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.