The permutation symbol (Evett 1966; Goldstein 1980, p. 172; Aris 1989, p. 16) is a three-index object sometimes called the Levi-Civita symbol (Weinberg 1972, p. 38; Misner et al. 1973, p. 87; Arfken 1985, p. 132; Chandrasekhar 1998, p. 68), Levi-Civita density (Goldstein 1980, p. 172), alternating tensor (Goldstein 1980, p. 172; Landau and Lifshitz 1986, p. 110; Chou and Pagano 1992, p. 182), or signature. It is defined by
(1)
The permutation symbol is implemented in the Wolfram Language as Signature[list].
There are several common notations for the symbol, the first of which uses the usual Greek epsilon character (Goldstein 1980, p. 172; Griffiths 1987, p. 139; Jeffreys and Jeffreys 1988, p. 69; Aris 1989, p. 16; Chou and Pagano 1992, p. 182), the second of which uses the curly variant (Weinberg 1972, p. 38; Misner et al. 1973, p. 87; Lightman et al. 1979, pp. 19-21 and 183-188; Arfken 1985, p. 132; Chandrasekhar 1998, p. 68), and the third of which uses a Latin lower case (Landau and Lifshitz 1986, p. 110; Green and Zerna 1992, p. 11).
The symbol can also be interpreted as a tensor, in which case it is called the permutation tensor.
The permutation symbol satisfies
where is the Kronecker delta (Arfken 1985, p. 136).
The symbol can be defined as the scalar triple product of unit vectors in a right-handed coordinate system,
(6)
The symbol can be generalized to an arbitrary number of elements, in which case the permutation symbol is , where is the number of transpositions of pairs of elements (i.e., permutation inversions) that must be composed to build up the permutation (Skiena 1990). This type of symbol arises in computation of determinants of matrices. The number of permutations on symbols having signature is , which is also the number of permutations having signature .
See alsoEven Permutation,
Odd Permutation,
Permutation,
Permutation Cycle,
Permutation Inversion,
Permutation Tensor,
Transposition Related Wolfram siteshttp://functions.wolfram.com/IntegerFunctions/Signature/ Explore with Wolfram|Alpha ReferencesArfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 132-133 and 136, 1985.Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989.Chandrasekhar, S. The Mathematical Theory of Black Holes. Oxford, England: Clarendon Press, 1998.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.Evett, A. A. "Permutation Symbol Approach to Elementary Vector Analysis." Amer. J. Phys. 34, 503-507, 1966.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd ed. New York: Dover, 1992.Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, 1987.Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 69-74, 1988.Landau, L. D. and Lifschitz, E. M. Theory of Elasticity, 3rd rev. enl. ed. Oxford, England: Pergamon Press, 1986.Lightman, A. P.; Price, R. H.; and Teukolsky, S. Problem Book in Relativity and Gravitation, 2nd pr. Princeton, NJ: Princeton University Press, 1979.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.Skiena, S. "Signature." §1.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 38, 1972. Referenced on Wolfram|AlphaPermutation Symbol Cite this as:Weisstein, Eric W. "Permutation Symbol." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PermutationSymbol.html
Subject classificationsRetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4