Antisymmetric and Symmetric Tensors
A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0.
For a general tensor U with components and a pair of indices i and j, U has symmetric and antisymmetric parts defined as:
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(symmetric part) (antisymmetric part).
Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in
Read more about this topic: Antisymmetric Tensor