Levi-Civita symbol: Difference between revisions
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The '''Levi-Civita symbol''', usually denoted as ε<sub>ijk</sub> | 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: | ||
* equal to 1, if the indices are pairwise distinct and in [[cyclic order]],<!--<ref name=cyclic> | |||
The term "cyclic order" imagines the items in a list, say ''a, b, c, ...'' arranged in a circle. Then all sequences that could be encountered by going once around the circle in the direction of the sequence ''a, b, c, ...'' are in cyclic order, regardless of the starting point. See {{cite book |title=An exercise book in algebra |author=Scoby McCurdy |url=http://books.google.com/books?id=0RMAAAAAYAAJ&pg=PA59 |pages=p. 59 |chapter=Cyclic order |year=1894 |publisher=D. C. Heath & Co.}} | The term "cyclic order" imagines the items in a list, say ''a, b, c, ...'' arranged in a circle. Then all sequences that could be encountered by going once around the circle in the direction of the sequence ''a, b, c, ...'' are in cyclic order, regardless of the starting point. See {{cite book |title=An exercise book in algebra |author=Scoby McCurdy |url=http://books.google.com/books?id=0RMAAAAAYAAJ&pg=PA59 |pages=p. 59 |chapter=Cyclic order |year=1894 |publisher=D. C. Heath & Co.}} | ||
</ref> | </ref>--> | ||
* 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 | |||
: <math> | |||
\varepsilon_{ijk} = \begin{cases} | |||
\ \ 1 & (ijk) = (123),(231),(312) \\ | |||
-1 & (ijk) = (132),(213),(321) \\ | |||
\ \ 0 & \text{else} | |||
\end{cases} | |||
</math> | |||
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, −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 of pairs, while an odd permutation requires an odd number.</ref> | |||
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. The generalized symbol equals the sign of the permutation (''ijk...r'') or, equivalently, the [[determinant]] of the corresponding unit vectors. Therefore the symbols also are called (Levi-Civita) ''permutation symbols''. | |||
===Levi-Civita tensor=== | ===Levi-Civita tensor=== | ||
The Levi-Civita symbol | 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 tensor|Levi-Civita (pseudo)tensor]] that conventionally also is denoted by ε<sub>''ijk''</sub>. | ||
The generalized symbol gives rise to an ''n''-dimensional completely antisymmetric (or alternating) pseudotensor. | |||
==Notes== | ==Notes== | ||
<references/> | <references/> | ||
Latest revision as of 11:35, 12 March 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, 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 also are called (Levi-Civita) permutation symbols.
Levi-Civita tensor
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
- ↑ 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 of pairs, while an odd permutation requires an odd number.