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Dyad product

  1. Oct 15, 2009 #1
    Ok I have seen the tensor double dot scalar product defined two ways and it all boils down to how the multiplication is defined. Does anyone know which is correct? I believe the first one is correct but I keep seeing the second one in various books on finite element methods.

    1. [tex]\nabla \vec{u} \colon \nabla \vec{v} = u_{i,j} v_{j,i}[/tex]

    or

    2. [tex]\nabla \vec{u} \colon \nabla \vec{v} = u_{i,j} v_{i,j}[/tex]


    Thank you in advance,
    dakg
     
    Last edited: Oct 15, 2009
  2. jcsd
  3. Oct 15, 2009 #2
    You mean outer multiplication between two vectors, right? The definition i have seen (using index notation) is, in [tex]D[/tex] dimensions,

    [tex]\vec{u} \otimes \vec{v}= a_{ij}=u_i v_j\;,\;1\leq i,j \leq D[/tex]
     
  4. Oct 15, 2009 #3
    sorry there is a [tex]\nabla[/tex] missing

    i'll edit it
     
  5. Oct 15, 2009 #4
    i have it in there but it isn't printing, let me try here

    [tex] \nabla \vec{u} \colon \nabla \vec{v} [/tex]
     
  6. Oct 15, 2009 #5

    lurflurf

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    The first one is more common, but it is a matter of convention.
     
  7. Oct 15, 2009 #6
    Do you know why? I found the first one in a Lightfoot book on transport.

    They make different results, so wouldn't one be correct and the other wrong?
     
  8. Oct 16, 2009 #7

    lurflurf

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    Not wrong just different.
    log(e)=1
    log(10)=1
    3*5+2=17
    3*5+2=21
    Here are examples of conventions that can lead to confusion.
    The convention here (using dyadic product for an example) is
    1) (ab):(cd)=(a.d)(b.c) the usual rule
    2) (ab):(cd)=(a.c)(b.d) the other rule
    The usual rule proably is choosen because of matrix algebra
    ie to be the same as matrix product
     
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