I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector fields and one-forms on Euclidean space. However, once I've got there I'm not too sure how to convince everyone that it was worth the journey. What down-to-earth uses could one cite to prove the worth of the construct?



Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness
The motivation that most appeals to me is very simple and can come up in a freshman vector calculus course.


We say that a vector field $F$ in $\mathbb{R}^3$ is *conservative* if $F = \nabla f$ for some scalar-valued function $f$. This has natural applications in physics (e.g. electric fields). It's easy to see this happens iff line integrals of $F$ are path independent, iff line integrals around closed loops vanish, etc. On a simply connected domain, $F$ is conservative iff $\nabla \times F = 0$ (use the freshman version of Stokes' theorem). On a non-simply connected domain, this may fail (e.g. $\mathbb{R}^3$ minus a line). The extent to which it fails is of course the de Rham cohomology of the domain. So this suggests that the de Rham cohomology is a good way to detect the "shape" of a domain, and one goes on from there.


Maybe this is too basic to be interesting, but I like it myself.


