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

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

    User Avatar
    Homework Helper

    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

    User Avatar
    Homework Helper

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Dyad product
  1. Wedge product (Replies: 8)

  2. Semidirect Product (Replies: 0)

  3. Geometric product (Replies: 5)

Loading...