Levi-Civita symbol: Difference between revisions
imported>John R. Brews (cyclic order) |
imported>Peter Schmitt (clarifications) |
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'''Remarks:''' | '''Remarks:''' | ||
The Levi-Civita symbol—named after the Italian mathematician and physicist [[Tullio Levi-Civita]]—mainly | |||
occurs in differential geometry and mathematical physics | |||
where it is used to define the components of the (three-dimensional) [[Levi-Civita tensor|Levi-Civita (pseudo)tensor]] | |||
that conventionally is also denoted by ε<sub>''ijk''</sub>. | |||
The symbol changes sign whenever two of the indices are interchanged. | The symbol changes sign whenever two of the indices are interchanged. | ||
The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). Therefore it is also called (Levi-Civita) ''permutation symbol''. | The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). Therefore it is also called (Levi-Civita) ''permutation symbol''. | ||
The symbol can be generalized to ''n''-dimensions, to become the ''n''-index symbol ε<sub>''ijk...r''</sub> completely antisymmetric in its indices, and with ε<sub>123...''n''</sub> = 1. More specifically, the symbol is has value 1 for even [[Permutation group|permutations]] of the ''n'' indices, value −1 for odd permutations, and value 0 otherwise.<ref name= Weber> | The symbol can be generalized to ''n''-dimensions, to become the ''n''-index symbol ε<sub>''ijk...r''</sub> completely antisymmetric in its indices, and with ε<sub>123...''n''</sub> = 1. More specifically, the symbol is has value 1 for even [[Permutation group|permutations]] of the ''n'' indices, value −1 for odd permutations, and value 0 otherwise.<ref name= Weber> | ||
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==Notes== | ==Notes== | ||
<references/> | <references/> | ||
Revision as of 19:01, 3 January 2011
The Levi-Civita symbol, usually denoted as εijk, is a notational convenience (similar to the Kronecker delta δij). Its value is:
- equal to 1, if the indices are pairwise distinct and in cyclic order,[1]
- equal to −1, if the indices are pairwise distinct but not in cyclic order, and
- equal to 0, if two of the indices are equal.
Thus
Remarks:
The Levi-Civita symbol—named after the Italian mathematician and physicist Tullio Levi-Civita—mainly occurs in differential geometry and mathematical physics where it is used to define the components of the (three-dimensional) Levi-Civita (pseudo)tensor that conventionally is also denoted by εijk.
The symbol changes sign whenever two of the indices are interchanged.
The Levi-Civita symbol equals the sign of the permutation (ijk). Therefore it is also called (Levi-Civita) permutation symbol.
The symbol can be generalized to n-dimensions, to become the n-index symbol εijk...r completely antisymmetric in its indices, and with ε123...n = 1. More specifically, the symbol is has value 1 for even permutations of the n indices, value −1 for odd permutations, and value 0 otherwise.[2]
Notes
- ↑ The term "cyclic order" imagines the items in a list, say a, b, c, ... arranged in a circle. Then all sequences that could be encountered by going once around the circle in the direction of the sequence a, b, c, ... are in cyclic order, regardless of the starting point. See Scoby McCurdy (1894). “Cyclic order”, An exercise book in algebra. D. C. Heath & Co., p. 59.
- ↑ For example, see Hans-Jurgen Weber, George Brown Arfken (2004). Essential mathematical methods for physicists, 5th ed. Academic Press, p. 164. ISBN 0120598779.