Prove the rotational invariance of the Laplace operator

[docnet]

Homework Statement

prove $\Delta$ is rotation invariant.

Relevant Equations

$\Delta$

Hello, please lend me your wisdom.

$\Delta u=\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u$

$Rx=\left<r_{11}x_1+...r_{1n}x_n+...+r_{n1}x_1+...+r_{nn}x_n\right>$

$(\Delta u)(Rx)=(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n, ...,r_{n1}x_1+...+r_{nn}x_n\right>$


$u\circ R=$

$\begin{pmatrix} (u)r_{11} & ... & (u)r_{1n} \\ ... & ... & ... \\ (u)r_{n1} & ... & (u)r_{nn} \end{pmatrix}$

$u\circ R x=\left<(u)r_{11}x_1+...(u)r_{1n}x_n+...+(u)r_{n1}x_1+...+(u)r_{nn}x_n\right>$

$\Delta u\circ R x= (\partial_{x1}^2+\partial_{x2}^2+...+\partial_{xn}^2)\left<(u)r_{11}x_1+...(u)r_{1n}x_n+...+(u)r_{n1}x_1+...+(u)r_{nn}x_n\right>=$

$(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n, ...,r_{n1}x_1+...+r_{nn}x_n\right>=(\Delta u)(Rx)$

I think $u\circ R$ does not mean $(u) (R)$ and I just showed the same calculation twice. How do I use the information $R^{-1}=R^T$ to prove this case?

 

[anuttarasammyak]

Let us take z-axis as axis of rotation of angle $\alpha$ and take cylindrical coordinates or spherical coordinates. For the rotation
$$\phi'=\phi + \alpha$$
, we get rotation invariance of Laplacian because
$$\frac{\partial^2}{\partial \phi^2}=\frac{\partial^2}{\partial \phi'^2}$$