 Solution

The coordinate curves are assumed to be geodesics as geometric curves; use their arc-length parametrization in the geodesic equations. On $u=c$, choose increasing arc length $s$, so $\dot{u}=0$ and $\dot{v}=1/\sqrt{G}$. The transverse equation becomesThus $(F/\sqrt{G}{)}_v=G_u/(2\sqrt{G})=(\sqrt{G}{)}_u$. On $v=d$, similarly $\dot{v}=0$, $\dot{u}=1/\sqrt{E}$, and the transverse equation gives $(F/\sqrt{E}{)}_u=E_v/(2\sqrt{E})=(\sqrt{E}{)}_v$. Hence the geodesic coordinate grid identities are