Question about what index notation is telling me

[AxiomOfChoice]

I'm trying to simplify the expression

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

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

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

But when I employ the contraction

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

and simplify, what I wind up with is this:

$$r_i \partial_k r_k - r_k\partial_ir_k.$$

I'm thinking that this first term becomes $\hat{r} (\nabla \cdot \hat{r})$...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]

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

$$\hat r = \frac{\vec r}{r} = \frac{x_i}{r} {\vec e_i}$$

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

$$r(x,y,z) = \sqrt{x^2 + y^2 + z^2}$$