Real functions Functions of several real variables Theorem

Stokes' theorem

Builds on Curl Line integral of a vector field Flux through a surface Orientable surface Parametrised surface Green's theorem Generalised Stokes theorem

Statement

Let $D \subseteq \mathbb{R}^2$ be the compact region bounded by a piecewise smooth simple closed curve $\gamma$ , oriented counterclockwise. Let $r$ be an injective regular parametrised surface of class $C^2$ on an open set that contains $D$ . Let $S = r(D)$ have the unit normal $n = (r_u \times r_v) / \| r_u \times r_v \|$ , and let $\partial S$ be the closed curve $r \circ \gamma$ . If $F$ is a vector field of class $C^1$ on an open set that contains $S$ , then $\oint_{\partial S} F \cdot d\mathbf{r} = \iint_S (\nabla \times F) \cdot n \, dS.$

Remarks

The left side is the circulation of $F$ around the boundary. The right side is the flux of the curl of $F$ through $S$ . The orientations match by the right-hand rule: with the thumb along $n$ , the fingers curl in the direction of $\partial S$ . Example: for the upper unit hemisphere and $F = (-y, x, 0)$ , $\nabla \times F = (0, 0, 2)$ , and both sides equal $2\pi$ . For a flat surface in the $xy$ -plane, the theorem reduces to Green's theorem. The proof pulls the integrals back to $D$ and applies Green's theorem there.

History

The theorem first appears in a letter from William Thomson to George Stokes in 1850. Stokes set it as a question on the Smith's Prize examination at Cambridge in 1854, and the result then got his name.

Reading path

To understand this page from the ground up, read these 59 pages in order. The foundations come first.

Euclidean space object
Function concept
Function composition concept
Infimum concept
Supremum concept
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Jordan measure definition
Limit of a function of several variables concept
Differentiable function of several variables concept
Parametrised curve concept
Iterated integral definition
Partial derivative concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Gradient concept
Chain rule in several variables statement
Differentiable on an open set definition
Double integral definition
Gradient definition
Higher partial derivatives definition
Jacobian matrix definition
Chain rule for a composition with a curve statement
Fubini's theorem statement
Iterated derivative definition
Triple integral definition
C^k function characterised by continuous derivatives statement
Differential form definition
Parametrised curve definition
Vector field definition
Wedge product of forms definition
Closed curve definition
Parametrised surface definition
Rectifiable curve definition
Conservative vector field definition
Curl definition
Tangent vector of a curve definition
Divergence definition
Line integral of a scalar field definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Regular curve definition
Green's theorem statement
One-form definition
Surface area definition
Arc length parametrisation definition
Surface integral definition
Divergence theorem on a rectangle in the plane statement
Line integral as an integral over the parameter interval statement
Exterior derivative definition
Orientable surface definition
Gradient theorem statement
Flux through a surface definition
Divergence theorem statement
Generalised Stokes theorem statement
Stokes' theorem statement

Where this leads

2 pages build on this one.

Tag	`0AKRKVHW94`
URI	`m2://0AKRKVHW94`
Aliases	`kelvin-stokes-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	26B20
Status	This copy of the site has no build data. Run `m2 status 0AKRKVHW94`.
Uses	Curl, Line integral of a vector field, Flux through a surface, Orientable surface, Parametrised surface, Green's theorem, Generalised Stokes theorem
Same as	Wikidata Q467756 Wikipedia Stokes' theorem

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