De Rham cohomology

Differential forms which are exterior derivatives are called exact and forms, whose exterior derivatives are 0 are called closed. Exact forms are closed, so the vector spaces of k-forms along with the exterior derivative are a cochain complex. Closed forms modulo exact forms are called the de Rham cohomology groups. Wedge product endows the direct sum of these groups with a ring structure.

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.