Einstein summation notation for magnetic dipole field

In summary: However, if you're looking for a more general introduction to the product rule and differentiation, I'd recommend looking into a calculus textbook. In summary, the student is struggling to do a derivation of the product rule for derivatives using Einstein summation notation. He is asking for help finding online resources that would provide more written out examples of such derivations.
  • #1
mmpstudent
16
0
I can do this derivation the old fashioned way, but am having trouble doing it with einstein summation notation.

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

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

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

but I don't really know how to get to that step
 
Last edited:
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  • #2
You may be interested in the The LaTeX guide for the forum. :smile: Link.
 
  • #3
Fredrik said:
You may be interested in the The LaTeX guide for the forum. :smile: Link.

You were too fast. Was trying to get it to work just needed to delete the spaces in brackets I guess.
 
  • #4
My first thought is that he's using the product rule for derivatives to evaluate ##\partial_j## acting on a product.
 
  • #5
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.
 
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  • #6
O wow, thanks. that makes much more sense now.
 
  • #7
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
mmpstudent said:
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
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).
 

1. What is Einstein summation notation for magnetic dipole field?

Einstein summation notation is a shorthand notation used to represent vector and tensor equations in physics. It is used to simplify and condense lengthy equations by representing repeated indices as a summation over all possible values.

2. How is Einstein summation notation used in the context of magnetic dipole field?

In the context of magnetic dipole field, Einstein summation notation is used to represent the components of the magnetic dipole moment, which is a vector quantity. This notation is particularly useful when dealing with multiple magnetic dipoles in a system.

3. What are the benefits of using Einstein summation notation for magnetic dipole field?

The main benefit of using Einstein summation notation for magnetic dipole field is that it simplifies and condenses equations, making them easier to read and work with. It also allows for more efficient calculations and reduces the chances of making errors.

4. Are there any limitations to using Einstein summation notation for magnetic dipole field?

One limitation of using Einstein summation notation for magnetic dipole field is that it may be difficult for beginners to understand and may require some practice to become familiar with. Additionally, it may not be suitable for all types of equations and may not always provide the most intuitive representation of the physical quantities involved.

5. Are there any other notations used to represent magnetic dipole field?

Yes, there are other notations used to represent magnetic dipole field, such as vector notation and index notation. However, Einstein summation notation is a widely used and accepted notation in the field of physics, particularly in the context of tensors and vector equations.

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