Stokes' theorem: Difference between revisions
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In [[vector analysis]] and [[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 | ==Vector analysis formulation== | ||
In vector analysis Stokes' theorem is commonly written as | |||
:<math> | :<math> | ||
\iint_S \,(\boldsymbol{\nabla}\times \mathbf{F})\cdot d\mathbf{S} = | \iint_S \,(\boldsymbol{\nabla}\times \mathbf{F})\cdot d\mathbf{S} = | ||
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</math> | </math> | ||
where '''∇''' × '''F''' is the [[curl]] of a [[vector field]] on <math>\scriptstyle \mathbb{R}^3</math>, the vector d'''S''' | 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''. | is a vector normal to the surface element d''S'', the contour integral is over a closed, non-intersecting path ''C'' bounding the open, two-sided surface ''S''. The direction of the vector d'''S''' is determined according to the [[right screw rule]] by the direction of integration along ''C''. | ||
In differential geometry the theorem is extended to integrals of [[exterior derivatives]] over [[oriented]], [[compact]], and [[differentiable]] [[manifolds]] of finite dimension. | ==Differential geometry formulation== | ||
In differential geometry the theorem is extended to integrals of [[exterior derivatives]] over [[oriented]], [[compact]], and [[differentiable]] [[manifold (geometry)|manifolds]] of finite dimension. It can be written as <math>\int_c d\omega=\int_{\partial c} \omega</math>, where <math>c</math> is a [[singular cube]], and <math>\omega</math> is a [[differential form]].[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:00, 22 October 2024
In vector analysis and differential geometry, Stokes' theorem is a statement that treats integrations of differential forms.
Vector analysis formulation
In vector analysis Stokes' theorem 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, non-intersecting path C bounding the open, two-sided surface S. The direction of the vector dS is determined according to the right screw rule by the direction of integration along C.
Differential geometry formulation
In differential geometry the theorem is extended to integrals of exterior derivatives over oriented, compact, and differentiable manifolds of finite dimension. It can be written as , where is a singular cube, and is a differential form.
