I am trying to show that the limit of $\int_{\partial B_\rho} u_x dy -u_y dx =0$ as $\rho \rightarrow 0$, where $\partial B_\rho$ is the boundary of a circle of radius $\rho$ centred at the origin, and $u$ is $C^1$.
I parametrized the path as $\gamma_\rho(t)=(\rho \cos t, \rho \sin t)$, and this gives me the integral as $\int u_x \rho \cos t + u_y \rho \sin t dt$. I am thinking that to show the integral goes to $0$, I should show the absolute value of this is less that $\rho$, or $| \int u_x \cos t+ u_y \sin t dt | \le 1$, which I can write as $ \int \nabla u • (\cos t, \sin t) dt \le 1 $. I am stuck here though with how to prove this.
Note: I do not think I’m able to use Green’s theorem, since u is only $C^1$ so $u_x,u_y$ isn’t necessarily $C^1$.
Any help is greatly appreciated.