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 ε<sub>''ijk''</sub>. | that conventionally is also denoted by ε<sub>''ijk''</sub>. | ||
The symbol | The symbol has been generalized to ''n'' dimensions, denoted as ε<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 ε<sub>123...''n''</sub> = 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 | 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, −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''. | |||
<!-- <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/> | ||
Revision as of 18:40, 3 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, 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.