A good reference sheet/manual about Einstein index notation?

[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!

 

[Orodruin]

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

 

[Gan_HOPE326]

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.

 

[DrGreg]

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".

 

[Gan_HOPE326]

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.