= Rotational invariance of the Laplacian
{title2=$\Delta_x(u\circ R)=(\Delta u)\circ R$}
For a constant <orthogonal matrix> $R$, the <Laplacian> is unchanged by the coordinate transformation $\xi=Rx$. If $u$ is twice continuously differentiable, the <chain rule> gives
$$
\sum_i\frac{\partial^2u}{\partial x_i^2}=\sum_{i,j,k}R_{ji}R_{ki}\frac{\partial^2u}{\partial\xi_j\partial\xi_k}=\sum_j\frac{\partial^2u}{\partial\xi_j^2},
$$
because $\sum_iR_{ji}R_{ki}=\delta_{jk}$ by <orthogonality> of the rows. Here $\delta_{jk}$ is the <Kronecker delta>. In two dimensions, the <rotation matrix> $\begin{pmatrix}a&-b\\b&a\end{pmatrix}$ with $a^2+b^2=1$ makes the mixed <partial derivatives> cancel explicitly. The same argument works for reflections and translations, so the <Laplacian> depends on the <Euclidean geometry> rather than the orientation or origin of the coordinates.
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