For a Differential k-Form with compact support on an oriented -dimensional
Manifold ,
|
(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 ,
|
(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
,
|
(3) |
where is the Hodge Star operator. If is a Vector Field on ,
|
(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
|
(5) |
which is the Gradient Theorem. If
is a Vector Field and an embedded compact
3-manifold with boundary in , then
|
(6) |
which is the Divergence Theorem. If is a Vector Field and is an oriented, embedded, compact
2-Manifold with boundary in , then
|
(7) |
which is the Curl Theorem.
Physicists generally refer to the Curl Theorem
|
(8) |
as Stokes' theorem.
See also Curl Theorem, Divergence Theorem, Gradient Theorem
© 1996-9 Eric W. Weisstein
1999-05-26