivatives on a suitable region.", "For the Green’s Theorem, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use a consistent orientation and coordinate system, and confirm the smoothness and boundary assumptions required by the theorem." ], "commonMistakes": [ "When copying Green’s Theorem, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Green’s Theorem with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use it to compute area with a line integral.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "covariance-matrix", "name": "Covariance Matrix", "latex": "\\Sigma=\\mathbb E[(X-\\mu)(X-\\mu)^T]", "category": "Statistics", "description": "Collects variances and pairwise covariances of a random vector.", "variables": [ "X: random vector", "μ: mean vector", "Σ: covariance matrix" ], "notes": [ "The matrix is symmetric and positive semidefinite." ], "examples": [ "Diagonal entries are variances; off-diagonal entries are covariances." ], "seo": { "title": "Covariance Matrix Formula — Meaning, Variables & LaTeX", "description": "Reference the covariance matrix, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "The matrix is symmetric and positive semidefinite.", "For the Covariance Matrix, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Covariance Matrix; map every symbol to its definition first.", "For the Covariance Matrix, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Diagonal entries are variances; off-diagonal entries are covariances.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "pca-eigenproblem", "name": "PCA Eigenvalue Problem", "latex": "\\Sigma v_k=\\lambda_k v_k", "category": "Statistics", "description": "Finds principal directions as covariance-matrix eigenvectors.", "variables": [ "Σ: covariance matrix", "vk: principal direction", "λk: explained variance" ], "notes": [ "Center variables before forming the covariance matrix." ], "examples": [ "Order components by decreasing eigenvalue." ], "seo": { "title": "PCA Eigenvalue Problem Formula — Meaning, Variables & LaTeX", "description": "Reference the pca eigenvalue problem, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Center variables before forming the covariance matrix.", "For the PCA Eigenvalue Problem, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in PCA Eigenvalue Problem; map every symbol to its definition first.", "For the PCA Eigenvalue Problem, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Order components by decreasing eigenvalue.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "logistic-sigmoid", "name": "Logistic Sigmoid", "latex": "\\sigma(z)=\\frac{1}{1+e^{-z}}", "category": "Statistics", "description": "Maps a real-valued score to a number between zero and one.", "variables": [ "z: linear predictor", "σ(z): transformed value" ], "notes": [ "A probability interpretation depends on the model and calibration." ], "examples": [ "At z=0, σ(z)=0.5." ], "seo": { "title": "Logistic Sigmoid Formula — Meaning, Variables & LaTeX", "description": "Reference the logistic sigmoid, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "A probability interpretation depends on the model and calibration.", "For the Logistic Sigmoid, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Logistic Sigmoid, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Logistic Sigmoid, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Logistic Sigmoid, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "At z=0, σ(z)=0.5.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "softmax", "name": "Softmax Normalization Formula", "latex": "\\operatorname{softmax}(z)_i=\\frac{e^{z_i}}{\\sum_j e^{z_j}}", "category": "Statistics", "description": "Converts a vector of scores into positive values summing to one.", "variables": [ "zi: score for class i" ], "notes": [ "Subtract max(z) before exponentiation for numerical stability." ], "examples": [ "Used for multiclass model outputs." ], "seo": { "title": "Softmax Normalization Formula Formula — Meaning, Variables & LaTeX", "description": "Reference the softmax normalization formula, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Subtract max(z) before exponentiation for numerical stability.", "For the Softmax Normalization Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Softmax Normalization Formula, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Softmax Normalization Formula, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Softmax Normalization Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Softmax Normalization Formula, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Used for multiclass model outputs.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "haversine-distance", "name": "Haversine Distance", "latex": "d=2R\\arcsin\\sqrt{\\sin^2\\frac{\\Delta\\varphi}{2}+\\cos\\varphi_1\\cos\\varphi_2\\sin^2\\frac{\\Delta\\lambda}{2}}", "category": "Geometry", "description": "Approximates great-circle distance between two latitude–longitude points on a sphere.", "variables": [ "R: sphere radius", "φ: latitude in radians", "λ: longitude in radians" ], "notes": [ "Earth is not a perfect sphere; ellipsoidal methods are more accurate." ], "examples": [ "Use a consistent radius and convert degrees to radians." ], "seo": { "title": "Haversine Distance: Formula, Worked Example & Checks", "description": "Learn the haversine distance with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "status": "published", "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "Earth is not a perfect sphere; ellipsoidal methods are more accurate.", "For the Haversine Distance, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit." ], "commonMistakes": [ "When copying Haversine Distance, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Haversine Distance, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "For two equatorial points separated by 90° longitude, a=sin²(π/4)=1/2 and d=2Rasin(1/√2)=πR/2.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Haversine Distance.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Haversine Distance only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Convert 90° to Δλ=π/2 and set both latitudes to zero.", "Compute the haversine term a=1/2.", "Evaluate 2Rasin(√a)=πR/2." ], "verification": "The points are one quarter of a great circle apart, so the distance must be one quarter of 2πR.", "boundaryCases": [ "For Haversine Distance, check zero, negative, and extreme input values before relying on the result.", "When using Haversine Distance, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form d=2R\\arcsin\\sqrt{\\sin^2\\frac{\\Delta\\varphi}{2}+\\cos\\varphi_1\\cos\\varphi_2\\sin^2\\frac{\\Delta\\lambda}{2}} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "two equatorial points separated by 90° longitude", "workedOutput": "d=πR/2", "notApplicableWhen": [ "Do not use Haversine Distance when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units." ] }, { "id": "multivariable-taylor", "name": "Multivariable Taylor Expansion", "latex": "f(x+h)\\approx f(x)+\\nabla f(x)^Th+\\frac12 h^T H_f(x)h", "category": "Analysis", "description": "Approximates a smooth multivariable function near a reference point.", "variables": [ "h: displacement vector", "Hf: Hessian matrix" ], "notes": [ "Higher-order terms are omitted in the quadratic approximation." ], "examples": [ "Used in optimization to derive Newton-type methods." ], "seo": { "title": "Multivariable Taylor Expansion Formula — Meaning, Variables & LaTeX", "description": "Reference the multivariable taylor expansion, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "fourier-transform", "inverse-fourier-transform" ], "assumptions": [ "Higher-order terms are omitted in the quadratic approximation.", "For the Multivariable Taylor Expansion, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Check the domain and the convergence, continuity, differentiability, or measurability conditions used by the result." ], "commonMistakes": [ "When copying Multivariable Taylor Expansion, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Multivariable Taylor Expansion with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Used in optimization to derive Newton-type methods.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "euler-characteristic-polyhedron", "name": "Euler Characteristic for Convex Polyhedra", "latex": "V-E+F=2", "category": "Topology", "description": "Relates vertices, edges, and faces of a convex polyhedron.", "variables": [ "V: number of vertices", "E: number of edges", "F: number of faces" ], "notes": [ "The constant changes for surfaces with different topology." ], "examples": [ "A cube has 8−12+6=2." ], "seo": { "title": "Euler Characteristic for Convex Polyhedra: Formula & Worked Example", "description": "Reference the euler characteristic for convex polyhedra, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [], "assumptions": [ "The constant changes for surfaces with different topology.", "State the underlying space and the separation, compactness, connectedness, or continuity conditions required." ], "commonMistakes": [ "Before substituting values into Euler Characteristic for Convex Polyhedra, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Euler Characteristic for Convex Polyhedra with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "A cube has 8−12+6=2.