LaTeX
\oint_C(P\,dx+Q\,dy)=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dAVariables
- C: positively oriented boundary
- D: enclosed region
How to use this formula
Relates a planar line integral to a double integral over its enclosed region.
Important notes
- Assume P and Q have continuous partial derivatives on a suitable region.
Quick example
Use it to compute area with a line integral.
Applicability, worked calculation, and verification
Assumptions and domain checks
- 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.
Worked example
Use it to compute area with a line integral.
Common mistakes
- 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.
Continue the workflow
Use Green’s Theorem in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\oint_C(P\,dx+Q\,dy)=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Green’s Theorem used for?
Relates a planar line integral to a double integral over its enclosed region.
Can I copy this formula as LaTeX?
Yes. Copy \oint_C(P\,dx+Q\,dy)=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.