= 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>:
$$
\boxed{f_{xx}=c^2f_{uu}-\frac{2sc}{u}f_{uv}+\frac{s^2}{u^2}f_{vv}+\frac{s^2}{u}f_u+\frac{2sc}{u^2}f_v,}
$$
$$
\boxed{f_{yy}=s^2f_{uu}+\frac{2sc}{u}f_{uv}+\frac{c^2}{u^2}f_{vv}+\frac{c^2}{u}f_u-\frac{2sc}{u^2}f_v.}
$$
In particular their sum is the <Laplacian in polar coordinates>, $f_{uu}+f_u/u+f_{vv}/u^2$.
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