How to Prove Vector Calculus Identity Involving Cross Product and Gradient?

[lylos]

Homework Statement

Prove the following:
$$(\vec{r}\times\nabla)\cdot(\vec{r}\times\nabla)=r^2\nabla^2-r^2 \frac{\partial^2}{\partial r^2}-2r\frac{\partial}{\partial r}$$

Homework Equations

$$(\hat{e_i}\times\hat{e_j})=\epsilon_{ijk}$$
$$(\hat{e_i}\cdot\hat{e_j})=\delta_{ij}$$

The Attempt at a Solution

$$(r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmn})(\hat{e_k}\cdot\hat{e_n})$$
$$(r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmn}\delta_{kn})$$
$$(r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmk})$$
$$(r_i\nabla_jr_l\nabla_m)(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})$$
$$(r_i\nabla_jr_i\nabla_j)-(r_i\nabla_ir_j\nabla_j)$$

At this point, I'm lost. Does the gradient operator work on all terms, should I rearrange?

 

[hunt_mat]

One of your equations is wrong,
$$(\hat{e_i}\times\hat{e_j})=\epsilon_{ijk}\hat{e}_{k}$$
I personally would write:
$$\mathbf{r}=x_{i}\hat{e}_{i},\quad\nabla =\hat{e}_{i}\partial_{i}$$
Hopefully this should help.

Mat