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Need help with index notation.

  1. Sep 17, 2010 #1
    1. The problem statement, all variables and given/known data
    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]

    2. Relevant equations
    [tex](\hat{e_i}\times\hat{e_j})=\epsilon_{ijk}[/tex]
    [tex](\hat{e_i}\cdot\hat{e_j})=\delta_{ij}[/tex]


    3. 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?
     
  2. jcsd
  3. Sep 17, 2010 #2

    hunt_mat

    User Avatar
    Homework Helper

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

    Mat
     
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