Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 11 / c / Solution

For polar coordinates with $u>0$, ${\partial }_x=\cos v{\partial }_u-(\sin v/u){\partial }_v$ and ${\partial }_y=\sin v{\partial }_u+(\cos v/u){\partial }_v$. Write $c=\cos v$, $s=\sin v$. Applying these operators twice, including their variable coefficients, gives the Cartesian second derivatives in polar coordinates:In particular their sum is the Laplacian in polar coordinates, $f_{uu}+f_u/u+f_{vv}/u^2$.