If $B_\rho = \{(x,y) : x^2 + y^2 \leq \rho\}$ and $u$ is $C^1$, how can I show that: $$\lim_{\rho \to 0}\int_{\partial B_{\rho}} u_x dy - u_ydx = 0$$
I know that by Green's theorem:
$$\int_{\partial B_{\rho}} u_x dy - u_ydx = \iint_{B_\rho} u_{xx} + u_{yy} dx dy$$
but I'm not sure what to do next...
This is simpler than you are expecting. Since $u$ is $C^1$ it and its derivatives are bounded. Parameterize $\partial B_\rho$ by $x(t) = \rho \cos t$ and $y(t) = \rho \sin t$. Then $$\int_{\partial B_\rho} u_x \, dy = \int_0^{2\pi} u_x(\rho \cos t,\rho \sin t) \rho \cos t \, dt$$ so that $$\left| \int_{\partial B_\rho} u_x \, dy \right| \le 2\pi M \rho$$ where $M = \sup |u_x|$. The other integral satisfies a similar estimate.