Stokes' Theorem

In differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.

This modern form of Stokes' theorem is a vast generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:

This classical statement, as well as the classical Divergence theorem and Green's Theorem, are simply special cases of the general formulation stated above.