# 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 03:45, 2 April 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 is a smooth (*k*−1)-form with compact support on *M*. Let ∂*M* denotes the boundary of *M* with its induced orientation, then

- ,

where *d* is the exterior derivative.