I was reading Spivak's Calculus on Manifolds and when I reached his proof of Stokes' Theorem, I noticed he states
Theorem: If $\omega$ is $(K-1)$ form on an open set $A\subset \mathbb{R}^{n}$ and if $C$ is a K-chain then $$\int_{\partial C} \omega = \int_{C} d\omega$$
If we take the example of $\omega = Pdx + Qdy$ where $P_{Y}=Q_{x}$ then we get Greene's theorem. However, when looking at the classical example of $$P=\frac{-y}{x^{2}+y^{2}}, Q=\frac{x}{x^{2}+y^{2}}$$ Greene's theorem does not hold meaning Stokes would not hold either. However, in looking through the proof of General Stokes, I could not see any reason for why this example fails the conditions needed for the theorem to hold. I can guess that maybe the reasons it does not hold is because Stokes needs $\omega$ to be exact, or perhaps the holes cause some issue with the boundary of $C$ but I have no real intuition over which of these, if any of these, are the problem.
Any help would be appreciated!