LaTeX
\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf rVariables
- S: oriented surface
- ∂S: boundary curve
How to use this formula
Relates surface curl to circulation around the oriented boundary.
Important notes
- Boundary orientation follows the right-hand rule.
Quick example
For a planar surface, it specializes to Green’s theorem.
Applicability, worked calculation, and verification
Assumptions and domain checks
- 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.
Worked example
For a planar surface, it specializes to Green’s theorem.
Common mistakes
- 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.
Continue the workflow
Use Stokes’ 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
\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf r. - 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 Stokes’ Theorem used for?
Relates surface curl to circulation around the oriented boundary.
Can I copy this formula as LaTeX?
Yes. Copy \iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf r 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.