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "fisher-information", "name": "Fisher Information", "latex": "I(\\theta)=\\mathbb E\\left[\\left(\\frac{\\partial}{\\partial\\theta}\\log p(X;\\theta)\\right)^2\\right]", "category": "Statistics", "description": "Measures how much an observable random variable tells us about an unknown parameter.", "variables": [ "θ: model parameter", "p(X;θ): likelihood" ], "notes": [ "Regularity conditions are required for equivalent second-derivative forms." ], "examples": [ "Higher Fisher information supports a lower Cramér–Rao variance bound." ], "seo": { "title": "Fisher Information Formula — Meaning, Variables & LaTeX", "description": "Reference the fisher information, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Regularity conditions are required for equivalent second-derivative forms.", "For the Fisher Information, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "For the Fisher Information, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Fisher Information, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Fisher Information, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Higher Fisher information supports a lower Cramér–Rao variance bound.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "cramer-rao-bound", "name": "Cramér–Rao Lower Bound", "latex": "\\operatorname{Var}(\\hat\\theta)\\ge\\frac{1}{I(\\theta)}", "category": "Statistics", "description": "Bounds the variance of an unbiased estimator using Fisher information.", "variables": [ "θ̂: unbiased estimator", "I(θ): Fisher information" ], "notes": [ "Biased estimators require a modified form." ], "examples": [ "An efficient estimator attains the bound under suitable conditions." ], "seo": { "title": "Cramér–Rao Lower Bound Formula — Meaning, Variables & LaTeX", "description": "Reference the cramér–rao lower bound, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "status": "published", "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "Biased estimators require a modified form.", "For the Cramér–Rao Lower Bound, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Cramér–Rao Lower Bound, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "When copying Cramér–Rao Lower Bound, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Cramér–Rao Lower Bound, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "An efficient estimator attains the bound under suitable conditions.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "distance-two-points-3d", "name": "Distance Between Two Points in 3D", "latex": "d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}", "category": "Geometry", "description": "Computes Euclidean distance between two points in three-dimensional space.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the distance between two points in 3d with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Distance Between Two Points in 3D: Formula & Worked Example", "description": "Reference the distance between two points in 3d, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "For the Distance Between Two Points in 3D, verify assumptions and units before applying the expression.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Do not drop the radical boundary in Distance Between Two Points in 3D; verify exactly which terms are inside the root and whether the chosen root is valid.", "For the Distance Between Two Points in 3D, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "Use the distance between two points in 3d with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "arc-length-circle", "name": "Circular Arc Length", "latex": "s=r\\theta", "category": "Geometry", "description": "Computes circular arc length when the angle is measured in radians.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the circular arc length with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Arc Length of a Circle: Formula & Example", "description": "Reference the circular arc length, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "pythagorean-theorem", "circle-area", "circle-circumference", "triangle-area", "sphere-volume" ], "assumptions": [ "For the Circular Arc Length, verify assumptions and units before applying the expression.", "Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula." ], "commonMistakes": [ "Before substituting values into Circular Arc Length, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Circular Arc Length, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step." ], "workedExample": "Use the circular arc length with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "double-angle-sine", "name": "Sine Double-Angle Identity", "latex": "\\sin(2x)=2\\sin x\\cos x", "category": "Trigonometry", "description": "Expresses the sine of twice an angle using sine and cosine.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the sine double-angle identity with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Double-Angle Formula for Sine: Worked Example", "description": "Reference the sine double-angle identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "sine-double-angle", "cosine-double-angle", "pythagorean-trig-identity", "double-angle-cosine", "half-angle-sine" ], "assumptions": [ "For the Sine Double-Angle Identity, verify assumptions and units before applying the expression.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "Do not mix degrees and radians in Sine Double-Angle Identity; use one angle convention throughout the calculation.", "Verify the result of Sine Double-Angle Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the sine double-angle identity with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "double-angle-cosine", "name": "Cosine Double-Angle Identity", "latex": "\\cos(2x)=\\cos^2x-\\sin^2x", "category": "Trigonometry", "description": "Expresses the cosine of twice an angle in squared trigonometric terms.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the cosine double-angle identity with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Double-Angle Formula for Cosine: Worked Example", "description": "Reference the cosine double-angle identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "sine-double-angle", "cosine-double-angle", "pythagorean-trig-identity", "double-angle-sine", "half-angle-sine" ], "assumptions": [ "For the Cosine Double-Angle Identity, verify assumptions and units before applying the expression.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "Do not mix degrees and radians in Cosine Double-Angle Identity; use one angle convention throughout the calculation.", "Verify the result of Cosine Double-Angle Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the cosine double-angle identity with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "half-angle-sine", "name": "Sine Half-Angle Identity", "latex": "\\sin^2\\frac{x}{2}=\\frac{1-\\cos x}{2}", "category": "Trigonometry", "description": "Relates a half-angle sine square to the cosine of the full angle.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the sine half-angle identity with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Sine Half-Angle Identity Formula — Meaning, Variables & LaTeX", "description": "Reference the sine half-angle identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "sine-double-angle", "cosine-double-angle", "pythagorean-trig-identity", "double-angle-sine", "double-angle-cosine" ], "assumptions": [ "For the Sine Half-Angle Identity, verify assumptions and units before applying the expression.", "For the Sine Half-Angle Identity, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The angle unit and the domain or branch of any inverse trigonometric function must be explicit.", "Use a consistent angle unit and verify the quadrant, periodicity, and domain restrictions of inverse functions." ], "commonMistakes": [ "When copying Sine Half-Angle Identity, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Sine Half-Angle Identity with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the sine half-angle identity with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "geometric-sequence-term", "name": "Geometric Sequence Term", "latex": "a_n=a_1r^{n-1}", "category": "Algebra", "description": "Computes the nth term of a geometric sequence.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the geometric sequence term with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Geometric Sequence Term — Formula, Variables, Steps & Example", "description": "Use the geometric sequence term with defined variables, applicability checks, worked steps, verification guidance, common mistakes, and copyable LaTeX." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Geometric Sequence Term, verify assumptions and units before applying the expression.", "For the Geometric Sequence Term, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Geometric Sequence Term, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Geometric Sequence Term, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Worked setup for Geometric Sequence Term: Use the geometric sequence term with a small known example, then verify the result independently. The Geometric Sequence Term calculation should preserve its original grouping and record the final unit or interpretation.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Geometric Sequence Term.