# A good reference sheet/manual about Einstein index notation?

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• Gan_HOPE326
In summary: I think I got the gist of it from the primer I read, but I'll be sure to check back with more detail when I have a chance. In summary, a reference sheet would be helpful to have if you want to understand more complex cases involving derivatives and integrals.
Gan_HOPE326
I'm not used to Einstein notation and I'm struggling a bit with the more complex examples of it. I got the general gist of it and can follow the basic cases but get sometimes a bit lost when there are a lot of indexes and calculus is involved. All primers I've found online for now only give the basic rules - sum over repeated indices, Kronecker delta, Levi-Civita symbol, and that's it. Is there some good 'reference sheet' I could use to look up more sophisticated cases like those involving derivation rules? I realize they can easily be derived from considering the represented sum, and in some cases I manage to do that, but the convenience of the notation should be to avoid having to write sums explicitly all the time, and knowing the rule beforehand I could at least work out why and how it applies and then trust it going forward. Thanks!

Can you give an example of the type of relation that you would like to have a reference for?

Orodruin said:
Can you give an example of the type of relation that you would like to have a reference for?

Mostly derivatives. I struggled quite a bit some days ago with understanding how you got from the relativistic EM Lagrangian

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

to the kernel of the action integral used in QFT

$$\frac{1}{2}A_\mu(\partial^2g^{\mu\nu}-\partial^\mu\partial^\nu)A_\nu$$

Part of this was because of not realising an integration by parts was happening in the process (I actually made a previous thread about it) but part of it was confusion about the meaning for example of ##\partial^2##, whether it was meant to represent ##\partial_\mu\partial_\mu## or ##\partial_\mu\partial^\mu##. Similarly today I ran into a case in which a derivative of a product of indexed quantities gives an additional factor of 2 - which is pretty obvious when carrying out the sum, but I would have probably missed if I didn't expand, and for more complex expressions that might become annoying (luckily for me, this one was simply a toy model of GR in 1+1 spacetime, so not many indices).

I guess what I'd hope for is some cheatsheet especially for derivation and integration. Which substitutions are legitimate to carry out, which prefactors appear and such. I imagine most people get this kind of knowledge through doing exercises in their relativity course, but unfortunately since I'm working on this on my own I don't get that luxury, and theory books I put my hands on tend to skim over all this. In alternative, a good reference for exercises with solutions I can carry out to learn more the basics and feel more confident with it would do the trick as well I guess.

Gan_HOPE326 said:
whether it was meant to represent ##\partial_\mu\partial_\mu## or ##\partial_\mu\partial^\mu##.
Well, ##\partial_\mu\partial_\mu## doesn't make sense because you're not allowed to repeat an index unless one is "downstairs" and the other is "upstairs".

DrGreg said:
Well, ##\partial_\mu\partial_\mu## doesn't make sense because you're not allowed to repeat an index unless one is "downstairs" and the other is "upstairs".

Fair enough, yeah.

## 1. What is Einstein index notation?

Einstein index notation, also known as Einstein summation convention, is a mathematical notation used to represent and manipulate tensors in a concise manner. It involves using Greek indices to represent repeated indices in a tensor equation, which simplifies the notation and makes it easier to perform calculations.

## 2. Why is Einstein index notation useful?

Einstein index notation is useful because it allows for a more compact representation of tensor equations. This makes it easier to write and understand complex equations, as well as perform calculations involving tensors. It also follows the principle of covariance, which means that the notation is independent of the coordinate system used.

## 3. How do you read Einstein index notation?

Einstein index notation is read by first identifying the indices that are repeated in an equation. These indices are then summed over all possible values, which is represented by the sigma symbol (∑). For example, the notation AijBij is read as the sum of all products of elements in the ith row and jth column of matrices A and B.

## 4. Can Einstein index notation be used for any type of tensor?

Yes, Einstein index notation can be used for any type of tensor, including vectors, matrices, and higher-order tensors. It is a general notation that can be applied to various mathematical and physical concepts, such as relativity, electromagnetism, and fluid dynamics.

## 5. Where can I find a good reference sheet/manual about Einstein index notation?

There are many online resources available for learning about Einstein index notation, including reference sheets and manuals. Some recommended sources include university websites, physics textbooks, and online educational platforms such as Khan Academy and Coursera.

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