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]] n and <math>\omega</math> is a smooth ''n''−1 form with compact support on ''M''. Let ∂''M'' denotes the boundary of ''M'' with its induced orientation, then | 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]] n and <math>\omega</math> is a smooth ''n''−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>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>, |
where ''d'' is the [[exterior derivative]]. | where ''d'' is the [[exterior derivative]]. | ||
[[category: mathematics]] | [[category: mathematics]] | ||
Revision as of 21:21, March 22, 2007
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 n and <math>\omega</math> is a smooth n−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.