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Geometric Sequence Term only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify every input required by Geometric Sequence Term and confirm that each value matches the variable definition and domain.", "Substitute the values into the complete expression a_n=a_1r^{n-1} without dropping parentheses, signs, powers, or branches.", "Evaluate Geometric Sequence Term in a stable order, preserve sufficient precision, and write the result together with its unit or mathematical interpretation.", "Verify the result independently by substitution, dimensional analysis, a limiting case, or an equivalent form appropriate to Algebra." ], "verification": "Check Geometric Sequence Term by substituting the result back into the defining relation when possible, confirming units or dimensions, and testing a simple limiting or known case.", "boundaryCases": [ "For Geometric Sequence Term, check zero, negative, and extreme input values before relying on the result.", "When using Geometric Sequence Term, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a_n=a_1r^{n-1} for copying; rearrange only after preserving equivalence and domain restrictions." ] }, { "id": "finite-geometric-sum", "name": "Finite Geometric Series Sum", "latex": "S_n=a_1\\frac{1-r^n}{1-r}", "category": "Algebra", "description": "Computes a finite geometric series when the ratio is not one.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the finite geometric series sum with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Finite Geometric Series Sum: Formula, Worked Example & Checks", "description": "Learn the Finite Geometric Series Sum, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Finite Geometric Series Sum, verify assumptions and units before applying the expression.", "For the Finite Geometric Series Sum, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Finite Geometric Series Sum, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Finite Geometric Series Sum, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Finite Geometric Series Sum, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For 2+6+18+54, a₁=2, r=3, n=4, so S₄=2(1-3⁴)/(1-3)=80.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Finite Geometric Series Sum.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Finite Geometric Series Sum only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute rⁿ=81.", "Evaluate the numerator 2(1-81)=-160.", "Divide by -2 to obtain 80." ], "verification": "Direct addition 2+6+18+54 also gives 80.", "boundaryCases": [ "For Finite Geometric Series Sum, check zero, negative, and extreme input values before relying on the result.", "When using Finite Geometric Series Sum, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form S_n=a_1\\frac{1-r^n}{1-r} for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a1=2, r=3, n=4", "workedOutput": "S4=80", "notApplicableWhen": [ "Do not use Finite Geometric Series Sum when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "infinite-geometric-sum", "name": "Infinite Geometric Series Sum", "latex": "S=\\frac{a_1}{1-r},\\quad |r|<1", "category": "Algebra", "description": "Computes the sum of a convergent infinite geometric series.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the infinite geometric series sum with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Infinite Geometric Series Sum: Formula, Worked Example & Checks", "description": "Learn the Infinite Geometric Series Sum, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Infinite Geometric Series Sum, verify assumptions and units before applying the expression.", "For the Infinite Geometric Series Sum, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Infinite Geometric Series Sum, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Infinite Geometric Series Sum, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Infinite Geometric Series Sum, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For 3+1.5+0.75+…, a₁=3 and r=1/2, so S=3/(1-1/2)=6.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Infinite Geometric Series Sum.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Infinite Geometric Series Sum only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Verify |r|=1/2<1.", "Compute 1-r=1/2.", "Divide 3 by 1/2 to obtain 6." ], "verification": "Partial sums 3, 4.5, 5.25, 5.625 approach 6 from below.", "boundaryCases": [ "For Infinite Geometric Series Sum, check zero, negative, and extreme input values before relying on the result.", "When using Infinite Geometric Series Sum, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form S=\\frac{a_1}{1-r},\\quad |r|<1 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a1=3, r=1/2", "workedOutput": "S=6", "notApplicableWhen": [ "Do not use Infinite Geometric Series Sum when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "arithmetic-sequence-term", "name": "Arithmetic Sequence Term", "latex": "a_n=a_1+(n-1)d", "category": "Algebra", "description": "Computes the nth term of an arithmetic sequence.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the arithmetic sequence term with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Arithmetic Sequence Term — Formula, Variables, Steps & Example", "description": "Use the arithmetic sequence term with defined variables, applicability checks, worked steps, verification guidance, common mistakes, and copyable LaTeX." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Arithmetic Sequence Term, verify assumptions and units before applying the expression.", "For the Arithmetic Sequence Term, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "Before substituting values into Arithmetic Sequence Term, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "For the Arithmetic Sequence Term, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "Worked setup for Arithmetic Sequence Term: Use the arithmetic sequence term with a small known example, then verify the result independently. The Arithmetic Sequence Term calculation should preserve its original grouping and record the final unit or interpretation.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Sequence Term.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Arithmetic Sequence Term only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify every input required by Arithmetic Sequence Term and confirm that each value matches the variable definition and domain.", "Substitute the values into the complete expression a_n=a_1+(n-1)d without dropping parentheses, signs, powers, or branches.", "Evaluate Arithmetic Sequence Term in a stable order, preserve sufficient precision, and write the result together with its unit or mathematical interpretation.", "Verify the result independently by substitution, dimensional analysis, a limiting case, or an equivalent form appropriate to Algebra." ], "verification": "Check Arithmetic Sequence Term by substituting the result back into the defining relation when possible, confirming units or dimensions, and testing a simple limiting or known case.", "boundaryCases": [ "For Arithmetic Sequence Term, check zero, negative, and extreme input values before relying on the result.", "When using Arithmetic Sequence Term, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form a_n=a_1+(n-1)d for copying; rearrange only after preserving equivalence and domain restrictions." ] }, { "id": "arithmetic-series-sum", "name": "Arithmetic Series Sum", "latex": "S_n=\\frac n2(a_1+a_n)", "category": "Algebra", "description": "Computes the sum of the first n terms of an arithmetic sequence.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the arithmetic series sum with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Arithmetic Series Sum: Formula, Worked Example & Checks", "description": "Learn the Arithmetic Series Sum, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Arithmetic Series Sum, verify assumptions and units before applying the expression.", "For the Arithmetic Series Sum, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Arithmetic Series Sum, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Arithmetic Series Sum, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Arithmetic Series Sum, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "The ten-term sequence 3,7,…,39 has S₁₀=10(3+39)/2=210.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Series Sum.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Arithmetic Series Sum only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify n=10, a₁=3, and aₙ=39.", "Add the first and last terms to obtain 42.", "Multiply by n/2=5 to obtain 210." ], "verification": "Pairing first with last gives five pairs of 42, also totaling 210.", "boundaryCases": [ "For Arithmetic Series Sum, check zero, negative, and extreme input values before relying on the result.", "When using Arithmetic Series Sum, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form S_n=\\frac n2(a_1+a_n) for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "a1=3, an=39, n=10", "workedOutput": "S10=210", "notApplicableWhen": [ "Do not use Arithmetic Series Sum when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "quadratic-vertex", "name": "Quadratic Vertex Formula", "latex": "x_v=-\\frac{b}{2a},\\quad y_v=f(x_v)", "category": "Algebra", "description": "Finds the vertex coordinates of a quadratic function.