 TOPICS  # Antisymmetric Tensor

An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that (1)

is antisymmetric.

The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies (2)

Furthermore, any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as (3)

The antisymmetric part of a tensor is sometimes denoted using the special notation (4)

For a general rank- tensor, (5)

where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example (6)

(Wald 1984, p. 26).

Alternating Multilinear Form, Exterior Algebra, Symmetric Tensor, Wedge Product

Portions of this entry contributed by Todd Rowland

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## References

Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.

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Antisymmetric Tensor

## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Antisymmetric Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntisymmetricTensor.html