Apparent counter example to Stoke's theorem? I think I found an apparent contradiction to Stoke's theorem with this 2-differential form $M= \overline{B^{2}}- \\{ 0 \\}$, $\partial M = S^1$, $$\omega = \frac{x~dy-y~dx}{x^2+y^2}$$ defined in $\mathbb{R}^2 - \\{0\\}$ and then pullbacked to $ \overline{B^{2}}- \\{ 0 \\}$ $d \omega = 0$ So that by Stoke's Theorem $$ \int\limits_{\partial M} \omega = \int\limits_{M} d \omega = 0 $$ But direct calculation shows that $\int\limits_{S^1} \omega \neq 0 $

Your $M$ is not compact, and $\omega$ is not compactly supported. So you have not contradicted Stokes's theorem.