For a Differential kForm 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
3manifold with boundary in , then

(6) 
which is the Divergence Theorem. If is a Vector Field and is an oriented, embedded, compact
2Manifold 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
© 19969 Eric W. Weisstein
19990526