Question about what index notation is telling me

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.
 

Ben Niehoff

Science Advisor
Gold Member
1,864
157
I would expand r in its Cartesian components first; otherwise you'll have to look up the correct formula for the curl in spherical coordinates, and things could get messy. So write

[tex]\hat r = \frac{\vec r}{r} = \frac{x_i}{r} {\vec e_i}[/tex]

Just remember that r (the radial length) is actually a function of x, y, and z:

[tex]r(x,y,z) = \sqrt{x^2 + y^2 + z^2}[/tex]
 

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