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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 fieldsfield]]s.
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
*When k=1, and the terms appearing in the theorem are translated into their simpler form, this is just the Fundamental Theorem of Calculus.
*When k=3, this is often called '''Gauss' Theorem''' or the '''Divergence Theorem''' and is useful in [[vector calculus]]:
:<math>\iiint_R (\nabla \cdot \vec w)\ \mathrm{d}V = \iint_S \vec w \cdot \vec{\mathrm{d}A}\,</math>
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]].
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.
[[Category:vector analysis]]
[[Category:calculus]]
[[Category:Mathematics]]
[[Category:Physics]]
