de Rham cohomology is a formal set-up for the analytic problem: If you have a differential k-form ω on a manifold M, is it the exterior derivative of another differential k-form ω'? Formally, if ω = d ω' then d ω = 0. This is more commonly stated as d°d = 0, meaning that if ω is to be the exterior derivative of a differential k-form, a necessary condition that ω must satisfy is that its exterior derivative is zero.

Alexander-Spanier cohomology | change of variables theorem | differential k-form | exterior derivative | vector space