For
a Differential k-Form with compact support on an oriented
-dimensional
Manifold
,
![\begin{displaymath}
\int_M dw = \int_{\partial M} w,
\end{displaymath}](s3_1020.gif) |
(1) |
where
is the Exterior Derivative of the differential form
. This connects to the ``standard''
Gradient, Curl, and Divergence Theorems
by the following relations. If
is a function on
,
![\begin{displaymath}
\mathop{\rm grad}(f) = c^{-1}\,df,
\end{displaymath}](s3_1023.gif) |
(2) |
where
(the dual space) is the duality isomorphism between a Vector Space
and its dual, given by the Euclidean Inner Product on
. If
is a Vector Field on a
,
![\begin{displaymath}
\mathop{\rm div}(f) = {}^*d{}^*c(f),
\end{displaymath}](s3_1025.gif) |
(3) |
where
is the Hodge Star operator. If
is a Vector Field on
,
![\begin{displaymath}
\mathop{\rm curl}(f) = c^{-1}{}^*dc(f).
\end{displaymath}](s3_1027.gif) |
(4) |
With these three identities in mind, the above Stokes' theorem in the three instances is transformed into the
Gradient, Curl, and Divergence Theorems
respectively as follows. If
is a function on
and
is a curve in
, then
![\begin{displaymath}
\int_\gamma \mathop{\rm grad}(f) \cdot d{\bf l} = \int_\gamma\, df = f(\gamma(1)) - f(\gamma(0)),
\end{displaymath}](s3_1028.gif) |
(5) |
which is the Gradient Theorem. If
is a Vector Field and
an embedded compact
3-manifold with boundary in
, then
![\begin{displaymath}
\int_{\partial M} f\cdot dA = \int_{\partial M} {}^*cf = \int_M d*cf = \int_M \mathop{\rm div}(f)\,dV,
\end{displaymath}](s3_1030.gif) |
(6) |
which is the Divergence Theorem. If
is a Vector Field and
is an oriented, embedded, compact
2-Manifold with boundary in
, then
![\begin{displaymath}
\int_{\partial M} f \, dl = \int_{\partial M} cf = \int_M dc(f) = \int_M \mathop{\rm curl}(f) \cdot dA,
\end{displaymath}](s3_1031.gif) |
(7) |
which is the Curl Theorem.
Physicists generally refer to the Curl Theorem
![\begin{displaymath}
\int_S(\nabla \times {\bf F})\cdot d{\bf a} = \int_{\partial S}{\bf F}\cdot d{\bf s}
\end{displaymath}](s3_1032.gif) |
(8) |
as Stokes' theorem.
See also Curl Theorem, Divergence Theorem, Gradient Theorem
© 1996-9 Eric W. Weisstein
1999-05-26