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Homework Help: Vector to Tensor properties

  1. Apr 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2.

    2. Relevant equations
    A tensor transforms under rotations (R) as a vector:
    Tij'=RinRjmTnm

    3. The attempt at a solution
    I wanted to just create the matrix, but I don't know how to prove that this is also a tensor.
     
  2. jcsd
  3. Apr 5, 2016 #2

    Dick

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    Well, how do the vectors ##A## and ##B## transform under rotations?
     
  4. Apr 6, 2016 #3
    Thanks a lot!
    I think I understand it now:
    $$A_i' B_j' = R_{in}A_n R_{jm}A_m$$
    $$A_i' B_j' = R_{in} R_{jm} (A_n A_m)$$
    $$A_i' B_j' = R_{in} R_{jm} Q_{nm}$$
    $$A_i' B_j' = Q'_{nm}$$
    So we for proving something is a tensor, we just apply some transformations to it, right?
     
  5. Apr 6, 2016 #4

    Dick

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    Yes, something is a tensor if it transforms like a tensor.
     
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