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The above formulation is also called as the "Curl Theorem," to distinguish it from the more general form of the Stokes' Theorem described below.
'''Stokes' Theorem''' expresses the surface integral of the [[curl]] of [[vector field]] in terms of its easier-to-compute [[circulation]]:
'''Stokes' Theorem''' expresses the surface integral of the [[curl]] of [[vector field]] in terms of its easier-to-compute [[circulation]]:
:<math>\iint_S (\nabla \times \vec w) \cdot \vec{\mathrm{d}A} = \oint_E \vec w \cdot \vec{\mathrm{d}l}\,</math>
:<math>\iint_S (\nabla \times \vec F) \cdot \vec{\mathrm{d}F} = \oint_E \vec F \cdot \vec{\mathrm{d}r}\,</math>
Stated another way, Stokes' Theorem equates the line integral of a vector fields to a surface integral of the same vector field. For this identity to be true, the ''direction'' of the vector normal ''n'' must obey the right-hand rule for the direction of the contour, ''i.e.'', when walking along the contour the surface must be on your left.
This is an extension of [[Green's Theorem]] to surface integrals, and is also the analog in two dimensions of the [[Divergence Theorem]]. The above formulation is also called as the "Curl Theorem," to distinguish it from the more general form of the Stokes' Theorem described below.
This is an extension of [[Green's Theorem]] to surface integrals, and is also the analog in two dimensions of the [[Divergence Theorem]]. The above formulation is also called as the "Curl Theorem," to distinguish it from the more general form of the Stokes' Theorem described below.
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.
== Examples ==
*[http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/stokes/stokes.html Calculation of both sides of Stokes' Theorem for a paraboloid, using a circle as the contour]
== General Form ==
== General Form ==