# Einstein summation notation for magnetic dipole field

1. Apr 30, 2013

### mmpstudent

I can do this derivation the old fashioned way, but am having trouble doing it with einstein summation notation.

Since $\vec{B}=\nabla \times \vec{a}$
$\vec{B}=\mu_{0}/4\pi (\nabla \times (m \times r)r^{-3}))$
$4\pi \vec{B}/\mu_{0}=\epsilon_{ijk} \nabla_{j}(\epsilon_{klm} m_{l} r_{m} r^{-3})$
$=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\partial_{j}m_{l}r_{m}r^{-3}$

here is where I am stumbling. My professor has for the next step

$=m_{l}(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})r^{-3} \delta_{jm}-3 r_{m}\hat{r}_{j}r^{-4})$

but I don't really know how to get to that step

Last edited: Apr 30, 2013
2. Apr 30, 2013

### Fredrik

Staff Emeritus
You may be interested in the The LaTeX guide for the forum. Link.

3. Apr 30, 2013

### mmpstudent

You were too fast. Was trying to get it to work just needed to delete the spaces in brackets I guess.

4. Apr 30, 2013

### Fredrik

Staff Emeritus
My first thought is that he's using the product rule for derivatives to evaluate $\partial_j$ acting on a product.

5. Apr 30, 2013

### WannabeNewton

First off, tell your professor that he is horribly butchering Einstein notation. Seriously, what was written down misses the entire point of the notation. Anyways, $\frac{4\pi}{\mu_{0}}B^{i} = \frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = \epsilon^{ijk}\epsilon_{klm}m^{l}[r^{-3}\partial_{j}r^{m} + r^{m}\partial_{j}(r^{-3})]$ hence $\frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = 2\delta^{[i}_{l}\delta^{j]}_{m}m^{l}[r^{-3}\partial_{j}r^{m} + r^{m}\partial_{j}(r^{-3})]$. Now, $\partial_{j}r^{m} = \delta^{m}_{j}$ and $\partial_{j}(r^{-3}) = -3(-r^i r_{i})^{-5/2}r_{k}\partial_{j}r^{k} = -3r^{-4}\hat{r}_{j}$ giving us $\frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = 2\delta^{[i}_{l}\delta^{j]}_{m}m^{l}[r^{-3}\delta^{m}_{j} -3r^{-4}\hat{r}_{j}r^{m}]$ as desired.

EDIT: By the way, in the above it should be $(r^i r_{i})^{-5/2}$ not $(-r^i r_{i})^{-5/2}$; I've gotten too used to General Relativity xD.

Last edited: Apr 30, 2013
6. Apr 30, 2013

### mmpstudent

O wow, thanks. that makes much more sense now.

7. Apr 30, 2013

### mmpstudent

Do you know of any materials online that would give more written out examples of such derivations with Einstein summation? I just need more practice

8. Apr 30, 2013

### WannabeNewton

I honestly can't think of any online resources off of the top of my head because I got used to the notation when learning special relativity (the text used was Schutz).