Differentiable Manifolds/Integration of forms

Stokes' theorem Let be an oriented, -dimensional smooth manifold with boundary and let . Then , where bears the orientation inherited from . 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 . Hence, and 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