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,


the pseudotensor is a generalization to an arbitrary basis defined by



 [alpha,beta,...,mu]={1   the arguments are an even permutation; -1   the arguments are an odd permutation; 0   two or more arguments are equal,

and g=det(g_(alphabeta)), where g_(alphabeta) is the metric tensor. epsilon(x_1,...,x_n) 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 epsilon^(alphabetadeltagamma) of rank four is important in general relativity, and has components defined as

 (epsilon)^(alphabetagammadelta)={+1   if alphabetagammadelta is an even permutation of 0123; -1   if alphabetagammadelta is an odd permutation of 0123; 0   otherwise

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


See also

Kronecker Delta, Permutation, Permutation Symbol, Scalar Triple Product

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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.

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