Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3 / b / Solution

Use the chain rule with $x=r\cos \theta$, $y=r\sin \theta$. Then $U_r=u_x\cos \theta +u_y\sin \theta$ and $V_{\theta }=-r{v}_x\sin \theta +r{v}_y\cos \theta$. Substituting the Cartesian Cauchy-Riemann equations gives $V_{\theta }=r{U}_r$. Similarly $V_r=v_x\cos \theta +v_y\sin \theta =-U_{\theta }/r$. Thus, for $r>0$, the polar equations areThe restriction $r>0$ is needed because the polar chart degenerates at the origin.