As you might know from Vector Calculus (or Geometry and Motion), the differential operators such as the divergence, the gradient and the curl are much better behaved on simply connected domains (where every loop can be continuously shrunk to a point); for example, on such domains, all vector fields with (called irrotational) are of the form for some scalar field . In particular, for such vector fields, the line integral around any closed loop is 0, and more generally any line integral of the form can just be written as . The failure of this when working with non-simply connected domains is one of the key observations that started the study of cohomology.

De Rham cohomology then allows us to view this situation in a much more general setting, that of manifolds (which are simply spaces built together from glueing together pieces of . This then explains exactly why the previous situation on held, and what happens more generally. This will lead us to the consideration of differential forms, the Poincaré Lemma and Stokes' theorem (pictured!).

Meet us in MS.04 at 7:30 for an exciting overview of de Rham cohomology and some of its applications - then pub.