Chapter 14.11, Problem 35MP

Show that the Cauchy-Riemann equations [see (2.2) and Problem 2.46] in a general orthogonal curvilinear coordinate system [see Chapter $10,\text{ Sections }8\text{ and }9]$ are $\frac{1}{h_1}\frac{\partial u}{\partial x_1}=\frac{1}{h_2}\frac{\partial v}{\partial x_2},\frac{1}{h_1}\frac{\partial v}{\partial x_1}=-\frac{1}{h_2}\frac{\partial u}{\partial x_2}$

where, as in Chapter 10, the variables are $x_1,x_2$ and the scale factors are $h_1,h_2$.

Hint: Consider the directional derivatives (Chapter 6, Section 6 ) in two perpendicular directions. (Compare Problem 2.46.) Also show that $u$ and $v$ satisfy Laplace’s equation, Chapter 10, equation (9.10) (drop the $x_3$ term and set $h_3=1$ ).