Learning Index Notation: Tensor/Cross Product Confusion

[Incand]

I just started learning index notation and I'm having some trouble understand what I'm allowed to do with it.
For example we can write the $\vec A \times (\vec B \times \vec C)$ as $\varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m$. I understand that $(\vec B \times \vec C) = \varepsilon_{klm}B_lC_m$
but why am I allowed to just throw in another vector and another $\varepsilon$-tensor to get another cross product? How would this write if i write out the summation symbols, do I sum over everything or does the order matter? Like this?
$\sum_{k=1}^3\sum_{j=1}^3\sum_{l=1}^3\sum_{m=1}^3 \varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m$
Does it matter in what order i sum this up? why not? The "tripple cross product" isn't commutative is it, so the order should matter? Why am I allowed to use the identity $\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$ here?

As you can se I'm really confused about pretty much everything having just started out the subject and I can't make much sense of the book I'm using sadly so some help clarifying it for me would be awesome!

 

[jedishrfu]

The only way you're going to understand this is to expand the tensor notation out, simplify the expression (ie remove canceled terms) and compare the result to a known vector identity.

Start with understanding the Levi-Civita symbol:

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Try to prove the $\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

and in your expansions look for terms with permuted indices and recall that $\varepsilon_{123} = \varepsilon_{231} = \varepsilon_{312}$ and that $\varepsilon_{123} = -\varepsilon_{213}$... and lastly when any two indices are equal as for example $\varepsilon_{iik} = 0$.

This will remove some terms and eventually you will be left with a componentized version of the identity.

This works for the vector identities too.

Here's a youtube video discussion on it:

 

[Incand]

Thanks for responding, that video were very usefull! Especially the end when he proved those statements .When I did it myself I wrote out a list of all the possibilities (sometimes eliminating some thanks to symmetry) and then verified them. His way of doing it is a lot cleaner (and faster).

I understand the identity $\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$ and I'm able to prove it (in fact the book I'm using does). What I was mainly confused about was how I was allowed to change order of the terms.
Mainly how I could write
$\varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m = \varepsilon_{ijk}\varepsilon_{klm}A_jB_lC_m$
Is this because everything here is a scalar and therefore I'm allowed to change the order? I guess that's the reason index notation is so powerful.

In that video he seems to make a point out of only swapping two terms in every step (when showing the identities in the last part) but that is just to make it easier to follow I assume? Since if you're allowed to swap any two elements however you want, you could sort the terms in any way you want (thinking about sorting algorithms here).

 

[jedishrfu]

Yes, everything is a scalar and so the ordinary laws of algebra apply.