## showing scalars are unchanged by rotation

Hello, just hoping someone can give me a hand here.

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

$$p_{ii}$$
$$p_{ij}p_{ji}$$
$$p_{ij}p_{jk}p_{ki}$$

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 $${p^'}_{ii} = \alpha_{ia} \alpha_{ib} p_{ab}$$

If it is isotropic then l.h.s = $$p_{ii}$$ & r.h.s = $$\alpha_{ia} \alpha_{ia}$$ by the substitution property. This is equal to $$p_{ii}$$ 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!
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 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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