= Scalar, symmetric-traceless and axial tensor decomposition
{title2=$P_{ij}=P\delta_{ij}+S_{ij}+\epsilon_{ijk}A_k$}
Every three-dimensional <Cartesian second-rank tensor> splits into scalar, symmetric-traceless, and antisymmetric parts. Their coefficients are $P=P_{kk}/3$, $S_{ij}=(P_{ij}+P_{ji})/2-P\delta_{ij}$, and $A_k=\epsilon_{kij}P_{ij}/2$. The epsilon contraction identity reconstructs the antisymmetric part and proves uniqueness. Under proper rotations $A$ behaves as a vector, but under reflections it is an <axial vector>.
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