A differential k-form can be integrated on an 
-dimensional
manifold. 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.
On a manifold 
covered by coordinate charts 
, there is a partition
of unity 
such that
1. 
has support in 
and
2. 
.
Then
 |
(2)
|
where the right-hand side is well-defined because each integration takes place in a coordinate chart.
The integral of the 
-form
is well-defined
because, under a change of coordinates 
, 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 
.
Since
 |
(5)
|
the pullback map of 
is
 |
(6)
|
the integral of 
on 
is
 |
(7)
|
Note that this computation is done more easily by Stokes' theorem, because 
.
