High Quality Content by WIKIPEDIA articles! In differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and 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. The theorem is often used in situations where ? is an embedded oriented submanifold of some bigger manifold on which the form ? is defined. A proof becomes particularly simple if the submanifold ? is a so-called "normal manifold", as in the figure on the r.h.s., which can be segmented into vertical stripes (e.g. parallel to the xn direction), such that after a partial integration concerning this variable, nontrivial contributions come only from the upper and lower boundary surfaces (coloured in yellow and red, repectively), where the complementary mutual orientations are visible through the arrows.

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