Global analysis, analysis on manifolds Theorem

Generalised Stokes theorem

Builds on Exterior derivative Differential form Orientable surface Gradient theorem Green's theorem Divergence theorem

Statement

Let $\Omega$ be an oriented smooth manifold of dimension $n$ with boundary $\partial \Omega$ , and give $\partial \Omega$ the induced orientation. Let $\omega$ be a smooth differential form of degree $n - 1$ on $\Omega$ with compact support, and let $d\omega$ be its exterior derivative. Then $\int_\Omega d\omega = \int_{\partial \Omega} \omega.$

Remarks

For $\Omega = [a, b]$ and a function $f$ , the theorem is the fundamental theorem of calculus, $\int_a^b f'(x) \, dx = f(b) - f(a)$ . The boundary point $a$ gets the negative orientation. For a curve and a 0-form, it is the gradient theorem. For a plane region and a 1-form, it is Green's theorem. For a surface in $\mathbb{R}^3$ and a 1-form, it is the classical Stokes' theorem. For a solid in $\mathbb{R}^3$ and a 2-form, it is the divergence theorem. The orientation and the compact support are needed so that both integrals are defined.

Reading path

To understand this page from the ground up, read these 58 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

Where this leads

3 pages build on this one.

Tag	`0HPHKCGEQ0`
URI	`m2://0HPHKCGEQ0`
Aliases	`stokes-cartan-theorem`
Kind	statement, theorem
Writing	W0 stub
MSC	58C35
Status	This copy of the site has no build data. Run `m2 status 0HPHKCGEQ0`.
Uses	Exterior derivative, Differential form, Orientable surface, Gradient theorem, Green's theorem, Divergence theorem
Same as	Wikidata Q10861254 Wikipedia Generalized Stokes theorem

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Used by

Stokes' theorem