Levi-Civita symbol: Difference between revisions

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The Levi-Civita symbol is used in the definiton of the [[Levi-Civita tensor]] that is also denoted as denoted as &epsilon;<sub>''ijk''</sub>.
The Levi-Civita symbol is used in the definiton of the [[Levi-Civita tensor]] that is also denoted as denoted as &epsilon;<sub>''ijk''</sub>.
The symbol can be generalized to ''n''-dimensions, as completely antisymmetric in its indices with &epsilon;<sub>123...''n''</sub>&nbsp;=&nbsp;1. More specifically, the symbol is one for even [[Permutation group|permutations]] of the indices, −1 for odd permutations, and 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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The Levi-Civita symbol, usually denoted as εijk, is a conventional abbreviation (similar to the Kronecker delta δij). It equals either 1, −1, or 0 depending on the values (1, 2, or 3) taken by the indices i, j, and k. It 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 is a special case (for n=3, because it involves three indices) of a more general notion:
   It equals the sign of the permutation (ijk). Therefore it is also called (Levi-Civita) permutation symbol.

The Levi-Civita symbol is used in the definiton of the Levi-Civita tensor that is also denoted as denoted as εijk.

The symbol can be generalized to n-dimensions, as completely antisymmetric in its indices with ε123...n = 1. More specifically, the symbol is one for even permutations of the indices, −1 for odd permutations, and 0 otherwise.

Both the symbol and the tensor are named after the Italian mathematician and physicist Tullio Levi-Civita.