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Irreducible Tensor

Given a general second Rank Tensor $A_{ij}$ and a Metric $g_{ij}$, define

$\displaystyle \theta$ $\textstyle \equiv$ $\displaystyle A_{ij}g^{ij} = A_i^i$ (1)
$\displaystyle \omega^i$ $\textstyle \equiv$ $\displaystyle \epsilon^{ijk}A_{jk}$ (2)
$\displaystyle \sigma_{ij}$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(A_{ij}+A_{ji})-{\textstyle{1\over 3}}g_{ij}A^k_k,$ (3)

where $\delta_{ij}$ is the Kronecker Delta and $\epsilon^{ijk}$ is the Levi-Civita Symbol. Then
$\sigma_{ij}+{\textstyle{1\over 3}}\theta g_{ij}+{\textstyle{1\over 2}}\epsilon_{ijk}\omega^k$
$\quad = {\textstyle{1\over 2}}(A_{ij}+A_{ji})+{\textstyle{1\over 2}}(\delta_i^\lambda \delta_j^\mu-\delta_i^\mu \delta_j^\lambda)A_{\lambda\mu}$
$\quad = {\textstyle{1\over 2}}(A_{ij}+A_{ji}) +{\textstyle{1\over 2}}(A_{ij}-A_{ji}) = A_{ij},$ (4)
where $\theta$, $\omega^i$, and $\sigma_{ij}$ are Tensors of Rank 0, 1, and 2.

See also Tensor

© 1996-9 Eric W. Weisstein