Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, $z\le0$.
$$\iint_S \nabla\times\mathbf{F}\cdot\mathbf{dS} = -2\pi.$$
So I want to calculate the integral
$$\oint_S \mathbf{F}\cdot\mathbf{dr}$$
But I am having trouble finding what line to integrate over. Any tips?