I'm stuck on the following problem:
Let $f(x,y) = -\frac{y}{x^2 + y^2}$ and $g(x,y) = \frac{x}{x^2 + y^2}$ for all $(x,y) \neq (0,0)$. Show that $$\oint_{\partial S} (f(x,y) \,dx + g(x,y)\,dy) = 2\pi $$ if $S$ is any open set in the plane containing $(0,0)$ with $\partial S$ as its boundary.
My attempt:
This looks like a Green's Theorem problem to me, so that was the first thing I tried:
\begin{align*}
\oint_{\partial S} (f(x,y) \,dx + g(x,y)\,dy) &= \iint_S \left(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \,dx \,dy.
\end{align*}
I found that \begin{align*} & \frac{\partial}{\partial x} \left(\frac{x}{x^2+y^2}\right) = \frac{\partial}{\partial y} \left(-\frac{y}{x^2 + y^2} \right) = \frac{y^2-x^2}{x^2+x^2}. \end{align*}
Thus, $\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} = 0$, and so the integral vanishes. Clearly I'm doing something wrong if the answer is supposed to be $2 \pi$...Where did I go wrong with this?