Generalized Stokes theorem

Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

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. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in ${\displaystyle \mathbb {R} ^{2}}$ or ${\displaystyle \mathbb {R} ^{3},}$ and the divergence theorem is the case of a volume in ${\displaystyle \mathbb {R} ^{3}.}$^[2] Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus.^[3]

Stokes' theorem says that the integral of a differential form ${\displaystyle \omega }$ over the boundary ${\displaystyle \partial \Omega }$ of some orientable manifold ${\displaystyle \Omega }$ is equal to the integral of its exterior derivative ${\displaystyle d\omega }$ over the whole of ${\displaystyle \Omega }$, i.e.,

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

Stokes' theorem was formulated in its modern form by Élie Cartan in 1945,^[4] following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.^[5][6]

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.^[7][8][9] 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.^[9][10] This classical case relates the surface integral of the curl of a vector field ${\displaystyle {\textbf {F}}}$ over a surface (that is, the flux of ${\displaystyle {\text{curl}}\,{\textbf {F}}}$) in Euclidean three-space to the line integral of the vector field over the surface boundary.