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

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

There are also alternate forms. If

$${\bf F} \equiv {\bf c}F,$$ (2)

then

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

and if

$${\bf F} \equiv {\bf c}\times {\bf P},$$ (4)

then

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

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

References

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