Show that $\nabla^{2}\left ( x_{i}x_{j}r^{n} \right ) = 2\delta _{ij}r^{n}+n\left ( n+5 \right )x_{i}x_{j}r^{n-2}$

Question

No matter what I can't seem to arrive at this answer. I've tried $\partial_{i}\left (\partial_{i}\left ( x_{i}x_{j}r^{n} \right ) \right )$ and $\delta _{ij}\partial_{i}\left (\partial_{j}\left ( x_{i}x_{j}r^{n} \right ) \right )$ and I never manage to get the final answer.

Can someone help?

Thanks!

Answer 1

First recall $\partial_i r=x_ir^{-1}$, so $\partial_ir^m=mx_ir^{m-2}$. This will come in handy in our calculation. I'll also assume you are working with $\mathbb{R}^3$, otherwise $\delta_{kk}$ will not be $3$.

We compute $$\require{cancel} \begin{align*} \nabla^2(x_ix_jr^n) &=\partial_k\partial_kx_ix_jr^n\\ &=\partial_k((\partial_kx_i)x_jr^n+x_i(\partial_kx_j)r^n+x_ix_j(\partial_kr^n))\\ &=\partial_k(\delta_{ik}x_jr^n+\delta_{jk}x_ir^n+nx_ix_jx_kr^{n-2})\\ &=\partial_ix_jr^n+\partial_jx_ir^n+n\partial_kx_ix_jx_kr^{n-2}\\ &=\delta_{ij}r^n+x_j\partial_ir^n+\delta_{ji}r^n+x_i\partial_jr^n\\ &\quad+n\left[\delta_{ik}x_jx_kr^{n-2}+x_i\delta_{jk}x_kr^{n-2}+x_ix_j\cancelto{3}{\color{red}{\delta_{kk}}}r^{n-2}+x_ix_jx_k\partial_kr^{n-2}\right]\\ &=\delta_{ij}r^n+nx_ix_jr^{n-2}+\delta_{ij}r^n+nx_ix_jr^{n-2}\\ &\quad+n\left[x_ix_jr^{n-2}+x_ix_jr^{n-2}+3x_ix_jr^{n-2}+(n-2)x_ix_j\cancelto{r^2}{\color{red}{x_kx_k}}r^{n-4}\right]\\ &=2\delta_{ij}r^n+n(n+5)x_ix_jr^{n-2}\\ \end{align*} $$

Answer 2

We have \begin{align*} \partial_k(x_ix_j r^n) &= \delta_{ik} x_j r^n + x_i \delta_{jk} r^n + x_i x_j nr^{n-1} \delta_k r\\ &= \delta_{ik} x_j r^n + x_i \delta_{jk} r^n + x_i x_j nr^{n-2}x_k \end{align*} as $\delta_k r = x_k r^{-1}$. We get from that \begin{align*} \delta_\ell \delta_k (x_i x_j r^n) &= \delta_\ell (\delta_{ik} x_j r^n + x_i \delta_{jk} r^n + x_i x_j nr^{n-2}x_k )\\ &= \delta_{ik}\delta_{j\ell} r^n + \delta_{ik}x_j x_\ell nr^{n-2} + \delta_{i\ell}\delta_{jk}r^n + \delta_{jk}x_ix_\ell nr^{n-2} + \delta_{i\ell} x_jx_k nr^{n-2} + \delta_{j\ell}x_ix_k nr^{n-2}\\ & \quad{} + \delta_{k\ell}x_ix_j nr^{n-2} + x_ix_jx_kx_\ell n(n-2)r^{n-4} \end{align*} Hence \begin{align*} \delta^{\ell k}\partial_\ell\partial_k(x_ix_jr^n) &= \delta_{ij} r^n +x_ix_jnr^{n-2} +\delta_{ij}r^n + 3x_ix_jnr^{n-2} + n d x_ix_j r^{n-2} + n(n-2)x_j x_jr^{n-2}\\ &= 2\delta_{ij} r^n + n(n+2+d)x_ix_j r^{n-2} \end{align*} If $d = 3$ (the dimension of the space in question), we are done.