# 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]

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 over the whole of Ω, i.e.,

${\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.}$

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] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 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: DR3 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 R3, then:[10][11][12]

${\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }$

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

${\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\\\end{pmatrix}}\cdot d\Gamma \to F_{x}\,dx+F_{y}\,dy+F_{z}\,dz}$
${\displaystyle \nabla \times {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}\cdot d\mathbf {S} ={\begin{pmatrix}\partial _{y}F_{z}-\partial _{z}F_{y}\\\partial _{z}F_{x}-\partial _{x}F_{z}\\\partial _{x}F_{y}-\partial _{y}F_{x}\\\end{pmatrix}}\cdot d\mathbf {S} \to }$
${\displaystyle d(F_{x}\,dx+F_{y}\,dy+F_{z}\,dz)=(\partial _{y}F_{z}-\partial _{z}F_{y})\,dy\wedge dz+(\partial _{z}F_{x}-\partial _{x}F_{z})\,dz\wedge dx+(\partial _{x}F_{y}-\partial _{y}F_{x})\,dx\wedge dy}$.

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).