Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What kind of product denotes :

  1. May 21, 2014 #1
    Hi,
    I'm reading a book about fluid dynamics and I found some strange product between tensors. It's written like this: v=S:∇I , where S and I are matrices and v is a vector. Symbol : usually denotes Frobenius inner product. In this case we have a product of a matrix with a tensor of rank 3 and the result is vector - so it's not classical Frobenius product.
    I would like to know what kind of product this is (especially useful would be index notation).
    Thanks for help.
     
  2. jcsd
  3. May 21, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    I would guess a double contraction, which would turn a rank 5 tensor into a vector:[tex]
    v_i = S_{ijk} \frac{\partial I_k}{\partial x_j}[/tex]
     
  4. May 21, 2014 #3

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I initially guessed something like this too, but then thought that

    might indicate that S only has two indices.
     
  5. May 21, 2014 #4

    George Jones

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Maybe something like ##\left( S \cdot \nabla \right) \cdot I## with component form

    $$v_k = S_{ij} \frac{\partial I_{ik}}{\partial x_j}?$$
     
  6. May 21, 2014 #5
    Yes, something like this. I and S only have 2 indices. How do you know over which indices to sum? Why wouldn't you sum over i and k instead of i and j?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook