= 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 becomes
$$
\frac d{ds}\left(\frac F{\sqrt G}\right)=\frac{G_u}{2G},\qquad \frac1{\sqrt G}\left(\frac F{\sqrt G}\right)_v=\frac{G_u}{2G}.
$$
Thus $(F/\sqrt G)_v=G_u/(2\sqrt G)=(\sqrt G)_u$. 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
$$
\boxed{(F/\sqrt G)_v=(\sqrt G)_u,\qquad(F/\sqrt E)_u=(\sqrt E)_v.}
$$
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