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Symmetric Tensor invariance

  1. Apr 8, 2017 #1
    1. The problem statement, all variables and given/known data
    The lecture notes states that if ##T_{ij}=T_{ji}## (symmetric tensor) in frame S, then ##T'_{ij}=T'_{ji}## in frame S'. The proof is shown as $$T'_{ij}=l_{ip}l_{jq}T_{pq}=l_{iq}l_{jp}T_{qp}=l_{jp}l_{iq}T_{pq}=T'_{ji}$$ where relabeling of p<->q was used in the second equality. Where I am confused is that after the 3rd equality, the order of ##l_{iq}## and ##l_{jp}## changes. Is this allowed in all tensors?
    Thanks in advance
     
  2. jcsd
  3. Apr 9, 2017 #2

    Orodruin

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    The ##l_{jp}## are just numbers so your question boils down to "is ##xy = yx##?" where ##x## and ##y## are real numbers.
     
  4. Apr 9, 2017 #3

    jedishrfu

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    Since the tensor is symmetric then you can interchange the i and j in the expression.

    Right?
     
  5. Apr 9, 2017 #4

    Orodruin

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    The idea was to show that a tensor with symmetric components in one coordinate system has symmetric components in all coordinate systems. Above this meand that the exchange of p and q in ##T_{pq}## is assumed to be fine and we want to show that this implies that i and j can be exchanged. The OP's question was regarding the validity of changing the order of the transformation coefficients.
     
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