1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Tensor identity - Help

  1. Jul 31, 2004 #1
    I'm having some trouble to prove the following tensor identity shown below in Einstein's summation convention:

    [tex](a_{ij}+a_{ji})x_{i}x_{j}=2a_{ij}x_{i}x_{j}[/tex]

    I expanded the terms but when I did group them I didn't get the identity. The only way I could get the identity is if

    [tex]a_{ij}=a_{ji}[/tex]

    and I don't see a reason why this would be so.

    Obviously I'm missing something. Can somebody tell what is it that I'm doing wrong?
     
  2. jcsd
  3. Jul 31, 2004 #2
    It's true that [tex]a_{ij}x_{i}x_{j} = a_{ji}x_{i}x_{j}[/tex]. Putting in the summation signs may help you to see this.
     
    Last edited: Jul 31, 2004
  4. Jul 31, 2004 #3

    robphy

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    To elaborate on Lonewolf's reply...
    [tex]
    \begin{align*}
    a_{ij}x_{i}x_{j}
    &= a_{ji}x_{j}x_{i}&&\text{relabel repeated [dummy] indices}\\
    &= a_{ji}x_{i}x_{j}&&\text{reorder the writing of the x factors}\\
    \end{align*}
    [/tex]

    You'll learn that [itex]\frac{1}{2}(a_{ij} + a_{ji})[/itex]
    is called "the symmetric part of [itex] a_{ij} [/itex]", and is written as [itex] a_{(ij)} [/itex].
    Hence, your identity can be written
    [tex] 2a_{(ij)}x_{i}x_{j} =2a_{ij}x_{i}x_{j}[/tex].

    Here is an "index gymnastics" proof, starting with half of your right-hand-side:
    [tex]
    \begin{align*}
    a_{ij}x_{i}x_{j}
    &= a_{ij}x_{(i}x_{j)}&&\text{since } x_{i}x_{j} = x_{(i}x_{j)} \\
    &= a_{(ij)}x_{i}x_{j}&&\text{since i and j are being symmetrized}
    \end{align*}
    [/tex]
     
  5. Jul 31, 2004 #4
    Guys, thanks to you help I got it right this time after some thinking and some calculations. I realized one of my problems was the fact that when I was looking at
    [tex]a_{ij}=a_{ji}[/tex]
    I was thinking of it as when we say
    [tex]a=b[/tex]
    The way I should look at that expression is not isolated from the rest of the other terms. The important thing, I think, is not the term itself isolated but how the summation terms comes up after the entire summation is expanded.
    I can continue now with the next page of my book :smile:
     
    Last edited: Jul 31, 2004
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Tensor identity - Help
  1. Tensors and Tensors (Replies: 7)

Loading...