= Isotropic contractions of a fourth-rank tensor
{title2=$\epsilon_{ijk}T_{ijkl}=0,\quad\delta_{ij}T_{ijkl}=\mu\delta_{kl},\quad\epsilon_{ijm}T_{ijkl}=\nu\epsilon_{klm}$}
Contract an <isotropic tensor> with the <Kronecker delta> or the <Levi-Civita symbol>, using invariance under proper rotations. The three displayed contractions have ranks one, two and three. The only isotropic vector is zero; an <isotropic second-rank tensor> is a scalar delta, and an <isotropic third-rank tensor> is a scalar epsilon. For $T_{ijkl}=\alpha\delta_{ij}\delta_{kl}+\beta\delta_{ik}\delta_{jl}+\gamma\delta_{il}\delta_{jk}$, direct contraction gives $\mu=3\alpha+\beta+\gamma$ and $\nu=\beta-\gamma$.
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