About Stokes' theorem

Question

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and integral. And now I want to find a example when the form is not compactly supported and Stokes' theorem won't hold anymore. I think I have to construct a special Partition of unity by functions with compact support. Can someone give an example?

Answer 1

The problem you underline is not about Stokes' theorem. It is about the integration of a form itself. If your form is not compactly supported, then you cannot define its integral over your manifold. Simply think of the manifold $\mathbb{R}$ and a $1$-form $f(x)dx$, where $f$ is a non-compactly supported function.