Stokes' theorem: Difference between revisions
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imported>Jitse Niesen m (Stokes', not Stoke's) |
imported>Paul Wormer (see talk) |
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In [[differential geometry]], '''Stokes' theorem''' is a statement that treats integrations of differential forms. | In [[vector analysis]] and [[differential geometry]], '''Stokes' theorem''' is a statement that treats integrations of differential forms. | ||
In vector analysis it is commonly written as | |||
:<math> | |||
\iint_S \,(\boldsymbol{\nabla}\times \mathbf{F})\cdot d\mathbf{S} = | |||
\oint_C \mathbf{F}\cdot d\mathbf{s} | |||
</math> | |||
where '''∇''' × '''F''' is the [[curl]] of a [[vector field]] on <math>\scriptstyle \mathbb{R}^3</math>, the vector d'''S''' | |||
is a vector normal to the surface element d''S'', the contour integral is over a closed path ''C'' bounding the surface ''S''. | |||
In differential geometry the theorem is extended to integrals of [[exterior derivatives]] over [[oriented]], [[compact]], and [[differentiable]] [[manifolds]] of finite dimension. | |||
Revision as of 08:42, 11 July 2008
In vector analysis and differential geometry, Stokes' theorem is a statement that treats integrations of differential forms.
In vector analysis it is commonly written as
where ∇ × F is the curl of a vector field on , the vector dS is a vector normal to the surface element dS, the contour integral is over a closed path C bounding the surface S.
In differential geometry the theorem is extended to integrals of exterior derivatives over oriented, compact, and differentiable manifolds of finite dimension.