Difference between revisions of "Stokes' Theorem"
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| − | Stoke's Theorem is a generalization of the [[Fundamental Theorem of Calculus]], which states that if ''M'' is an oriented piece-wise smooth [[manifold]] of [[dimension]] k and <math>\omega</math> is a smooth (''k''−1)-form with compact support on ''M''. Let ∂''M'' denotes the boundary of ''M'' with its induced orientation, then | + | Stoke's Theorem is the fundamental theorem of [[Exterior Calculus]] and a generalization of the [[Fundamental Theorem of Calculus]], which states that if ''M'' is an oriented piece-wise smooth [[manifold]] of [[dimension]] k and <math>\omega</math> is a smooth (''k''−1)-form with compact support on ''M''. Let ∂''M'' denotes the boundary of ''M'' with its induced orientation, then |
:<math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>, | :<math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>, | ||
Revision as of 08:45, April 2, 2007
Stoke's Theorem is the fundamental theorem of Exterior Calculus and a generalization of the Fundamental Theorem of Calculus, which states that if M is an oriented piece-wise smooth manifold of dimension k and <math>\omega</math> is a smooth (k−1)-form with compact support on M. Let ∂M denotes the boundary of M with its induced orientation, then
- <math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>,
where d is the exterior derivative.