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the quadratic vertex formula with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Quadratic Vertex Formula: Formula, Worked Example & Checks", "description": "Learn the Quadratic Vertex Formula, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "For the Quadratic Vertex Formula, verify assumptions and units before applying the expression.", "For the Quadratic Vertex Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Quadratic Vertex Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Quadratic Vertex Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Quadratic Vertex Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For f(x)=x²-6x+5, xᵥ=-(-6)/(2·1)=3 and yᵥ=f(3)=-4.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Vertex Formula.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Quadratic Vertex Formula only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Identify a=1 and b=-6.", "Compute xᵥ=3.", "Substitute x=3 into f to obtain -4." ], "verification": "Completing the square gives f(x)=(x-3)²-4, confirming the vertex.", "boundaryCases": [ "For Quadratic Vertex Formula, check zero, negative, and extreme input values before relying on the result.", "When using Quadratic Vertex Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form x_v=-\\frac{b}{2a},\\quad y_v=f(x_v) for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "f(x)=x²-6x+5", "workedOutput": "vertex (3,-4)", "notApplicableWhen": [ "Do not use Quadratic Vertex Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "derivative-power-rule", "name": "Derivative Power Rule", "latex": "\\frac{d}{dx}x^n=nx^{n-1}", "category": "Calculus", "description": "Differentiates a power of x.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the derivative power rule with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Derivative Power Rule: Reference & Example", "description": "Reference the derivative power rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For the Derivative Power Rule, verify assumptions and units before applying the expression.", "For the Derivative Power Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative Power Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Use the derivative power rule with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-product-rule", "name": "Derivative Product Rule", "latex": "(fg)\\prime=f\\prime g+fg\\prime", "category": "Calculus", "description": "Differentiates the product of two differentiable functions.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the derivative product rule with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Derivative Product Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative product rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For the Derivative Product Rule, verify assumptions and units before applying the expression.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "Before substituting values into Derivative Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Use the derivative product rule with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-quotient-rule", "name": "Derivative Quotient Rule", "latex": "\\left(\\frac fg\\right)\\prime=\\frac{f\\prime g-fg\\prime}{g^2}", "category": "Calculus", "description": "Differentiates the quotient of two functions.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the derivative quotient rule with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Derivative Quotient Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative quotient rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For the Derivative Quotient Rule, verify assumptions and units before applying the expression.", "For the Derivative Quotient Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative Quotient Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Use the derivative quotient rule with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "derivative-chain-rule", "name": "Derivative Chain Rule", "latex": "\\frac{d}{dx}f(g(x))=f\\prime(g(x))g\\prime(x)", "category": "Calculus", "description": "Differentiates a composite function.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the derivative chain rule with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Derivative Chain Rule Formula — Meaning, Variables & LaTeX", "description": "Reference the derivative chain rule, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "derivative-definition", "product-rule", "chain-rule", "fundamental-theorem-calculus", "integration-by-parts" ], "assumptions": [ "For the Derivative Chain Rule, verify assumptions and units before applying the expression.", "For the Derivative Chain Rule, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used." ], "commonMistakes": [ "When copying Derivative Chain Rule, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks." ], "workedExample": "Use the derivative chain rule with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-calculus-volume-1" ] }, { "id": "newton-method", "name": "Newton Method Iteration", "latex": "x_{n+1}=x_n-\\frac{f(x_n)}{f\\prime(x_n)}", "category": "Numerical Methods", "description": "Iteratively approximates a root of a differentiable function.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the newton method iteration with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Newton Method Iteration Formula — Meaning, Variables & LaTeX", "description": "Reference the newton method iteration, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule" ], "assumptions": [ "For the Newton Method Iteration, verify assumptions and units before applying the expression.", "For the Newton Method Iteration, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Newton Method Iteration, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "Use the newton method iteration with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "total-probability", "name": "Law of Total Probability", "latex": "P(B)=\\sum_iP(B\\mid A_i)P(A_i)", "category": "Probability", "description": "Computes an event probability over a partition of cases.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the law of total probability with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Total Probability Rule: Formula & Example", "description": "Reference the law of total probability, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "For the Law of Total Probability, verify assumptions and units before applying the expression.", "For the Law of Total Probability, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not omit the index or bounds in Law of Total Probability; changing either one changes which terms are included.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Use the law of total probability with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "expected-value-discrete", "name": "Discrete Expected Value", "latex": "\\mathbb E[X]=\\sum_x xP(X=x)", "category": "Probability", "description": "Computes the probability-weighted average of a discrete random variable.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the discrete expected value with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Discrete Expected Value Formula — Meaning, Variables & LaTeX", "description": "Reference the discrete expected value, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "For the Discrete Expected Value, verify assumptions and units before applying the expression.", "For the Discrete Expected Value, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not omit the index or bounds in Discrete Expected Value; changing either one changes which terms are included.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Use the discrete expected value with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "variance-identity", "name": "Variance Identity", "latex": "\\operatorname{Var}(X)=\\mathbb E[X^2]-\\mathbb E[X]^2", "category": "Probability", "description": "Expresses variance using first and second moments.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the variance identity with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Variance Identity Formula — Meaning, Variables & LaTeX", "description": "Reference the variance identity, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "For the Variance Identity, verify assumptions and units before applying the expression.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Variance Identity; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Use the variance identity with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "matrix-inverse-2x2", "name": "Inverse of a 2×2 Matrix", "latex": "A^{-1}=\\frac1{ad-bc}\\begin{bmatrix}d&-b\\\\-c&a\\end{bmatrix}", "category": "Linear Algebra", "description": "Computes the inverse of a nonsingular two-by-two matrix.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the inverse of a 2×2 matrix with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Inverse of a 2×2 Matrix Formula — Meaning, Variables & LaTeX", "description": "Reference the inverse of a 2×2 matrix, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "For the Inverse of a 2×2 Matrix, verify assumptions and units before applying the expression.", "For the Inverse of a 2×2 Matrix, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "When copying Inverse of a 2×2 Matrix, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Use the inverse of a 2×2 matrix with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "projection-vector", "name": "Vector Projection", "latex": "\\operatorname{proj}_{u}v=\\frac{v\\cdot u}{u\\cdot u}u", "category": "Linear Algebra", "description": "Projects one vector onto the direction of another.