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''&minus;1)-form with compact support on ''M''.  Let ∂''M'' denotes the boundary of ''M'' with its induced orientation, then  
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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''&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>,

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.