the left-hand side then leads to equivalent differential formulations (see below). The theorem is often used in situations where is an embedded oriented submanifold of some bigger manifold, often , on which the form is defined. Topological preliminaries; integration over chains Let be a smooth manifold. A (smooth) singular -simplex in is defined as a smooth map from the standard simplex in to . The group of singular -chains on is defined to be the free abelian group on the set of singular -simplices in . These groups, together with the boundary map, , define a chain complex. The corresponding homology (resp. cohomology) group is isomorphic to the usual singular homology group (resp. the singular cohomology group ), defined using continuous rather than smooth simplices in . On the other hand, the differential forms, with exterior derivative, , as the connecting map, form a cochain complex, which defines the de Rham cohomology groups . Differential -forms can be integrated over a -simplex in a natural way, by pulling back to . Extending by linearity allows one to integrate over chains. This gives a linear map from the space of -forms to the th group of singular cochains, , the linear functionals on . In other words, a -form defines a functional on the -chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, , behaves like the dual of on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means: closed forms, i.e., , have zero integral over boundaries, i.e. over manifolds that can be written as , and exact forms, i.e., , have zero integral over cycles, i.e. if the boundaries sum up to the empty set: . De Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to 1 and 2 above hold true. In other words, if are cycles generating the th homology group, then for any corresponding real numbers, , there exist a closed form, , such that and this form is unique up to exact forms. Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa. Formally stated, the latter reads: Underlying principle To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an