Below is a problem which states a fact about "Tchebyshef net". I don't understand meaning of bolded part.
The coordinate curves of a parametrization $x(u, v)$ constitute a Tchebyshef net if the lengths of the opposite sides of any quadrilateral formed by them are equal. Show that a necessary and sufficient condition for this is $$\frac{\partial E}{\partial v} =\frac{\partial G}{\partial u}=0.$$
Reference: Differential Geometry of Curves and Surfaces [Manfredo P.do carmo] Page 100 Problem 7.
Best Answer
The definition is equivalent to
two curves $x(u_1,v)$, $x(u_2,v)$ determine segments of equal lengths on all curves $x(u,\mathrm{const})$.
two curves $x(u,v_1)$, $x(u,v_2)$ determine segments of equal lengths on all curves $x(\mathrm{const},v)$.
For statement 1, it means that, for any constant $v$, the curve $\alpha:(u_1,u_2)\to \mathbb{R}^3$ given by $\alpha(u)=x(u,v)$ has the same length on the surface. That is$$\int_{u_1}^{u_2} \sqrt{E(u,v)} \, du =\mathrm{const} \quad \forall \, \mathrm{const} \, v $$ Differentiate above equation with respect to $v$, we get $$\int_{u_1}^{u_2} \partial_v\sqrt{E(u,v)} \, du =0 \quad \forall \, \mathrm{const} \, v $$
Hence we get $\frac{\partial E}{\partial v}=0$. Similarly, we can get the second condition.