# Generalized Stokes theorem

In vector calculus and differential geometry the **generalized Stokes theorem** (sometimes with apostrophe as **Stokes' theorem** or **Stokes's theorem**), also called the **Stokes–Cartan theorem**,[1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.[2]

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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.,

Stokes' theorem was formulated in its modern form by Élie Cartan in 1945,[3] following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.[4][5]

This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850.[6][7][8] Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861.[8][9] This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field **F** over a surface (that is, the flux of curl **F**) in Euclidean three-space to the line integral of the vector field over its boundary (also known as the loop integral).

**Simple classical vector analysis example**

Let *γ*: [*a*, *b*] → **R**^{2} be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides **R**^{2} into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose *ψ*: *D* → **R**^{3} is smooth, with *S* := *ψ*(*D*). If Γ is the space curve defined by Γ(*t*) = *ψ*(*γ*(*t*))[note 1] and **F** is a smooth vector field on **R**^{3}, then:[10][11][12]

This classical statement, is a special case of the general formulation stated above after making an identification of vector field with a 1-form and its curl with a two form through

- .

Other classical generalisations of the fundamental theorem of calculus like the divergence theorem, and Green's theorem are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).