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Dot product of tensors?

  1. Jan 25, 2009 #1
    Hello,

    I was trying to follow a proof that uses the dot
    product of two rank 2 tensors, as in A dot B.

    How is this dot product calculated?

    A is 3x3, Aij, and B is 3x3, Bij, each a rank 2 tensor.

    Any help is greatly appreciated.

    Thanks!

    sugarmolecule
     
  2. jcsd
  3. Jan 26, 2009 #2

    Fredrik

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    I've never heard of a dot product of tensors. Can you give us more details? Tip: If this is from a book, check if it's available at books.google.com. You might even be able to show us the specific page where you found this.
     
  4. Jan 26, 2009 #3
    Hi,

    I found this reference online that lists a potential intepretation:

    www.math.mtu.edu/~feigl/courses/CFD-script/tensor-review.pdf

    It lists the dot product of two rank-2 tensors U, V in 3-space as:

    UikVkj

    Does that look right?

    Thanks,

    sugarmolecule
     
  5. Jan 26, 2009 #4
    nevermind i was thinking of something else.
     
    Last edited: Jan 27, 2009
  6. Jan 27, 2009 #5

    Fredrik

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    I suspected that. I didn't know that anyone uses term "dot product" about rank 2 tensors, but if they do, it's logical that they mean precisely that. I don't see a reason to call it a dot product though. To me, that's just the definition of matrix multiplication, and if we insist on thinking of U and V as tensors, then the operation would usually be described as a ''contraction" of two indices of the rank 4 tensor that you get when you take what your text calls the "dyadic product" of U and V.
     
  7. Oct 6, 2009 #6
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