- #1

docnet

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- 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?