Levi-Civita symbol: Difference between revisions
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The '''Levi-Civita symbol''', usually denoted as ε<sub>''ijk''</sub>, is a | 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: | ||
(similar to the [[Kronecker delta]] δ<sub>''ij''</sub>). | |||
* equal to 1, if the indices are pairwise distinct and in [[cyclic order]], | * 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 −1, if the indices are pairwise distinct but not in cyclic order, and | ||
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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 | The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). Therefore it is also called (Levi-Civita) ''permutation symbol''. | ||
The Levi-Civita symbol is used in the | The Levi-Civita symbol is used in the definition of the [[Levi-Civita tensor]] that has components denoted as ε<sub>''ijk''</sub>. | ||
The symbol can be generalized to ''n''-dimensions, | 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. | ||
Both the symbol and the tensor are named after the Italian mathematician and physicist [[Tullio Levi-Civita]]. | Both the symbol and the tensor are named after the Italian mathematician and physicist [[Tullio Levi-Civita]]. | ||
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Revision as of 22:14, 2 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
Remarks:
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 is used in the definition of the Levi-Civita tensor that has components denoted as εijk.
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.
Both the symbol and the tensor are named after the Italian mathematician and physicist Tullio Levi-Civita.