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# Permutation Tensor

The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. Recalling the definition of the permutation symbol in terms of a scalar triple product of the Cartesian unit vectors,

 (1)

the pseudotensor is a generalization to an arbitrary basis defined by

 (2) (3)

where

 (4)

and , where is the metric tensor. is nonzero iff the vectors are linearly independent.

When viewed as a tensor, the permutation symbol is sometimes known as the Levi-Civita tensor. The permutation tensor of rank four is important in general relativity, and has components defined as

 (5)

(Weinberg 1972, p. 38). The rank four permutation tensor satisfies the identity

 (6)

Kronecker Delta, Permutation, Permutation Symbol, Scalar Triple Product

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## References

Chou, P. C. and Pagano, N. J. "The Alternating Tensor." §8.7 in Elasticity: Tensor, Dyadic, and Engineering Approaches. New York: Dover, pp. 182-186, 1992.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

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Permutation Tensor

## Cite this as:

Weisstein, Eric W. "Permutation Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationTensor.html