**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 C^{1}. If ∂ *M* denotes the boundary of *M* with its induced orientation, then

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:

Likewise the Ostrogradsky-Gauss theorem

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 than the special cases, of course, although the latter are more accessible and are often considered more convenient by practicing scientists and engineers.