showing scalars are unchanged by rotation

MathematicalPhysics

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!

MathematicalPhysics

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.