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Easy tensor question

  1. Sep 22, 2013 #1

    Ibix

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    Science Advisor

    Another trivial question from me.

    1. The problem statement, all variables and given/known data

    Which (if any) of the following are valid tensor expressions:
    (a)[itex]A^\alpha+B_\alpha[/itex]
    (b)[itex]R^\alpha{}_\beta A^\beta+B^\alpha=0[/itex]
    (c)[itex]R_{\alpha\beta}=T_\gamma[/itex]
    (d)[itex]A_{\alpha\beta}=B_{\beta\alpha}[/itex]

    2. Relevant equations

    Nothing relevant - these are generic tensors.

    3. The attempt at a solution

    (a) and (c) are not valid because the indices don't match up. (d) is valid - in matrix notation, A=BT.

    I'm not sure about (b), though. The left hand side is valid; summing over the dummy index makes it a sum of two vectors. I'm not quite sure how to interpret the equality, though. I can see it as a vector equaling a scalar - which is not valid. Alternatively, I can read an implicit [itex]\forall \alpha[/itex] - in other words, that each element of the vector on the left hand side is identically zero.

    I lean towards the first interpretation - but I'm not sure.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 22, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    For an equation like (b), the zero on the right would normally be interpreted as the zero vector (rather than the zero scalar); or more precisely, it would be the ##\alpha## component of the zero vector (which would of course have a value of 0 in any coordinate system and any reference frame). So, I think your second interpretation is better.
     
  4. Sep 22, 2013 #3

    Ibix

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    Science Advisor

    Thank you very much.

    This is self-study. When I downloaded the problem sheet the answers were available - now term has started and they've gone... Shouldn't have been conscientious.
     
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