Stokes' theorem states that 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
(k-1)-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 theorem, curl theorem, and divergence
theorem by the following relations. If 
is a function on 
,
 |
(2)
|
where 
(the dual vector space) is the duality isomorphism
between a vector space and its dual
vector space, given by the Euclidean inner product
on 
.
If 
is a vector field on 
,
 |
(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 theorem, curl theorem, and divergence
theorem 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-dimensional manifold
with boundary in 
,
then
 |
(6)
|
which is the divergence theorem. If 
is a vector field and 
is an oriented,
embedded, compact
2-dimensional manifold
with boundary in 
,
then
 |
(7)
|
which is the curl theorem.
de Rham cohomology is defined using differential k-forms. When 
is an oriented compact manifold embedded
as 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.
