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Showing scalars are unchanged by rotation

  1. Oct 10, 2004 #1
    Hello, just hoping someone can give me a hand here.

    I have a second-order tensor P, which has components [tex]p_{ij}[/tex] and I want to show that the following scalar quantities are unchanged by rotation:

    [tex]p_{ii}[/tex]
    [tex]p_{ij}p_{ji}[/tex]
    [tex]p_{ij}p_{jk}p_{ki}[/tex]

    Now, I know scalars are zero'th order tensors, I know im going to have to use the tensor transformation law, I know I must keep in mind the orthogonality of the rotation matrix and I must use the substitution property.

    This is what i've done but im not happy that its valid as a solution to my problem.

    The transformation law tells us that [tex]{p^'}_{ii} = \alpha_{ia} \alpha_{ib} p_{ab}[/tex]

    If it is isotropic then l.h.s = [tex]p_{ii}[/tex] & r.h.s = [tex]\alpha_{ia} \alpha_{ia}[/tex] by the substitution property. This is equal to [tex]p_{ii}[/tex] by the orthogonality of the rotation matrix.

    Im not happy with this, any help is much appreciated! Thanks, Matt.

    p.s. this is only the first quantity!
     
  2. jcsd
  3. Oct 15, 2004 #2
    I worked out how to do it now, if anyone wants to know..

    p'(ii) = alpha (ia) alpha (ib) p (ab)

    p'(ii) = delta (ab) p (ab)

    p'(ii) = p (aa)

    which in this case can be rewritten to look like

    p'(ii) = p(ii)

    This implies p(ii) is invariant, or unchanged by rotation.
     
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