Levi-Civita symbol: Difference between revisions

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The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric.
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, &minus;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 symbol has been generalized to ''n'' dimensions, denoted as &epsilon;<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 &epsilon;<sub>123...''n''</sub>&nbsp;=&nbsp;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 also called (Levi-Civita) ''permutation symbols''.


'''Remarks:'''
===Remarks===


The Levi-Civita symbol&mdash;named after the Italian mathematician and physicist [[Tullio Levi-Civita]]&mdash;mainly  
The Levi-Civita symbol&mdash;named after the Italian mathematician and physicist [[Tullio Levi-Civita]]&mdash;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 &epsilon;<sub>''ijk''</sub>.
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 &epsilon;<sub>''ijk''</sub>.


The symbol has been generalized to ''n'' dimensions, denoted as &epsilon;<sub>''ijk...r''</sub> and
The generalized symbol gives rise to an ''n''-dimensional completely antisymmetric (or alternating) pseudotensor.
depending on ''n'' indices taking values from 1 to ''n''.
It is determined by being antisymmetric in the indices and by &epsilon;<sub>123...''n''</sub>&nbsp;=&nbsp;1.
<br>
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'')<ref name=permutation>The sign of a permutation is 1 for even, &minus;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> or, equivalently, the [[determinant]] of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) ''permutation symbols''.
 
 
<!-- <ref name= Weber>
For example, see {{cite book |title=Essential mathematical methods for physicists |author=Hans-Jurgen Weber, George Brown Arfken |url=http://books.google.com/books?id=k046p9v-ZCgC&pg=PA164 |pages=p. 164 |isbn=0120598779 |edition=5th ed |year=2004 |publisher=Academic Press}}
</ref> -->


==Notes==
==Notes==
<references/>
<references/>

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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 are also called (Levi-Civita) permutation symbols.

Remarks

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

  1. 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.