= Laplacian in planar parabolic coordinates
{title2=$\Delta_{xy}f=(g_{uu}+g_{vv})/(u^2+v^2)$}
For <planar parabolic coordinates> and $g(u,v)=f(x(u,v),y(u,v))$, the <multivariable chain rule> gives $g_u=uf_x+vf_y$ and $g_v=-vf_x+uf_y$. Differentiating once more and adding cancels the first <partial derivatives> and the <mixed partial derivatives>:
$$
g_{uu}+g_{vv}=(u^2+v^2)(f_{xx}+f_{yy}).
$$
Consequently, on every local coordinate patch away from the origin, $f$ is a <harmonic function> if and only if $g$ is a <harmonic function>. The <Laplacian> changes by a positive scale factor, so solutions of the <Laplace equation> are preserved. The restriction away from the origin is needed to divide by the <Jacobian determinant>.
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