Problem
Problem 6.41 Let $X_{1}$ and $X_{2}$ be the coordinate vector fields for a set of orthogonal coordinates on a surface. Prove that there are isothermal coordinates (also called conformal coordinates) with the same domain of definition and the same coordinate curves (as images) if and only if
$$
X_{2} X_{1}\left(\log \frac{g_{11}}{g_{22}}\right)=0
$$
where $g=\sum_{i, j=1}^{2} g_{i j} \mathrm{~d} x^{i} \otimes \mathrm{d} x^{j}$ is the metric.