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Tensors questions.

  1. Jun 13, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data
    I recently started delving into tensor calculus and am quite the stuck with the following:
    Given the tensor Ai = (x+y, y-x, z)i in cartesian coordinates, what would be the second covariant coordinate in cylindrical coordinates?
    AND
    Given the tensor Aij = (-1 0, -1 1)ij and the metric gij = (2 3, 3 4)ij, what would be A21?


    2. Relevant equations



    3. The attempt at a solution
    First, aren't I actually expected to find y-x in cylindrical coordinates, which is rsinθ - rcosθ? I have found the metric to be (1 0 0, 0 r2 0, 0 0 1), but I am really not sure how to put all the pieces together and how to proceed.
    Next, for finding A21 won't I actually need to multiply the given matrix by the metric and its inverse, thus yielding a similar matrix as the original?
    I could use some guidance, please.
     
  2. jcsd
  3. Jun 13, 2013 #2
    I have managed to answer the first part. Now I am mainly stymied by the second, viz. how to find A21. Could anyone please help?
     
  4. Jun 13, 2013 #3
    Contract twice?: $$ g^l_i g^j_k A^k_l = A^i_j$$ not entirely sure I understand the problem you've posed.

    Which is what I think you said, you'll have ## g_{il} g^{jk} ## i.e. the metric and its inverse times the (1,1) tensor. I think I'd write out the summation instead of matrix multiplication, though, because ##g_{il}g^{jk} = \left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right) ## seems almost too easy and it's 4 in the morning and I'm ready for sleep, not more math.
     
    Last edited: Jun 13, 2013
  5. Jun 13, 2013 #4
    What I first got, albeit did not write it here, was g1jg2jAjAj. Does that agree with what you wrote as the solution?
     
  6. Jun 13, 2013 #5
    I don't think so, but I'm inclined to say no because you need the metric acting on the (1,1) tensor twice. Your notation alone reverts it to two (0,1) tensors with only one summation. I need sleep but I'm off tomorrow, so I'll give it some thought if you haven't figured it out all the way, I can write write it out..

    cheers
     
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