## Stokes' Theorem

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.