- #1
lylos
- 79
- 0
Homework Statement
Prove the following:
[tex](\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}[/tex]
Homework Equations
[tex](\hat{e_i}\times\hat{e_j})=\epsilon_{ijk}[/tex]
[tex](\hat{e_i}\cdot\hat{e_j})=\delta_{ij}[/tex]
The Attempt at a Solution
[tex](r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmn})(\hat{e_k}\cdot\hat{e_n})[/tex]
[tex](r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmn}\delta_{kn})[/tex]
[tex](r_i\nabla_j\epsilon_{ijk}r_l\nabla_m\epsilon_{lmk})[/tex]
[tex](r_i\nabla_jr_l\nabla_m)(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})[/tex]
[tex](r_i\nabla_jr_i\nabla_j)-(r_i\nabla_ir_j\nabla_j)[/tex]
At this point, I'm lost. Does the gradient operator work on all terms, should I rearrange?