Stokes' theorem

Let M be an oriented piecewise smooth compact manifold of dimension n and let ω be a n-1 differential form on M of class C¹. If ∂ M denotes the boundary of M with its induced orientation, then

<math>\int_M d\omega = \int_{\partial M} \omega</math>

Here d is the exterior derivative, which is defined using the manifold structure only. The theorem is to be considered as a generalisation of the fundamental theorem of calculus and indeed easily proved using this theorem.

The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form ω is defined.

The theorem easily extends to linear combinations of piecewise smooth submanifolds, so called chains. Stokes theorem then shows that closed forms defined up to an exact form[?] can be integrated over chains defined only up to a boundary[?]. This is the basis for the pairing between homology groups and de Rham cohomology.

The classical Kelvin-Stokes theorem:

<math> \int_{\Sigma} {\rm rot} \; \mathbf{v} \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} \mathbf{v} \cdot d \mathbf{r} </math>

relating the integral of the rotation of a vector field over a surface Σ in Euclidean 3 space to the integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form using the metric on Euclidean 3 space. The first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in his letter to Stokes.

Likewise the Ostrogradsky-Gauss theorem

<math>\int_{\mathrm{Vol}} \operatorname{div}\ v\ d\mathrm{Vol} = \int_{\partial \mathrm{Vol}} v \cdot d\Sigma</math>

is a special case if we identify a vector field with the n-1 form obtained by contracting the the vector field with the Euclidean volume form.

The Fundamental Theorem of Calculus and Green's theorem are also special cases of the general Stokes theorem.

The general form of the Stokes theorem using differential forms is more powerful and generally easier to work with.