Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by $F(\theta)=(\sin\theta, \cos\theta)$, then $\int_{S^1}w=\int_{[0,2\pi]} F^{\ast}w=\int_{[0,2\pi]}(-\cos^2\theta \,\mathrm d\theta+\sin^2\theta \,\mathrm d\theta)=-2\pi$.
Then I want to evaluate $\int_{S^1}w$ by using Stoke's theorem (This uses the assupmtion that $S^1$ is the boundary of some compact manifold with boundary in $\mathbb R^2-{(0,0)}$).
If the result of this integral is different from $-2\pi$, then I can conclude the assumption is false, thus proving the result. However, I don't know how to evaluate by using Stoke, I am stuck with how to change the 1-form into 0-form and evaluate it. Thanks for any help!
If $S^1$ is the only boundary component, then such a compact manifold would have to be both open and (relatively) closed in $\mathbb R^2-{(0,0)}$, and so it would have to be either the interior or the exterior of the circle, neither of which is a compact manifold. Here one doesn't need to use the Jordan curve theorem because your circle is standard.