Levi-Civita symbol: Difference between revisions
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The '''Levi-Civita symbol''', usually denoted as ε<sub>''ijk''</sub>, is a notational convenience (similar to the [[Kronecker delta]] δ<sub>''ij''</sub>). Its value is: | The '''Levi-Civita symbol''', usually denoted as ε<sub>''ijk''</sub>, is a notational convenience (similar to the [[Kronecker delta]] δ<sub>''ij''</sub>). Its value is: | ||
* equal to 1, if the indices are pairwise distinct and in [[cyclic order]],<!--<ref name=cyclic> | * equal to 1, if the indices are pairwise distinct and in [[cyclic order]],<!--<ref name=cyclic> | ||
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\end{cases} | \end{cases} | ||
</math> | </math> | ||
The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric. In different words, the Levi-Civita symbol with three indices equals the ''sign'' of the [[permutation]] (''ijk'').<ref name=permutation>The sign of a permutation is 1 for even, −1 for odd permutations and 0 if two indices are equal. An ''even'' permutation is a sequence (''ijk...r'') that can be restored to (123...''n'') using an even number of interchanges, while an odd permutation requires an odd number.</ref> | The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric. In different words, the Levi-Civita symbol with three indices equals the ''sign'' of the [[permutation]] (''ijk'').<ref name=permutation>The sign of a permutation is 1 for even, −1 for odd permutations and 0 if two indices are equal. An ''even'' permutation is a sequence (''ijk...r'') that can be restored to (123...''n'') using an even number of interchanges of pairs, while an odd permutation requires an odd number.</ref> | ||
The symbol has been generalized to ''n'' dimensions, denoted as ε<sub>''ijk...r''</sub> and depending on ''n'' indices taking values from 1 to ''n''. | The symbol has been generalized to ''n'' dimensions, denoted as ε<sub>''ijk...r''</sub> and depending on ''n'' indices taking values from 1 to ''n''. | ||
It is determined by being antisymmetric in the indices and by ε<sub>123...''n''</sub> = 1. The generalized symbol equals the sign of the permutation (''ijk...r'') or, equivalently, the [[determinant]] of the corresponding unit vectors. Therefore the symbols are | It is determined by being antisymmetric in the indices and by ε<sub>123...''n''</sub> = 1. The generalized symbol equals the sign of the permutation (''ijk...r'') or, equivalently, the [[determinant]] of the corresponding unit vectors. Therefore the symbols also are called (Levi-Civita) ''permutation symbols''. | ||
=== | ===Levi-Civita tensor=== | ||
The Levi-Civita symbol—named after the Italian mathematician and physicist [[Tullio Levi-Civita]]—occurs mainly 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 also is denoted by ε<sub>''ijk''</sub>. | The Levi-Civita symbol—named after the Italian mathematician and physicist [[Tullio Levi-Civita]]—occurs mainly 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 also is denoted by ε<sub>''ijk''</sub>. | ||
Latest revision as of 11:35, 12 March 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,
- 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
The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric. In different words, the Levi-Civita symbol with three indices equals the sign of the permutation (ijk).[1]
The symbol has been generalized to n dimensions, denoted as εijk...r and depending on n indices taking values from 1 to n. It is determined by being antisymmetric in the indices and by ε123...n = 1. The generalized symbol equals the sign of the permutation (ijk...r) or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols also are called (Levi-Civita) permutation symbols.
Levi-Civita tensor
The Levi-Civita symbol—named after the Italian mathematician and physicist Tullio Levi-Civita—occurs mainly in differential geometry and mathematical physics where it is used to define the components of the (three-dimensional) Levi-Civita (pseudo)tensor that conventionally also is denoted by εijk.
The generalized symbol gives rise to an n-dimensional completely antisymmetric (or alternating) pseudotensor.
Notes
- ↑ The sign of a permutation is 1 for even, −1 for odd permutations and 0 if two indices are equal. An even permutation is a sequence (ijk...r) that can be restored to (123...n) using an even number of interchanges of pairs, while an odd permutation requires an odd number.