Levi-Civita symbol: Difference between revisions
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The Levi-Civita symbol changes sign whenever two of the indices are interchanged, | The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric. | ||
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The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). | The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). | ||
Likewise, the generalized symbol equals the sign of the permutation (''ijk...r'') | |||
< | Likewise, the generalized symbol equals the sign of the permutation (''ijk...r'')<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 one that can be restored to 123...''n'' using an even number of interchanges, while an odd permutations requires an odd number.</ref> or, equivalently, the [[determinant]] of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) ''permutation symbols''. | ||
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or, equivalently, the [[determinant]] of the corresponding unit vectors. | |||
Therefore the symbols are also called (Levi-Civita) ''permutation | |||
Revision as of 00:32, 4 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,
- 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.
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 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.
It give rise to an n-dimensional completely antisymmetric (or alternating) pseudotensor.
The Levi-Civita symbol equals the sign of the permutation (ijk).
Likewise, the generalized symbol equals the sign of the permutation (ijk...r)[1] or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) permutation symbols.
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 one that can be restored to 123...n using an even number of interchanges, while an odd permutations requires an odd number.