Stokes' Theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem), discovered in its modern form by É. Cartan and first published in 1945, 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. Lord Kelvin communicated it to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name, even though it was actually first published by Hermann Hankel in 1861. 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:

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