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The operator $\partial_i \partial_j$ is symmetric by switching $i$ and $j$. All tensors can be decomposed the following way :
$T_{ab} = T_{[ab]} + T_{\{ab\}}$
With $T_{\{ab\}}$ the symmetrized tensor ($\frac{1}{2}(T_{ab} + T_{ba})$)and $T_{[ab]}$ the antisymmetrized tensor ($\frac{1}{2}(T_{ab}...