Solution
= 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 $V_\theta=-rv_x\sin\theta+rv_y\cos\theta$. Substituting the Cartesian <Cauchy-Riemann equations> gives $V_\theta=rU_r$. Similarly $V_r=v_x\cos\theta+v_y\sin\theta=-U_\theta/r$. Thus, for $r>0$, the polar equations are
$$
\boxed{U_r=\frac1rV_\theta,\qquad V_r=-\frac1rU_\theta.}
$$
The restriction $r>0$ is needed because the polar chart degenerates at the origin.