Form integration assigns an integral to a differential k-form over an oriented 
-dimensional submanifold
or 
-chain.
In particular, integration over an 
-dimensional manifold
requires a differential n-form. The
basic example is an 
-form
in the open unit ball in 
. Since 
is a top-dimensional
form, it can be written 
and so
 |
(1)
|
where the integral is the Lebesgue integral, provided 
is absolutely integrable.
On an oriented manifold 
, assume that 
has compact support
or is absolutely integrable. If 
is covered by coordinate charts 
, there is a partition
of unity 
such that
1. 
has support in 
and
2. 
.
Then
 |
(2)
|
where each term is evaluated in a coordinate chart. Absolute integrability makes the sum well-defined.
The integral is independent of the coordinate
charts and partition of unity because, under
an orientation-preserving coordinate
transformation 
,
the integral transforms according to the determinant
of the Jacobian, while an 
-form pulls back by the determinant
of the Jacobian. Hence,
 |
(3)
|
is the same integral in either coordinate chart.
For example, it is possible to integrate the 2-form
 |
(4)
|
on the sphere 
. Because a point has measure
zero, it is enough to integrate 
on 
, which can be covered by stereographic
projection 
.
This map is
 |
(5)
|
where 
.
The pullback map of 
is
 |
(6)
|
The integral of 
on 
is
 |
(7)
|
Note that this computation is done more easily by Stokes' theorem, because 
.
