(i) What are geodesic polar coordinates at a point $P$ on a surface $M$ with a Riemannian metric $d s^{2}$ ?

Assume that

$$d s^{2}=d r^{2}+H(r, \theta)^{2} d \theta^{2}$$

for geodesic polar coordinates $r, \theta$ and some function $H$. What can you say about $H$ and $d H / d r$ at $r=0$ ?

(ii) Given that the Gaussian curvature $K$ may be computed by the formula $K=-H^{-1} \partial^{2} H / \partial r^{2}$, show that for small $R$ the area of the geodesic disc of radius $R$ centred at $P$ is

$$\pi R^{2}-(\pi / 12) K R^{4}+a(R),$$

where $a(R)$ is a function satisfying $\lim _{R \rightarrow 0} a(R) / R^{4}=0$.