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''&minus;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''&minus;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>
  
 
where ''d'' is the [[exterior derivative]].
 
where ''d'' is the [[exterior derivative]].
  
 
  [[category: mathematics]]
 
  [[category: mathematics]]

Revision as of 16:21, 22 March 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 is a smooth n−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.