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the vector projection with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Projection of One Vector onto Another: Formula", "description": "Reference the vector projection, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "For the Vector Projection, verify assumptions and units before applying the expression.", "For the Vector Projection, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "When copying Vector Projection, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Use the vector projection with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "least-squares-normal", "name": "Least-Squares Normal Equations", "latex": "X^{\\mathsf T}X\\hat\\beta=X^{\\mathsf T}y", "category": "Linear Algebra", "description": "Characterizes ordinary least-squares coefficients through normal equations.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the least-squares normal equations with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Least-Squares Normal Equations Formula — Meaning, Variables & LaTeX", "description": "Reference the least-squares normal equations, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "For the Least-Squares Normal Equations, verify assumptions and units before applying the expression.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Before substituting values into Least-Squares Normal Equations, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Use the least-squares normal equations with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "determinant-eigenvalues", "name": "Determinant as Eigenvalue Product", "latex": "\\det(A)=\\prod_{i=1}^{n}\\lambda_i", "category": "Linear Algebra", "description": "Relates a square matrix determinant to its eigenvalues with multiplicity.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the determinant as eigenvalue product with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Determinant as Eigenvalue Product: Formula & Worked Example", "description": "Reference the determinant as eigenvalue product, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "dot-product", "two-by-two-determinant", "matrix-multiplication", "three-by-three-determinant", "inverse-two-by-two" ], "assumptions": [ "For the Determinant as Eigenvalue Product, verify assumptions and units before applying the expression.", "For the Determinant as Eigenvalue Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.", "Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold." ], "commonMistakes": [ "Do not omit the index or bounds in Determinant as Eigenvalue Product; changing either one changes which terms are included.", "Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established." ], "workedExample": "Use the determinant as eigenvalue product with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "kinetic-energy-classical", "name": "Classical Kinetic Energy", "latex": "K=\\frac12mv^2", "category": "Physics", "description": "Computes translational kinetic energy in classical mechanics.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the classical kinetic energy with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Classical Kinetic Energy Formula — Meaning, Variables & LaTeX", "description": "Reference the classical kinetic energy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the Classical Kinetic Energy, verify assumptions and units before applying the expression.", "For the Classical Kinetic Energy, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Classical Kinetic Energy, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Classical Kinetic Energy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the classical kinetic energy with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "continuous-compounding", "name": "Continuous Compounding", "latex": "A=Pe^{rt}", "category": "Financial Mathematics", "description": "Computes accumulated value under continuous compounding.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the continuous compounding with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Continuous Compounding Formula — Meaning, Variables & LaTeX", "description": "Reference the continuous compounding, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "simple-interest", "present-value", "future-value-annuity", "present-value-annuity", "loan-payment" ], "assumptions": [ "For the Continuous Compounding, verify assumptions and units before applying the expression.", "Match the interest rate to the compounding period, convert percentages to decimals, and keep cash-flow timing consistent." ], "commonMistakes": [ "Do not mix annual, monthly, nominal, and effective rates in Continuous Compounding; convert the rate and number of periods to the same time basis.", "Verify the result of Continuous Compounding with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the continuous compounding with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-algebra-trigonometry" ] }, { "id": "coulomb-law", "name": "Coulomb Law", "latex": "F=k\\frac{|q_1q_2|}{r^2}", "category": "Physics", "description": "Computes electrostatic force magnitude between point charges.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the coulomb law with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Coulomb Law Formula — Meaning, Variables & LaTeX", "description": "Reference the coulomb law, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the Coulomb Law, verify assumptions and units before applying the expression.", "For the Coulomb Law, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Coulomb Law, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Coulomb Law with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the coulomb law with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "capacitor-energy", "name": "Energy Stored in a Capacitor", "latex": "U=\\frac12CV^2", "category": "Physics", "description": "Computes energy stored in an ideal capacitor.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the energy stored in a capacitor with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Energy Stored in a Capacitor Formula — Meaning, Variables & LaTeX", "description": "Reference the energy stored in a capacitor, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the Energy Stored in a Capacitor, verify assumptions and units before applying the expression.", "For the Energy Stored in a Capacitor, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying Energy Stored in a Capacitor, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Energy Stored in a Capacitor with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the energy stored in a capacitor with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "photon-energy", "name": "Photon Energy", "latex": "E=hf", "category": "Physics", "description": "Relates photon energy to frequency through Planck constant.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the photon energy with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Photon Energy Formula — Meaning, Variables & LaTeX", "description": "Reference the photon energy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the Photon Energy, verify assumptions and units before applying the expression.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Photon Energy; perform a dimensional check before accepting the result.", "Verify the result of Photon Energy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the photon energy with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "de-broglie-wavelength", "name": "de Broglie Wavelength", "latex": "\\lambda=\\frac hp", "category": "Physics", "description": "Relates particle wavelength to momentum.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the de broglie wavelength with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "de Broglie Wavelength Formula — Meaning, Variables & LaTeX", "description": "Reference the de broglie wavelength, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the de Broglie Wavelength, verify assumptions and units before applying the expression.", "For the de Broglie Wavelength, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "When copying de Broglie Wavelength, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of de Broglie Wavelength with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the de broglie wavelength with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "boltzmann-entropy", "name": "Boltzmann Entropy", "latex": "S=k_B\\ln\\Omega", "category": "Physics", "description": "Relates thermodynamic entropy to multiplicity.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the boltzmann entropy with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Boltzmann Entropy Formula — Meaning, Variables & LaTeX", "description": "Reference the boltzmann entropy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the Boltzmann Entropy, verify assumptions and units before applying the expression.