= Solution
Write $c=\cos\theta$, $s=\sin\theta$. Since $\theta$ is constant, the <multivariable chain rule> gives $\partial_x=c\partial_u-s\partial_v$ and $\partial_y=s\partial_u+c\partial_v$. Applying these operators twice to a twice continuously differentiable <function> gives
$$
g_{xx}=c^2g_{uu}-2csg_{uv}+s^2g_{vv},\qquad g_{yy}=s^2g_{uu}+2csg_{uv}+c^2g_{vv}.
$$
The mixed terms cancel and $c^2+s^2=1$, so
$$
\boxed{g_{xx}+g_{yy}=g_{uu}+g_{vv}.}
$$
This is <rotational invariance of the Laplacian>: an <orthogonal change of coordinates> preserves the <trace> of the <Hessian matrix>. A nonconstant angle would introduce additional <derivatives> and is not covered by this constant rotation calculation.
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