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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.

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  

Here S is a surface, E is the boundary path of S, and the single integral denotes path integration around E with <math>\vec{\mathrm{d}l}</math> as the length element. The <math>\nabla \times</math> on the left side is the [[curl]] operator.

Here S is a surface, E is the boundary path of S, and the single integral denotes path integration around E with <math>\vec{\mathrm{d}l}</math> as the length element. The <math>\nabla \times</math> on the left side is the [[curl]] operator.

These last two examples (and Stokes' theorem in general) are somewhat esoteric, and are the subject of vector calculus.  They play important roles in [[electrodynamics]].  The divergence and curl operations are cornerstones of [[Maxwell's Equations]].

These last two examples (and Stokes' theorem in general) are are the subject of vector calculus.  They play important roles in [[electrodynamics]].  The divergence and curl operations are cornerstones of [[Maxwell's Equations]].

Stokes' Theorem is a lower-dimension version of the Divergence Theorem, and a higher-dimension version of [[Green's Theorem]].  Green’s Theorem relates a line integral to a double integral over a region, while Stokes' Theorem relates a line integral to a surface integral.

Stokes' Theorem is a lower-dimension version of the Divergence Theorem, and a higher-dimension version of [[Green's Theorem]].  Green’s Theorem relates a line integral to a double integral over a region, while Stokes' Theorem relates a surface integral of the curl of a function to its line integral.

[[Category:vector analysis]]

[[Category:vector analysis]]

[[Category:calculus]]

[[Category:calculus]]

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Physics]]

[[Category:Physics]]