Stokes' Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes (1819-1903). The theorem aquired its name after Stokes' habit of including it on the prize examinations. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... The silver Anglia knight, commissioned as a trophy in 1850, intended to represent the Black Prince. ... Sir George Gabriel Stokes, 1st Baronet (August 13, 1819 – February 1, 1903) was an Irish mathematician and physicist. ... 1819 was a common year starting on Friday (see link for calendar). ... 1903 (MCMIII) was a common year starting on Thursday (see link for calendar). ...

Let M be an oriented piecewise smooth manifold of dimension n and let ω be a n−1 compactly supported differential form on M of class C¹. If ∂M denotes the boundary of M with its induced orientation, then This page is about a higher mathematics topic. ... In common usage, the dimensions (from Latin measured out) of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This page is about a higher mathematics topic. ...

Here d is the exterior derivative, which is defined using the manifold structure only. The Stokes Theorem can be considered as a generalisation of the fundamental theorem of calculus; and the latter indeed follows easily from the former. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ...

The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form ω is defined.

The theorem easily extends to linear combinations of piecewise smooth submanifolds, so-called chains. The Stokes Theorem then shows that closed forms defined up to an exact form can be integrated over chains defined only up to a boundary. This is the basis for the pairing between homology groups and de Rham cohomology. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... The word Boundary has a variety of meanings. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

The classical Kelvin-Stokes Theorem:

which relates the surface integral of the curl of a vector field over a surface Σ in Euclidean 3 space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with n = 2) once we identify a vector field with a 1 form using the metric on Euclidean 3 space. The first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in his letter to Stokes. In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... The Right Honourable William Thomson, 1st Baron Kelvin, GCVO, OM, PC, PRS (26 June 1824–17 December 1907) was a Scottish-Irish mathematical physicist and engineer, an outstanding leader in the physical sciences of the 19th century. ...

Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem) In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

is a special case if we identify a vector field with the n-1 form obtained by contracting the vector field with the Euclidean volume form.

The fundamental theorem of calculus and Green's theorem are also special cases of the general Stokes theorem. The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special case of the more...

The general form of the Stokes Theorem using differential forms is more powerful than the special cases, of course, although the latter are more accessible and are often considered more convenient by practicing scientists and engineers.