= Solution
{c}
The <multivariable chain rule> first gives
$$
g_u=uf_x+vf_y,\qquad g_v=-vf_x+uf_y.
$$
Taking a second <partial derivative> in each variable, with the <partial derivatives> of $f$ evaluated at $(x(u,v),y(u,v))$, gives
$$
\begin{aligned}
g_{uu}&=f_x+u^2f_{xx}+uv(f_{xy}+f_{yx})+v^2f_{yy},\\
g_{vv}&=-f_x+v^2f_{xx}-uv(f_{xy}+f_{yx})+u^2f_{yy}.
\end{aligned}
$$
The <mixed partial derivatives> cancel even without using their equality. Hence the <Laplacians> satisfy
$$
\boxed{g_{uu}+g_{vv}=(u^2+v^2)(f_{xx}+f_{yy}).}
$$
Away from the origin the factor $u^2+v^2$ is positive, so \b[$g$ is a <harmonic function> if and only if $f$ is a <harmonic function>] on the corresponding local coordinate patch. This is the <Laplacian in planar parabolic coordinates>.
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