Chapter 13.5, Problem 53E

Cauchy-Riemann Equations Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann equations

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

can be written in polar coordinate form as

$\frac{\partial u}{\partial r}=\frac{1}{r}·\frac{\partial v}{\partial \theta }$ and $\frac{\partial v}{\partial r}=-\frac{1}{r}·\frac{\partial u}{\partial \theta }$