I just started learning index notation and I'm having some trouble understand what I'm allowed to do with it.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Tensor/Index notation

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