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'''Stokes' Theorem''' holds that the double integral of the [[curl]] of a vector field with respect to a surface is equal to its line integral with respect to a simple curve enclosing the surface.  This is the analog in two dimensions of the [[Divergence Theorem]].  Stokes' Theorem is useful in calculating [[circulation]] in mechanical engineering.  Stokes' Theorem has no application to [[conservative field]]s.

'''Stokes' Theorem''' holds that the double integral of the [[curl]] of a vector field with respect to a surface is equal to its line integral with respect to a simple curve enclosing the surface.  This is the analog in two dimensions of the [[Divergence Theorem]].   

 

Stokes' Theorem is useful in calculating [[circulation]] in mechanical engineering.  A [[conservative field]] has a circulation (line integral on a simple, closed curve) of zero, and application of the Stokes' Theorem to such a field proves that the curl of a conservative field over the enclosed surface must also be zero.

In its most general form, this theorem is the fundamental theorem of [[Exterior Calculus]], and is a generalization of the [[Fundamental Theorem of Calculus]]. It states that if ''M'' is an oriented piecewise smooth [[manifold]] of [[dimension]] k and <math>\omega</math> is a smooth (''k''&minus;1)-[[differential form|form]] with compact support on ''M'', and ∂''M'' denotes the boundary of ''M'' with its induced orientation, then  

In its most general form, this theorem is the fundamental theorem of [[Exterior Calculus]], and is a generalization of the [[Fundamental Theorem of Calculus]]. It states that if ''M'' is an oriented piecewise smooth [[manifold]] of [[dimension]] k and <math>\omega</math> is a smooth (''k''&minus;1)-[[differential form|form]] with compact support on ''M'', and ∂''M'' denotes the boundary of ''M'' with its induced orientation, then