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Line 1: − 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''−1)-form with compact support on ''M''. Let ∂''M'' denotes the boundary of ''M'' with its induced orientation, then + Line 5:
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Major expansion -- applications to vector calc/physics/electrodynamics
'''Stokes' Theorem''', in its most general form, 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
:<math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>,
:<math>\int_M \mathrm{d}\omega = \oint_{\partial M} \omega\!\,</math>,
where ''d'' is the [[exterior derivative]].
where ''d'' is the [[exterior derivative]].
[[category: mathematics]]
There are a number of well-known special cases of Stokes' theorem:
*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 '''Green's 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>
Where R is some region of 3-space, S is the boundary surface of R, the triple integral denotes volume integration over R with dV as the volume element, and the double integral denotes surface integration over S with <math>\vec{\mathrm{d}A}</math> as the oriented normal of the surface element. The <math>\nabla \cdot</math> on the left side is the [[divergence]] operator, and the <math>\cdot</math> on the right side is the vector [[inner product|dot product]].
*When k=2, this is often also called '''Stokes' Theorem''' (the less general form):
:<math>\iint_S (\nabla \times \vec w) \cdot \vec{\mathrm{d}A} = \oint_E \vec w \cdot \vec{\mathrm{d}l}\,</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]].
[[Category:Mathematics]]
[[Category:Physics]]