1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Proving tensor symmetry under transformation

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Using indical notation, prove that a 2nd order symmetric tensor D remains symmetric when transformed into any other coordinate system.

    2. Relevant equations
    Tensor law of transformation (2nd order):
    [tex]D'_{pq} = a_{pr}a_{qs}D_{rs}[/tex]

    3. The attempt at a solution
    I think I'm required to prove that [tex]D'_{pq} = D'_{qp}[/tex] (where D is a symmetric 2nd order tensor)

    [tex]D'_{pq} = a_{pr}a_{qs}D_{rs}[/tex]
    [tex]D'^{T}_{pq} = (a_{pr}a_{qs}D_{rs})^T[/tex]
    [tex]D'_{qp} = a_{qs}a_{pr}D_{sr}[/tex] (can someone please explain why when you transpose this, the a's swaps position but the D swaps indices?)

    [tex]D_{rs} = D_{sr}[/tex] (as D is symmetric)

    =>[tex]D'_{pq} = a_{pr}a_{qs}D_{rs} = a_{qs}a_{pr}D_{rs} = D'_{qp}[/tex]

    Thus [tex]D'_{pq} = D'_{qp}[/tex]

    Did I prove it correctly?

    Thank you
    Last edited: Mar 17, 2010
  2. jcsd
  3. Mar 18, 2010 #2
    Did I not write my question properly or does no one know?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook