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better intro
'''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''' 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>
This is an extension of [[Green's Theorem]] to surface integrals, and is also the analog in two dimensions of the [[Divergence Theorem]].
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.