= Isotropic third-rank tensor
{title2=$T_{ijk}=b\epsilon_{ijk}$}
A rank-three Cartesian <tensor> invariant under every proper <rotation matrix> in three dimensions is a multiple of the <Levi-Civita symbol>. Half-turns about coordinate axes force every nonzero component to contain each coordinate label once, and quarter-turns make these components alternating. Conversely, the <determinant> formula $R_{ia}R_{jb}R_{kc}\epsilon_{abc}=(\det R)\epsilon_{ijk}$ proves invariance when $\det R=1$. Requiring invariance under reflections as an ordinary tensor forces $b=0$; invariance under proper rotations alone permits nonzero $b$.
Back to article page
