For 
a differential (k-1)-form with compact
support on an oriented 
-dimensional manifold
with boundary 
,
 |
(1)
|
where 
is the exterior derivative of the differential
form 
.
When 
is a compact manifold without boundary, then
the formula holds with the right hand side zero.
Stokes' theorem 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.
de Rham cohomology is defined using differential k-forms. When 
is a submanifold (without
boundary), it represents a homology class. Two closed forms represent the same cohomology class if they differ by an exact
form, 
.
Hence,
 |
(8)
|
Therefore, the evaluation of a cohomology class on a homology class is well-defined.
Physicists generally refer to the curl theorem
 |
(9)
|
as Stokes' theorem.
