A.k.a. Gauss's Theorem. Let
be a region in space with boundary
. Then
![\begin{displaymath}
\int_V (\nabla\cdot{\bf F})\,dV = \int_{\partial V} {\bf F}\cdot d{\bf a}.
\end{displaymath}](d2_1082.gif) |
(1) |
Let
be a region in the plane with boundary
.
![\begin{displaymath}
\int_S \nabla\cdot{\bf F}\, dA = \int_{\partial S}{\bf F}\cdot {\bf n} \,ds.
\end{displaymath}](d2_1085.gif) |
(2) |
If the Vector Field
satisfies certain constraints, simplified forms can be used. If
where
is a constant vector
, then
![\begin{displaymath}
\int_S {\bf F}\cdot d{\bf a} = {\bf c}\cdot \int_S v\,d{\bf a}.
\end{displaymath}](d2_1089.gif) |
(3) |
But
![\begin{displaymath}
\nabla \cdot(f{\bf v}) = (\nabla f)\cdot {\bf v}+f(\nabla\cdot {\bf v}),
\end{displaymath}](d2_1090.gif) |
(4) |
so
![\begin{displaymath}
\int_V \nabla \cdot({\bf c}v)\,dV = {\bf c}\cdot\int_V(\nabla v+v\nabla\cdot {\bf c})\,dV = {\bf c}\cdot\int_V \nabla v\,dV
\end{displaymath}](d2_1091.gif) |
(5) |
![\begin{displaymath}
{\bf c}\cdot\left({\int_S v\,d{\bf a}-\int_V\nabla v\,dV}\right)= 0.
\end{displaymath}](d2_1092.gif) |
(6) |
But
, and
must vary with
so that
cannot
always equal zero. Therefore,
![\begin{displaymath}
\int_S v\,d{\bf a} = \int_V \nabla v\,dV.
\end{displaymath}](d2_1096.gif) |
(7) |
If
, where
is a constant vector
, then
![\begin{displaymath}
\int_S d{\bf a}\times {\bf P} = \int_V \nabla \times {\bf P}\,dV.
\end{displaymath}](d2_1098.gif) |
(8) |
See also Curl Theorem, Gradient, Green's Theorem
References
Arfken, G. ``Gauss's Theorem.'' §1.11 in
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.
© 1996-9 Eric W. Weisstein
1999-05-24