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

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)

See also 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. Referenced on Wolfram|Alpha Permutation Tensor
Cite this as:
Weisstein, Eric W. "Permutation Tensor."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationTensor.html

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