 Solution

For a coordinate curve $v=v_0$, the length between $u_1$ and $u_2$ iswhere $E$ is a coefficient of the first fundamental form. Thus $\partial E/\partial v=0$ makes the two $u$-directed sides of every coordinate rectangle equally long. Similarly,so $\partial G/\partial u=0$ makes the two $v$-directed sides equally long.

Conversely, suppose every pair of opposite sides has equal length. Fix $v_0,v_1$. Equality of the first pair for every interval $[u_1,u_2]$ givesThe fundamental theorem of calculus implies that the continuous integrand vanishes, so $E$ is independent of $v$. The same argument shows that $G$ is independent of $u$. Hence the stated equal-length property is equivalent to $E_v=G_u=0$.