Given a sperically symmetric problem, I am asked to show that its Cauchy stress tensor, in spherical coordinates will assume the form:

$${\overline{\overline{\sigma}}}=\sigma_{RR}(r)\overline{e_r}\otimes\overline{e_r}+\sigma_{\theta\theta}(r)(\overline{e_\theta}\otimes\overline{e_\theta}+\overline{e_\phi}\otimes\overline{e_\phi}) $$ My idea was to say that that tensor must be invariant under all the O(3) transfromation since the physics of this problem must be invariant under rotation. The problem is that I should express the O(3) transformation in spherical coordinates and this is not immediate for me...

  • 1
    $\begingroup$ Just search for Killing vectors on the sphere. In most literature, these will have been derived in spherically coordinates. Use these to determine how $O(3)$ transformations act on the spherical coordinates. $\endgroup$
    – Prahar
    Dec 27 '15 at 19:13