Theorem (Stokes' theorem):
Let be an oriented, -dimensional smooth manifold with boundary and let . Then
where bears the orientation inherited from .
Proof: We first prove the special cases where or . In the first case, we restrict attention to a form of the form , so that . Now has no boundary, so that
by the definition of the integral of forms. Indeed, we have the empty atlas for , which is oriented. Also, we have, using Fubini's theorem,
since and hence have compact support, proving the statement for . We proceed to the half-space, ie. we set . A general ()-form may be written as
so that we have
and the two integrals coincide. Having dealt with the two special cases, we may proceed to the general case. Hence, suppose we have an oriented manifold with oriented atlas such that each is equal to either or , and let , where . Then by definition of the integral of a top form over an oriented manifold, whenever is a partition of unity subordinate to , we have