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Curl Theorem

A special case of Stokes' Theorem in which $F$ is a Vector Field and $M$ is an oriented, compact embedded 2-Manifold with boundary in $\Bbb{R}^3$, given by

\int_S(\nabla \times {\bf F})\cdot d{\bf a} = \int_{\partial S}{\bf F}\cdot d{\bf s}.
\end{displaymath} (1)

There are also alternate forms. If

{\bf F} \equiv {\bf c}F,
\end{displaymath} (2)

\int_S d{\bf a}\times \nabla F = \int_C Fd{\bf s}.
\end{displaymath} (3)

and if
{\bf F} \equiv {\bf c}\times {\bf P},
\end{displaymath} (4)

\int_S (d{\bf a}\times \nabla)\times {\bf P} = \int_C d{\bf s}\times {\bf P}.
\end{displaymath} (5)

See also Change of Variables Theorem, Curl, Stokes' Theorem


Arfken, G. ``Stokes's Theorem.'' §1.12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 61-64, 1985.

© 1996-9 Eric W. Weisstein