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Boltzmann Entropy; perform a dimensional check before accepting the result.", "Verify the result of Boltzmann Entropy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the boltzmann entropy with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "first-law-thermodynamics", "name": "First Law of Thermodynamics", "latex": "\\Delta U=Q-W", "category": "Physics", "description": "Relates internal-energy change, heat, and work under a stated sign convention.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the first law of thermodynamics with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "First Law of Thermodynamics Formula — Meaning, Variables & LaTeX", "description": "Reference the first law of thermodynamics, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "newton-second-law", "mass-energy-equivalence", "kinetic-energy", "gravitational-potential-energy", "momentum-formula" ], "assumptions": [ "For the First Law of Thermodynamics, verify assumptions and units before applying the expression.", "Use dimensionally consistent units and the sign, coordinate, and reference-frame conventions stated for the problem." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in First Law of Thermodynamics; perform a dimensional check before accepting the result.", "Verify the result of First Law of Thermodynamics with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the first law of thermodynamics with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-college-physics" ] }, { "id": "exponential-decay", "name": "Exponential Decay", "latex": "N(t)=N_0e^{-\\lambda t}", "category": "Differential Equations", "description": "Models decay at a constant proportional rate.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the exponential decay with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Exponential Decay Formula — Meaning, Variables & LaTeX", "description": "Reference the exponential decay, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation" ], "assumptions": [ "For the Exponential Decay, verify assumptions and units before applying the expression.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "Before substituting values into Exponential Decay, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "Use the exponential decay with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "logistic-solution", "name": "Logistic Growth Solution", "latex": "N(t)=\\frac{K}{1+Ae^{-rt}}", "category": "Differential Equations", "description": "Describes bounded growth toward a carrying capacity.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the logistic growth solution with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Logistic Growth Solution Formula — Meaning, Variables & LaTeX", "description": "Reference the logistic growth solution, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "first-order-linear-ode", "exponential-decay-ode", "logistic-differential-equation", "harmonic-oscillator-ode", "heat-equation" ], "assumptions": [ "For the Logistic Growth Solution, verify assumptions and units before applying the expression.", "For the Logistic Growth Solution, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method." ], "commonMistakes": [ "When copying Logistic Growth Solution, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not present a general solution as the requested solution before applying all initial or boundary conditions." ], "workedExample": "Use the logistic growth solution with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "binomial-pmf", "name": "Binomial Probability Mass Function", "latex": "P(X=k)=\\binom nkp^k(1-p)^{n-k}", "category": "Probability", "description": "Computes exact successes in independent Bernoulli trials.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the binomial probability mass function with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Binomial Probability Mass Function: Formula & Worked Example", "description": "Reference the binomial probability mass function, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "bayes-theorem", "conditional-probability", "binomial-probability", "permutation-formula", "combination-formula" ], "assumptions": [ "For the Binomial Probability Mass Function, verify assumptions and units before applying the expression.", "The event model, conditioning information, independence assumptions, and probability range must match the problem." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Binomial Probability Mass Function; map every symbol to its definition first.", "Do not assume events are independent or mutually exclusive unless the problem states or proves that condition." ], "workedExample": "Use the binomial probability mass function with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "covariance-definition", "name": "Covariance Definition", "latex": "\\operatorname{Cov}(X,Y)=\\mathbb E[(X-\\mu_X)(Y-\\mu_Y)]", "category": "Statistics", "description": "Measures joint linear variation of two random variables.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the covariance definition with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Covariance Definition Formula — Meaning, Variables & LaTeX", "description": "Reference the covariance definition, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "For the Covariance Definition, verify assumptions and units before applying the expression.", "For the Covariance Definition, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions." ], "commonMistakes": [ "Do not substitute sample and population quantities interchangeably in Covariance Definition; map every symbol to its definition first.", "For the Covariance Definition, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Use the covariance definition with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "mean-confidence-interval", "name": "Mean Confidence Interval", "latex": "\\bar{x}\\pm t_{\\alpha/2,n-1}\\frac{s}{\\sqrt n}", "category": "Statistics", "description": "Gives a t-based confidence interval for a population mean.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the mean confidence interval with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Confidence Interval for a Mean: Formula & Example", "description": "Reference the mean confidence interval, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "sample-mean", "sample-variance", "z-score", "standard-error-mean", "population-variance" ], "assumptions": [ "For the Mean Confidence Interval, verify assumptions and units before applying the expression.", "For the Mean Confidence Interval, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible." ], "commonMistakes": [ "Do not collapse the ± in Mean Confidence Interval into one value; calculate each admissible branch and check it in the original problem.", "For the Mean Confidence Interval, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions." ], "workedExample": "Use the mean confidence interval with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "openstax-intro-statistics" ] }, { "id": "convolution-definition", "name": "Convolution Definition", "latex": "(f*g)(t)=\\int_{-\\infty}^{\\infty}f(\\tau)g(t-\\tau)\\,d\\tau", "category": "Transforms", "description": "Combines functions through shifted overlap integration.", "variables": [ "Define every symbol and unit before substitution.", "Check the domain, shape, and convention required by the formula." ], "notes": [ "Verify assumptions and units before applying the expression.", "Keep exact values until the final rounding step when possible." ], "examples": [ "Use the convolution definition with a small known example, then verify the result independently." ], "status": "published", "seo": { "title": "Convolution Definition Formula — Meaning, Variables & LaTeX", "description": "Reference the convolution definition, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [], "assumptions": [ "For the Convolution Definition, verify assumptions and units before applying the expression.", "Integration limits, the differential, and any constant of integration must be included where required.", "State the transform convention, normalization, sign, domain, and convergence conditions." ], "commonMistakes": [ "Keep the integration bounds and differential attached to Convolution Definition, and include a constant of integration for an indefinite integral.", "Verify the result of Convolution Definition with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Use the convolution definition with a small known example, then verify the result independently.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "sum-first-integers", "name": "Sum of the First Integers", "latex": "\\sum_{k=1}^{n}k=\\frac{n(n+1)}2", "category": "Algebra", "description": "Computes the sum of the first n positive integers.", "variables": [ "Define all variables, domains, and units before substitution.", "result: the quantity produced after the stated inputs and conditions are applied" ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the sum of the first integers to a small known case and verify the result." ], "status": "published", "seo": { "title": "Sum of the First Integers: Formula, Worked Example & Checks", "description": "Learn the Sum of the First Integers, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Sum of the First Integers, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First Integers, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sum of the First Integers, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Sum of the First Integers, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sum of the First Integers, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For n=10, 10·11/2=55.