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.

Read more about Stokes' Theorem: Introduction, General Formulation, Topological Reading; Integration Over Chains, Underlying Principle, Special Cases

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“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)