2 typo

Perhaps even simpler than the examples from electromagnetism in $\mathbb{R}^3$ minus some points is the following:

The angle "function" $\varphi\colon S^1 \longrightarrow \mathbb{R}$ is not really globally defined as turning aroung one time gives a discontinuity. It jumps by $2\pi$. Neverhtheless, the differential $\mathrm{d}\varphi$ is a perfectly global one-form on $S^1$. It is the usual volume form, not being exact but closed for dimensional reasons. So the non-trivial first deRham cohomology of $S^1$ is responsible for counting angles and the fact that $0 \ne 2\pi$ ;)

This can be upgraded to the more interesting statement that on a orientable compact manifold without boundary you have a non-trivial top-degree deRham cohomology: again, the reason is that we can integrate a volume for form resulting in a non-zero volume. Thus (by Stokes theorem) the volume form can not be exact. It is closed without thinking about it, simply for dimensional reasons.

1

Perhaps even simpler than the examples from electromagnetism in $\mathbb{R}^3$ minus some points is the following:

The angle "function" $\varphi\colon S^1 \longrightarrow \mathbb{R}$ is not really globally defined as turning aroung one time gives a discontinuity. It jumps by $2\pi$. Neverhtheless, the differential $\mathrm{d}\varphi$ is a perfectly global one-form on $S^1$. It is the usual volume form, not being exact but closed for dimensional reasons. So the non-trivial first deRham cohomology of $S^1$ is responsible for counting angles and the fact that $0 \ne 2\pi$ ;)

This can be upgraded to the more interesting statement that on a orientable compact manifold without boundary you have a non-trivial top-degree deRham cohomology: again, the reason is that we can integrate a volume for resulting in a non-zero volume. Thus (by Stokes theorem) the volume form can not be exact. It is closed without thinking about it, simply for dimensional reasons.