Levi-Civita symbol: Difference between revisions

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The Levi-Civita symbol changes sign whenever two of the indices are interchanged, i.e., it is antisymmetric.




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that conventionally is also denoted by &epsilon;<sub>''ijk''</sub>.
that conventionally is also denoted by &epsilon;<sub>''ijk''</sub>.


The symbol changes sign whenever two of the indices are interchanged.
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.
<br>
It give rise to an ''n''-dimensional completely antisymmetric (or alternating) pseudotensor.


The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk''). Therefore it is also called (Levi-Civita) ''permutation symbol''.
The Levi-Civita symbol equals the sign of the [[permutation]] (''ijk'').
Likewise, the generalized symbol equals the sign of the permutation (''ijk...r''),
<br>
(i.e., 1 for even, &minus;1 for odd permutions and 0 if two indices are equal)
<br>
or, equivalently, the [[determinant]] of the corresponding unit vectors.
Therefore the symbols are also called (Levi-Civita) ''permutation symbol''.


The symbol can be generalized to ''n''-dimensions, to become the ''n''-index symbol &epsilon;<sub>''ijk...r''</sub> completely antisymmetric in its indices, and with &epsilon;<sub>123...''n''</sub>&nbsp;=&nbsp;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.<ref name= Weber>


<!-- <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}}
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> -->
</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, i.e., 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),
(i.e., 1 for even, −1 for odd permutions and 0 if two indices are equal)
or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) permutation symbol.


Notes