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# Stokes' theorem

Stokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes (1819-1903).

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

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:

which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean 3 space to the line 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.