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Proving tensor symmetry under transformation

  • Thread starter Hendrick
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Homework Statement


Using indical notation, prove that a 2nd order symmetric tensor D remains symmetric when transformed into any other coordinate system.


Homework Equations


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


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:

Answers and Replies

  • #2
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Did I not write my question properly or does no one know?
 

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