Hello, just hoping someone can give me a hand here.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Showing scalars are unchanged by rotation

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