Let $S$ be a surface with Riemannian metric having first fundamental form $d u^{2}+G(u, v) d v^{2}$. State a formula for the Gauss curvature $K$ of $S$.

Suppose that $S$ is flat, so $K$ vanishes identically, and that $u=0$ is a geodesic on $S$ when parametrised by arc-length. Using the geodesic equations, or otherwise, prove that $G(u, v) \equiv 1$, i.e. $S$ is locally isometric to a plane.