Index Gymnastics Genral

>eta
In the way it's written, paying some heed to up/down indices, it's written as if you'd be working with a space equipped with a metric to move between the vector space and its dual. Briefly, it's used to raise/lower indices, but the upstairs/downstairs (contravariant, covariant, or mixed indices) Kronecker deltas are exactly the same anyway.
>f, other indices out of nowhere
If you're working in Einstein notation, which this is (please Google it if unfamiliar, it's pretty straightforward), repeated indices are dummy indices--so it could be replaced with any variable you wish, as these are summed over and not actual indices of your final expression.
>rules with upper/lower
This explains the previous confusion, upper and lower matter if you're working with a metric. If you're not sure what a metric is, it's essentially a way to relate a vector with upstairs indices to a mathematical objects containing the same information with downstairs indices. Please refer to Einstein notation again, I think this clears up the confusion with kronecker delters eliminating variables (indices?).

>index forms for grad, etc.
They're straightforward to understand/work out yourself once you see the rules. The epsilon symbols you're seeing are referred to as Levi-Civita symbols (sometimes permutation symbol, antisymmetric symbol) and have properties closely related to the definition of determinants and cross products. The other index notations will be obvious once you spend some time working with it.

You'll do well to look up Einstein notation and maybe some older mechanics books that have a section on it.

I'm not 100% sure what exactly the context for this stuff is but I might be able to help.

>eta a variable?
Eta could refer to the metric tensor. That's usually what we reserve it for. It is not a variable though

>f out of nowhere
I think they may have meant for that to be an "l" but you should really be able to follow along at that step to see what's happening.

>kronecker delta
that's literally what it does

>help with the indexing for the diff operators
I'm pretty sure the book Div Grad Curl and All That goes into this notation. I learned it before reading the book so I don't remember.

>>7806059
>Eta could refer to the metric tensor.

Specifically, eta usually means the Minkowski metric tensor.


>>7805995
Is you just want to learn this type of index stuff, go through a relativity course. Probably the best place to pick it up.
http://theoreticalminimum.com/courses/special-relativity-and-electrodynamics/2012/spring