Past exam of the geometry course of the University of Oxford / 2011 / 3 / a / Solution

For a regular parametrization $\mathbf{r}(u,v)$, the first fundamental form is the induced Riemannian metricIt measures squared lengths of tangent vectors and therefore lengths of curves. In the plane put $x=\rho \cos \theta$, $y=\rho \sin \theta$. The derivatives are orthogonal, of lengths one and $\rho$, soThese polar coordinates are regular for $\rho >0$.