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Quotient Theorem (Tensors)

  1. Apr 23, 2009 #1
    Prove that [itex]b_{ijkl}=\int_{r<a} dV x_i x_j \frac{\partial^2}{\partial_k \partial_l} (\frac{1}{r})[/itex] where [itex]r=|x|[/itex] is a 4th rank tensor.

    i've had a couple of bashes and got nowhere other than to establish that its quotient theorem.

    can i just pick a tensor of rank 3 to multiply it with or something?
     
  2. jcsd
  3. Apr 23, 2009 #2

    tiny-tim

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    hmm … do you mean …
     
  4. Apr 23, 2009 #3
    yep.
     
  5. Apr 2, 2011 #4
    I have no idea how to solve this too, can you give me some idea please?
     
  6. Apr 2, 2011 #5

    tiny-tim

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    welcome to pf!

    hi nhanle! welcome to pf! :wink:

    ok, what is the test for something being a tensor? :smile:
     
  7. Apr 4, 2011 #6
    hi tiny-tim,
    thank you for your reply. This is how vague the definition of tensor I am holding at the moment.
    I am also confused about the Affine connection. Can you help me clarify this?

    Thank you
     
  8. Apr 4, 2011 #7

    tiny-tim

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    ?? :confused:

    i'm not going to type out a lecture on tensors and connections :redface:

    please go back to your book or your lecture notes, and read up about tensors
     
  9. Apr 5, 2011 #8
    those appears on my lecture notes and also my book (general relativity - M.P.Hobson, G. Efstathiou, A.N. Lasenby) with very vague definitions.

    From my understanding, if one is to be a rank N-tensor, it should expect to have N derivative summations under coordinate transformation. Is that right?
     
  10. Apr 5, 2011 #9

    tiny-tim

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    does your book show why the Christoffel symbols aren't tensors?

    if so, that should show you how to do it :smile:
     
  11. Apr 5, 2011 #10
    it does but only with a few special case. So, I stumpled on this question "Prove that bijkl = ∫r<a dV xi xj ∂2(1/r)/∂k∂l, where r=|x|, is a 4th rank tensor."

    How to transform the partial derivatives? Thank you for being so patient with me
    I also have question about the affine connection https://www.physicsforums.com/showthread.php?t=189456 which was raised long ago but no one seems to be interested in answering :(
     
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