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sum of the First Integers.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Sum of the First Integers only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute n+1=11.", "Multiply 10×11=110.", "Divide by 2 to obtain 55." ], "verification": "Direct addition 1+2+…+10 gives 55.", "boundaryCases": [ "For Sum of the First Integers, check zero, negative, and extreme input values before relying on the result.", "When using Sum of the First Integers, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\sum_{k=1}^{n}k=\\frac{n(n+1)}2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "n=10", "workedOutput": "55", "notApplicableWhen": [ "Do not use Sum of the First Integers when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "sum-first-squares", "name": "Sum of the First Squares", "latex": "\\sum_{k=1}^{n}k^2=\\frac{n(n+1)(2n+1)}6", "category": "Algebra", "description": "Computes the sum of the first n square numbers.", "variables": [ "Define all variables, domains, and units before substitution.", "result: the quantity produced after the stated inputs and conditions are applied" ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the sum of the first squares to a small known case and verify the result." ], "status": "published", "seo": { "title": "Sum of the First Squares: Formula, Worked Example & Checks", "description": "Learn the Sum of the First Squares, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Sum of the First Squares, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First Squares, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sum of the First Squares, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Sum of the First Squares, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sum of the First Squares, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For n=5, 5·6·11/6=55.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sum of the First Squares.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Sum of the First Squares only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute n+1=6 and 2n+1=11.", "Multiply 5×6×11=330.", "Divide by 6 to obtain 55." ], "verification": "Direct addition 1+4+9+16+25 gives 55.", "boundaryCases": [ "For Sum of the First Squares, check zero, negative, and extreme input values before relying on the result.", "When using Sum of the First Squares, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\sum_{k=1}^{n}k^2=\\frac{n(n+1)(2n+1)}6 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "n=5", "workedOutput": "55", "notApplicableWhen": [ "Do not use Sum of the First Squares when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "sum-first-cubes", "name": "Sum of the First Cubes", "latex": "\\sum_{k=1}^{n}k^3=\\left[\\frac{n(n+1)}2\\right]^2", "category": "Algebra", "description": "Computes the sum of the first n cube numbers.", "variables": [ "Define all variables, domains, and units before substitution.", "result: the quantity produced after the stated inputs and conditions are applied" ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the sum of the first cubes to a small known case and verify the result." ], "status": "published", "seo": { "title": "Sum of the First Cubes: Formula, Worked Example & Checks", "description": "Learn the Sum of the First Cubes, define its variables and assumptions, follow a worked input-to-output calculation, verify the result, and copy the LaTeX form." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Sum of the First Cubes, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Sum of the First Cubes, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Sum of the First Cubes, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Sum of the First Cubes, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Sum of the First Cubes, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "For n=4, [4·5/2]²=10²=100.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Sum of the First Cubes.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Sum of the First Cubes only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Compute n(n+1)/2=4·5/2=10.", "Square the triangular number.", "Obtain 100." ], "verification": "Direct addition 1+8+27+64 gives 100.", "boundaryCases": [ "For Sum of the First Cubes, check zero, negative, and extreme input values before relying on the result.", "When using Sum of the First Cubes, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form \\sum_{k=1}^{n}k^3=\\left[\\frac{n(n+1)}2\\right]^2 for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "n=4", "workedOutput": "100", "notApplicableWhen": [ "Do not use Sum of the First Cubes when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "harmonic-number", "name": "Harmonic Number", "latex": "H_n=\\sum_{k=1}^{n}\\frac1k", "category": "Algebra", "description": "Defines the nth harmonic number.", "variables": [ "Define all variables, domains, and units before substitution.", "result: the quantity produced after the stated inputs and conditions are applied" ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the harmonic number to a small known case and verify the result." ], "status": "published", "seo": { "title": "Harmonic Number: Formula, Worked Example & Checks", "description": "Learn the harmonic number with variable definitions, applicability checks, a complete numeric input, step-by-step calculation, verified output, LaTeX, and common mistakes." }, "relatedFormulas": [ "quadratic-formula", "slope-intercept-form", "distance-formula", "arithmetic-sequence", "geometric-sequence" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Harmonic Number, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Harmonic Number, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "For the Harmonic Number, all variables must lie in the stated domain, and every denominator or inverse operation must be defined." ], "commonMistakes": [ "When copying Harmonic Number, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "For the Harmonic Number, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted." ], "workedExample": "H₄=1+1/2+1/3+1/4=25/12≈2.0833.", "reviewedAt": "2026-07-26", "sourceSet": "standard-math-reference", "qualityTier": "core", "reviewMethod": "Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Harmonic Number.", "sourceIds": [ "openstax-algebra-trigonometry" ], "verifiedAgainst": "OpenStax Algebra and Trigonometry", "domain": "Apply Harmonic Number only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.", "workedSteps": [ "Use the common denominator 12.", "Add 12/12+6/12+4/12+3/12.", "Simplify to 25/12 and convert to a decimal if needed." ], "verification": "The value lies between 2 and 2.25, consistent with the four positive terms.", "boundaryCases": [ "For Harmonic Number, check zero, negative, and extreme input values before relying on the result.", "When using Harmonic Number, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values." ], "alternativeForms": [ "Keep the canonical LaTeX form H_n=\\sum_{k=1}^{n}\\frac1k for copying; rearrange only after preserving equivalence and domain restrictions." ], "workedInput": "n=4", "workedOutput": "H4=25/12≈2.0833", "notApplicableWhen": [ "Do not use Harmonic Number when its variable definitions, domain restrictions, or structural assumptions differ from the problem." ] }, { "id": "euler-totient-product", "name": "Euler Totient Product", "latex": "\\varphi(n)=n\\prod_{p\\mid n}\\left(1-\\frac1p\\right)", "category": "Number Theory", "description": "Computes Euler totient from the distinct prime divisors of n.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the euler totient product to a small known case and verify the result." ], "status": "published", "seo": { "title": "Euler Totient Product Formula — Meaning, Variables & LaTeX", "description": "Reference the euler totient product, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "euclidean-algorithm", "euler-totient", "chinese-remainder" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Euler Totient Product, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "For the Euler Totient Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Apply the formula over the intended integer domain and check divisibility, coprimality, parity, or modulus conditions." ], "commonMistakes": [ "When copying Euler Totient Product, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Euler Totient Product with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the euler totient product to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "chinese-remainder", "name": "Chinese Remainder Reconstruction", "latex": "x\\equiv\\sum_i a_iM_iN_i\\pmod M", "category": "Number Theory", "description": "Reconstructs a simultaneous congruence solution under coprime moduli.