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Tensor/Index notation

  1. Jul 14, 2015 #1
    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!
  2. jcsd
  3. Jul 15, 2015 #2


    Staff: Mentor

    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:


    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:

    Last edited: Jul 15, 2015
  4. Jul 15, 2015 #3
    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).
  5. Jul 15, 2015 #4


    Staff: Mentor

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