In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing that I remember being puzzled by, and which I felt was never answered during the course even though I asked the professor about it, was what the theorem could be used for. More specifically, what I was hoping to see was an application of the de Rham theorem to proving a result that was "elementary" (meaning that it could be understood, and seen to be interesting, by someone who had not already studied the material in that course).
Is there a good motivating problem of this type for the de Rham theorem?
To give you a better idea of what exactly I'm asking for, here's what I consider to be a good motivating problem for the Lebesgue integral. It is Exercise 10 in Chapter 2 of Rudin's Real and Complex Analysis. If $\lbrace f_n\rbrace$ is a sequence of continuous functions on $[0,1]$ such that $0\le f_n \le 1$ and such that $f_n(x)\to 0$ as $n\to\infty$ for every $x\in[0,1]$, then $$\lim_{n\to\infty}\int_0^1 f_n(x)\thinspace dx = 0.$$
This problem makes perfect sense to someone who only knows about the Riemann integral, but is rather tricky to prove if you're not allowed to use any measure theory.
If it turns out that there are lots of answers then I might make this community wiki, but I'll hold off for now.
Best Answer
Here is a really "trivial" application. Since a volume form (say from a Riemannian metric) for a compact manifold $M$ is clearly closed (it has top degree) and not exact (by Stoke's Theorem), it follows that the cohomology is non-trivial, so $M$ cannot be contractible.