- #1
AxiomOfChoice
- 533
- 1
I'm trying to simplify the expression
[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.
[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.