Take $\Sigma=\mathbb{D}$ to be the unit disk with metric $g=\frac{4}{(1+|z|^2)^2}\,|dz|^2$. If $\phi$ is a nice enough function on $\mathbb{D}$, then I want to compute $$\int_{\partial \Sigma} k_g \phi\,ds_g$$ and $$\int_{\partial\Sigma} \partial_n \phi\,ds_g,$$ where $\partial_n$ is the outer normal derivative, $k$ is the curvature of the boundary and $ds$ is an element of arclength. How exactly does one go about computing these quantities?

This post imported from StackExchange Physics at 2015-10-11 18:30 (UTC), posted by SE-user AndyCee