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Stokes' Theorem is useful in calculating circulation in mechanical engineering. Stokes' Theorem has no application to conservative fields.
'''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''' 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 fields]].
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''−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''−1)-[[differential form|form]] with compact support on ''M'', and ∂''M'' denotes the boundary of ''M'' with its induced orientation, then