A differential kform can be integrated on an dimensional manifold. The basic example is an form in the open unit ball in . Since is a topdimensional 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 righthand side is welldefined because each integration takes place in a coordinate chart. The integral of the form is welldefined 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 2form
(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 .