- #1

- 533

- 1

[tex]

(\hat{r} \times \vec{\nabla}) \times \hat{r},

[/tex]

where [itex]\hat{r}[/itex] is the radial unit vector, using index notation. I think I'm right to write this as:

[tex]

((\hat{r} \times \vec{\nabla}) \times \hat{r})_i = \varepsilon_{ijk}(\varepsilon_{jmn}r_m\partial_n)r_k.

[/tex]

But when I employ the contraction

[tex]

\varepsilon_{ijk}\varepsilon_{jmn} = \delta_{im}\delta_{kn} - \delta_{in}\delta_{km}

[/tex]

and simplify, what I wind up with is this:

[tex]

r_i \partial_k r_k - r_k\partial_ir_k.

[/tex]

I'm thinking that this first term becomes [itex]\hat{r} (\nabla \cdot \hat{r})[/itex]...is that right? And what about the second term? I'm kind of clueless as to what to do with that.

I might have made other mistakes here, though, so I'd appreciate someone pointing them out. Thanks.