Cartesian second derivatives in polar coordinates

For $x=r\cos \theta$, $y=r\sin \theta$ and $r>0$, set $c=\cos \theta$, $s=\sin \theta$. The differential operators are ${\partial }_x=c{\partial }_r-(s/r){\partial }_{\theta }$ and ${\partial }_y=s{\partial }_r+(c/r){\partial }_{\theta }$. Applying them twice givesTheir sum is the Laplacian in polar coordinates. The first-derivative terms come from differentiating the coordinate coefficients, as required for second derivatives under a two-variable coordinate change. The origin is excluded from the coordinate formulas, even when the underlying Cartesian function is smooth there.