In my differential geometry lecture, my professor stated Stokes' theorem as follows:
Let $M$ be a $n$-dimensional, piecewise continuously differentiable submanifold with boundary of $\mathbb R^m$. Then, for every continuously differentiable $(n-1)$-form $\omega$ with compact support on $M$, $$\int_M d\omega = \int_{\partial M}\omega$$
Now I want to find a counterexample for when $\omega$ is not compactly supported, but what is meant with "compact support" when talking about forms? For example, the form $$\omega = x \,dy \wedge dz + y \,dz \wedge dx + z \,dx \wedge dy$$ on $S^2$ was used as an application of Stokes' theorem, but I don't see how this has compact support? $x$ is unbounded on $\mathbb R^m$. On $S^2$, it of course is not, but how am I supposed to find something unbounded on a compact set?
Well, the support of a $k-$form $\omega\in \Omega^k(M)$ on a manifold $M$ is the set $$\text{supp}(\omega)=\overline{\{x\in M: \omega_x\ne 0\}}. $$ The support is said to be compact when it is compact as a topological space. In your particular example we have that any form $\omega$ on a compact manifold $M=S^2$ is compactly supported, since $\text{supp}(\omega)$ is a closed set, and a closed subset of a compact topological space is compact.