Vector Integral

The following vector integrals are related to the Curl Theorem. If

$${\bf F} \equiv {\bf c}\times{\bf P}(x,y,z),$$ (1)

then

$$\int_C d{\bf s}\times{\bf P} = \int_S (d{\bf a}\times\nabla)\times{\bf P}.$$ (2)

If

$${\bf F} \equiv {\bf c}F,$$ (3)

then

$$\int_C F\,ds = \int_S d{\bf a}\times\nabla{\bf F}.$$ (4)

The following are related to the Divergence Theorem. If

$${\bf F} \equiv {\bf c}\times{\bf P}(x,y,z),$$ (5)

then

$$\int_V \nabla\times{\bf F}\,dV = \int_S d{\bf a}\times{\bf F}.$$ (6)

Finally, if

$${\bf F} \equiv {\bf c}F,$$ (7)

then

$$\int_V \nabla F\,dV = \int _S F\,d{\bf a}.$$ (8)

See also Curl Theorem, Divergence Theorem, Gradient Theorem, Green's Identities, Line Integral, Surface Integral, Vector Derivative, Volume Integral