I read on Wikipedia about that the Gauss-Bonnet theorem for compact two-dimensional Riemannian manifolds $M$ with boundary $\partial M$ with Gaussian curvature $K$ of $M$ and geodesic curvature $k_g$ of $\partial M$ states:
$$ \int_M Kd\sigma+\int_{\partial M}k_g ds=2\pi\chi(M) $$ where $d\sigma$ and $ds$ are volume forms and $\chi(M)$ the Euler characterstic of $M$. Does anyone have a references in which this proven?