If I have the tensor equation $$T_{kij}+T_{jik} = S_{ijk}$$ where $T$ is symmetric on the last two indices $T_{ijk}=T_{i(jk)},$ then is there a way to write $T_{ijk}$ in terms of only the tensor $S?$
Rewriting tensor equation.
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tensors
1 Answers
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You easily observe that $S$ is also symmetric in the last two indices, $S_{ijk}= S_{i(jk)}$.
The solution is $$T_{kij}= \frac12 (S_{ijk} + S_{jik} - S_{kij}).$$ You can check by inserting in your equation;
You obtain $$T_{kij}+ T_{jik} =\frac12 (S_{ijk} + \underline{S_{jik}} - \underline{\underline{S_{kij}}}) + \frac12 (S_{ikj} + \underline{\underline{S_{kij}}} - \underline{S_{jik}}) = \frac12(S_{ijk} + S_{ikj}) = S_{ijk}.$$