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Kronecer delta tensor transformation

  1. Apr 20, 2010 #1

    Matterwave

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    1. The problem statement, all variables and given/known data
    Show that [tex]\delta_a^b[/tex] is a in fact a mixed tensor of valence (1,1).


    2. Relevant equations

    Definition of a (1,1) tensor:

    [tex]\delta'_a^b=\frac{\partial x'^b}{\partial x^c}\frac{\partial x^d}{\partial x'^a}\delta_c^d[/tex]


    3. The attempt at a solution

    So, I just explicitly put back the summations and I get a sum over c from 1 to 4 and a sum over d from 1 to 4 of the above expression. I did the sum over c first and got:

    [tex]\sum_{d=1}^4 (\delta_1^d\frac{\partial x'^b}{\partial x^1}\frac{\partial x^d}{\partial x'^a}+...)[/tex]

    Where the ... had the [tex]\delta_2^d[/tex] terms etc. I did the summation over d, and I seem to be getting:

    [tex]\frac{\partial x'^b}{\partial x^1}\frac{\partial x^1}{\partial x'^a}+\frac{\partial x'^b}{\partial x^2}\frac{\partial x^2}{\partial x'^a}+\frac{\partial x'^b}{\partial x^3}\frac{\partial x^3}{\partial x'^a}+\frac{\partial x'^b}{\partial x^4}\frac{\partial x^4}{\partial x'^a}[/tex]

    Which equals:
    [tex]4\frac{\partial x'^b}{\partial x'^a}[/tex]

    I get the 4 because there's 4 instances where the kronecker deltas don't equal 0 (when d=1 the first term remains, when d=2 the second term remains, etc)...but I really shouldn't have that 4 there. What happened? =(
     
  2. jcsd
  3. Apr 20, 2010 #2

    gabbagabbahey

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    This should look a lot like the chain rule to you :wink:

    [itex]x'^b[/itex] can be thought of as being a function of [itex]x_1[/itex], [itex]x_2[/itex], [itex]x_3[/itex], and [itex]x_4[/itex]; and each of those coordinates can be thought of as being a function of [itex]x'^a[/itex] (as well as the other 3 primed coordinates, but it's a partial derivative so that doesn't matter)

    [tex]\frac{\partial x'^b}{\partial x^1}\frac{\partial x^1}{\partial x'^a}\neq\frac{\partial x'^b}{\partial x'^a}[/tex]
     
  4. Apr 20, 2010 #3

    Matterwave

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    Ok...but then I don't see how to do that summation then...>_>

    So, is it true that:


    [tex]\frac{\partial x'^b}{\partial x^1}\frac{\partial x^1}{\partial x'^a}+\frac{\partial x'^b}{\partial x^2}\frac{\partial x^2}{\partial x'^a}+\frac{\partial x'^b}{\partial x^3}\frac{\partial x^3}{\partial x'^a}+\frac{\partial x'^b}{\partial x^4}\frac{\partial x^4}{\partial x'^a}=\frac{\partial x'^b}{\partial x'^a}[/tex]

    ?

    I don't see how that can be true...yet if I've done the steps right, that must be true...oh wait, is that just the reverse chain rule? If so, shouldn't it then be changed to a total derivative or something?
     
  5. Apr 20, 2010 #4

    gabbagabbahey

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    Yes, this is the chain rule.

    [tex]\frac{\partial}{\partial x'^a} x'^b(x^1,x^2,x^3,x^4)=\frac{\partial x'^b}{\partial x^1}\frac{\partial x^1}{\partial x'^a}+\frac{\partial x'^b}{\partial x^2}\frac{\partial x^2}{\partial x'^a}+\frac{\partial x'^b}{\partial x^3}\frac{\partial x^3}{\partial x'^a}+\frac{\partial x'^b}{\partial x^4}\frac{\partial x^4}{\partial x'^a}[/tex]
     
  6. Apr 20, 2010 #5

    Matterwave

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    Right, I looked it up on Wiki, and found the same. Thanks! =D

    (looks like my astro-induced habit of canceling out differentials has backfired this time).
     
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