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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?
     
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