Question about what index notation is telling me

[AxiomOfChoice]

I'm trying to simplify the expression

<br /> (\hat{r} \times \vec{\nabla}) \times \hat{r},<br />

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

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

But when I employ the contraction

<br /> \varepsilon_{ijk}\varepsilon_{jmn} = \delta_{im}\delta_{kn} - \delta_{in}\delta_{km}<br />

and simplify, what I wind up with is this:

<br /> r_i \partial_k r_k - r_k\partial_ir_k.<br />

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}