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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.
:<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 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]].
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.