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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
Here d is the exterior derivative, which is defined using the manifold structure only. The theorem is to be considered as a generalisation of the fundamental theorem of calculus and indeed easily proved using this theorem.
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 combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end ofs of piecewise smooth submanifolds, so-called chains. 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 groupsIn mathematics (especially algebraic topology and abstract algebra), homology 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). See homology theory and de Rham cohomologyIn 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 repre.
The classical Kelvin-Stokes theorem:
which relates the surface integral of the curlThis article is about curl in mathematics, see also Curl programming language and cURL, the Unix command line tool for transferring files. In vector calculus, curl is a vector operator that shows a vector field's tendency to rotate about a point. A vector of a vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the 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.
Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem)
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 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.
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