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Tensor Analysis

  1. Feb 25, 2010 #1
    1. The problem statement, all variables and given/known data
    For ease of writing, a covariant tensor [tex]\bf G..[/tex] will be written as [tex]\bf G[/tex] and a,b,c,d are vectors.

    Let [tex]\bf S[/tex] and [tex]\bf G[/tex] be two non-zero symmetric covariant tensors in a four-dimensional vector space. Furthermore, let S and G satisfy the identity:

    [tex][\bf G \otimes \bf S](\vec a, \vec b, \vec c, \vec d) - [\bf G \otimes \bf S](\vec a, \vec d, \vec b, \vec c) + [\bf G \otimes \bf S](\vec b, \vec c, \vec a, \vec d) - [\bf G \otimes \bf S](\vec c, \vec d, \vec a, \vec b) \equiv 0[/tex]

    for all a,b,c,d in V. Prove that there must exist a scalar [tex]\lambda \neq 0[/tex] such that

    [tex]\bf G = \lambda \bf S[/tex]

    2. The attempt at a solution
    First we write it as:
    [tex]\bf G(\vec a, \vec b)\bf S(\vec c, \vec d) - \bf G(\vec a, \vec d)\bf S(\vec b, \vec c) + \bf G(\vec b, \vec c)\bf S(\vec a, \vec d) - \bf G(\vec c, \vec d)\bf S(\vec a, \vec b)[/tex]

    My first thought it to set [tex]\alpha = \bf G(\vec a, \vec b) \ and \ \beta = \bf G(\vec a, \vec d) \ and \ \gamma = \bf G(\vec b, \vec c) \ and \ \delta = \bf G(\vec c, \vec d)[/tex] since they are just scalars. After utilizing the symmetric properties of the tensors we get:

    [tex]\alpha \bf S(\vec c, \vec d) - \beta \bf S(\vec c, \vec b) + \gamma \bf S(\vec a, \vec d) - \delta \bf S(\vec a, \vec b)[/tex]

    [tex]\bf S(\vec c, \alpha \vec d - \beta \vec b) + \bf S(\vec a, \gamma \vec d - \delta \vec b)[/tex]

    Which doesn't seem to get us anywhere.

    I next tried to use a different substitution(same alpha and beta but this time gamma and delta I set to be S(...) and I get:

    [tex]\bf S(\vec c, \alpha \vec d - \beta \vec b) = \bf G(\vec c, \delta \vec d - \gamma \vec b)[/tex]

    This looks slightly more promising as far as I can tell but I don't know where to go.
    I tried to do this:
    [tex]\alpha \vec d - \beta \vec b = \bf G(\vec a, \vec b)\vec d - \bf G(\vec a, \vec d) \vec b = [/tex]
    [tex] = \bf G(\vec a, \vec b)d_{i} - \bf G(\vec a, \vec d)b_{i}[/tex]

    But that doesn't seem to pan out (I took a few more steps in this direction but it doesn't seem to go anywhere useful).

    Any suggestions?
  2. jcsd
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