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Rank 3 tensor created by taking the derivative of electromagnetic field tensor

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the rank 3 tensor [tex]S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta}[/tex] is completely antisymmetric.

    I just don't feel comfortable doing this stuff at all. Each of the three terms seems like they should be exactly the same to me. Could someone show me how I would start doing something like this please? Furthermore, if this is a rank 3 tensor, what would it mean if this tensor equals 0?

    Thanks. :-\
     
  2. jcsd
  3. Oct 20, 2009 #2

    gabbagabbahey

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    If [itex]S_{\alpha\beta\gamma}[/itex] is completely antisymmetric, then

    [tex]S_{\alpha\beta\gamma}=-S_{\alpha\gamma\beta}[/tex]
    [tex]S_{\alpha\beta\gamma}=-S_{\beta\alpha\gamma}[/tex]

    and

    [tex]S_{\alpha\beta\gamma}=-S_{\gamma\beta\alpha}[/tex]

    That is, [itex]S_{\alpha\beta\gamma}[/itex] is antisymmetric on all pairs of indices....

    So, start by comparing [itex]S_{\alpha\beta\gamma}[/itex] to [itex]S_{\alpha\gamma\beta}[/itex] using the definition you posted...
     
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