info prev up next book cdrom email home

Vector Integral

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

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

then
\begin{displaymath}
\int_C d{\bf s}\times{\bf P} = \int_S (d{\bf a}\times\nabla)\times{\bf P}.
\end{displaymath} (2)

If
\begin{displaymath}
{\bf F} \equiv {\bf c}F,
\end{displaymath} (3)

then
\begin{displaymath}
\int_C F\,ds = \int_S d{\bf a}\times\nabla{\bf F}.
\end{displaymath} (4)


The following are related to the Divergence Theorem. If

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

then
\begin{displaymath}
\int_V \nabla\times{\bf F}\,dV = \int_S d{\bf a}\times{\bf F}.
\end{displaymath} (6)

Finally, if
\begin{displaymath}
{\bf F} \equiv {\bf c}F,
\end{displaymath} (7)

then
\begin{displaymath}
\int_V \nabla F\,dV = \int _S F\,d{\bf a}.
\end{displaymath} (8)

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




© 1996-9 Eric W. Weisstein
1999-05-26