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the chinese remainder reconstruction to a small known case and verify the result." ], "status": "published", "seo": { "title": "Chinese Remainder Reconstruction: Formula & Worked Example", "description": "Reference the chinese remainder reconstruction, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "euclidean-algorithm", "euler-totient", "euler-totient-product" ], "assumptions": [ "Verify assumptions and conventions before use.", "The plus-minus sign represents separate branches; evaluate and verify both when both are admissible.", "For the Chinese Remainder Reconstruction, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.", "Apply the formula over the intended integer domain and check divisibility, coprimality, parity, or modulus conditions." ], "commonMistakes": [ "Do not collapse the ± in Chinese Remainder Reconstruction into one value; calculate each admissible branch and check it in the original problem.", "Verify the result of Chinese Remainder Reconstruction with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the chinese remainder reconstruction to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "planck-relation", "name": "Planck Relation", "latex": "E=\\hbar\\omega", "category": "Quantum Mechanics", "description": "Relates energy to angular frequency.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the planck relation to a small known case and verify the result." ], "status": "published", "seo": { "title": "Planck Relation Formula — Meaning, Variables & LaTeX", "description": "Reference the planck relation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "time-dependent-schrodinger", "heisenberg-uncertainty", "commutator", "uncertainty-principle", "schrodinger-equation" ], "assumptions": [ "Verify assumptions and conventions before use.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "Do not combine values with inconsistent units or sign conventions in Planck Relation; perform a dimensional check before accepting the result.", "Verify the result of Planck Relation with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the planck relation to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "uncertainty-principle", "name": "Heisenberg Uncertainty Principle", "latex": "\\sigma_x\\sigma_p\\ge\\frac{\\hbar}{2}", "category": "Quantum Mechanics", "description": "Bounds simultaneous position and momentum uncertainty.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the heisenberg uncertainty principle to a small known case and verify the result." ], "status": "published", "seo": { "title": "Heisenberg Uncertainty Principle: Formula & Worked Example", "description": "Reference the heisenberg uncertainty principle, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "time-dependent-schrodinger", "heisenberg-uncertainty", "commutator", "planck-relation", "schrodinger-equation" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Heisenberg Uncertainty Principle, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "When copying Heisenberg Uncertainty Principle, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Heisenberg Uncertainty Principle with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the heisenberg uncertainty principle to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "schrodinger-equation", "name": "Time-Dependent Schrödinger Equation", "latex": "i\\hbar\\frac{\\partial}{\\partial t}\\Psi=\\hat H\\Psi", "category": "Quantum Mechanics", "description": "Describes unitary time evolution of a quantum state.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the time-dependent schrödinger equation to a small known case and verify the result." ], "status": "published", "seo": { "title": "Schrödinger Equation: Time-Dependent Form & Example", "description": "Reference the time-dependent schrödinger equation, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "time-dependent-schrodinger", "heisenberg-uncertainty", "commutator", "planck-relation", "uncertainty-principle" ], "assumptions": [ "Verify assumptions and conventions before use.", "For the Time-Dependent Schrödinger Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "State the Hilbert-space, normalization, operator, basis, and unit conventions used in the calculation." ], "commonMistakes": [ "When copying Time-Dependent Schrödinger Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Verify the result of Time-Dependent Schrödinger Equation with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the time-dependent schrödinger equation to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "binary-cross-entropy", "name": "Binary Cross Entropy", "latex": "L=-[y\\log p+(1-y)\\log(1-p)]", "category": "Machine Learning", "description": "Measures probabilistic classification loss for a binary target.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the binary cross entropy to a small known case and verify the result." ], "status": "published", "seo": { "title": "Binary Cross Entropy Formula — Meaning, Variables & LaTeX", "description": "Reference the binary cross entropy, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [], "assumptions": [ "Verify assumptions and conventions before use.", "Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.", "State the data, loss, regularization, normalization, and optimization assumptions used by the model." ], "commonMistakes": [ "Before substituting values into Binary Cross Entropy, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Verify the result of Binary Cross Entropy with a known case, inverse operation, dimensional check, or independent calculation before publishing it." ], "workedExample": "Apply the binary cross entropy to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "gradient-descent-update", "name": "Gradient Descent Update", "latex": "\\theta_{t+1}=\\theta_t-\\eta\\nabla J(\\theta_t)", "category": "Optimization", "description": "Updates parameters in the negative gradient direction.", "variables": [ "Define all variables, domains, and units before substitution." ], "notes": [ "Verify assumptions and conventions before use." ], "examples": [ "Apply the gradient descent update to a small known case and verify the result." ], "status": "published", "seo": { "title": "Gradient Descent Step Rule: Formula & Example", "description": "Reference the gradient descent update, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "gradient-descent", "newton-optimization", "lagrange-multiplier-condition", "kkt-stationarity", "jensen-inequality" ], "assumptions": [ "Verify assumptions and conventions before use.", "Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result." ], "commonMistakes": [ "Before substituting values into Gradient Descent Update, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.", "Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions." ], "workedExample": "Apply the gradient descent update to a small known case and verify the result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "forward-difference", "name": "Forward Difference Derivative", "latex": "f\\prime(x)\\approx\\frac{f(x+h)-f(x)}{h}", "category": "Numerical Methods", "description": "Approximates a derivative with a first-order forward difference.", "variables": [ "Define all variables and the step size before use." ], "notes": [ "Check convergence and truncation assumptions." ], "examples": [ "Apply the forward difference derivative to a smooth test function and compare with a known result." ], "status": "published", "seo": { "title": "Forward Difference Derivative Formula — Meaning, Variables & LaTeX", "description": "Reference the forward difference derivative, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule" ], "assumptions": [ "Check convergence and truncation assumptions.", "For the Forward Difference Derivative, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Forward Difference Derivative, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "Apply the forward difference derivative to a smooth test function and compare with a known result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] }, { "id": "central-difference", "name": "Central Difference Derivative", "latex": "f\\prime(x)\\approx\\frac{f(x+h)-f(x-h)}{2h}", "category": "Numerical Methods", "description": "Approximates a derivative with a symmetric second-order difference.", "variables": [ "Define all variables and the step size before use." ], "notes": [ "Check convergence and truncation assumptions." ], "examples": [ "Apply the central difference derivative to a smooth test function and compare with a known result." ], "status": "published", "seo": { "title": "Central Difference Derivative Formula — Meaning, Variables & LaTeX", "description": "Reference the central difference derivative, its variables, notation, assumptions, common mistakes, and copyable LaTeX expression." }, "relatedFormulas": [ "euler-method", "runge-kutta-four", "newton-raphson", "trapezoidal-rule", "simpson-rule" ], "assumptions": [ "Check convergence and truncation assumptions.", "For the Central Difference Derivative, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.", "The iteration, step size, tolerance, and stopping rule must be suitable for the problem and the method must converge in the chosen region." ], "commonMistakes": [ "When copying Central Difference Derivative, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.", "Do not report a numerical approximation without a tolerance, convergence check, and an estimate of rounding or truncation error." ], "workedExample": "Apply the central difference derivative to a smooth test function and compare with a known result.", "reviewedAt": "2026-07-23", "sourceSet": "standard-math-reference", "qualityTier": "supporting", "sourceIds": [ "nist-dlmf" ] } ]