imported>John R. Brews |
imported>Peter Schmitt |
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| The '''Levi-Civita symbol''', usually denoted as ε<sub>ijk</sub> equals one if ''i,j,k = 1,2,3'' or any permutation that keeps the same cyclic order,<ref name=cyclic>
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| 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 '''Levi-Civita symbol''', usually denoted as ε<sub>''ijk''</sub>, is a conventional abbreviation |
| | (similar to the [[Kronecker delta]] δ<sub>''ij''</sub>). |
| | 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 |
| | : <math> |
| | \varepsilon_{ijk} = \begin{cases} |
| | \ \ 1 & (ijk) = (123),(231),(312) \\ |
| | -1 & (ijk) = (132),(213),(321) \\ |
| | \ \ 0 & \text{else} |
| | \end{cases} |
| | </math> |
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| </ref> or minus one if the order is different, or zero if any two of the indices are the same. It is named after the Italian mathematician and physicist [[Tullio Levi-Civita]].
| | '''Remarks:''' |
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| The symbol can be generalized to ''n''-dimensions, as completely antisymmetric in its indices with ε<sub>123...''n''</sub> = 1. More specifically, the symbol is one for even [[Permutation group|permutations]] of the indices, −1 for odd permutations, and 0 otherwise. | | The symbol changes sign whenever two of the indices are interchanged. |
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| ===Levi-Civita tensor=== | | The Levi-Civita symbol is a special case (for ''n''=3, because it involves three indices) of a more general notion: |
| | <br> It equals the sign of the [[permutation]] (''ijk''). Therefore it is also called (Levi-Civita) ''permutation symbol''. |
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| The Levi-Civita symbol also is used to denote the components of the ''Levi-Civita tensor'', sometimes called the ''Levi-Civita form'', and in ''n'' dimensions this tensor is an invariant of the special unitary group [[SU(n)]].<ref name=Vaughn> | | The Levi-Civita symbol is used in the definiton of the [[Levi-Civita tensor]] that is also denoted as denoted as ε<sub>''ijk''</sub>. |
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| {{cite book |title=Introduction to mathematical physics |author=Michael T. Vaughn |pages=p. 484 |url=http://books.google.com/books?id=E6_DiJDIptoC&pg=PA484 |isbn=3527406271 |publisher=Wiley-VCH |year=2007}}
| | Both the symbol and the tensor are named after the Italian mathematician and physicist [[Tullio Levi-Civita]]. |
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| </ref> It flips sign under reflections, and physicists call it a ''pseudo''-tensor.<ref name=Felsager> | | <!-- |
| | | {{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.}} |
| {{cite book |title=Geometry, particles, and fields |author=Bjørn Felsager |pages=p. 358 |url=http://books.google.com/books?id=R1XkarKY7AwC&pg=PA358 |year=1998 |isbn=0387982671 |publisher=Springer}} | | --> |
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| </ref> It also is called the ''alternating tensor''<ref name=Sharma>
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| {{cite book |title=Matrix Methods and Vector Spaces in Physics |author=Vinod K. Sharma |url=http://books.google.com/books?id=Kg2ZjUmOB9EC&pg=PT386 |pages=p. 370|chapter=§9.2 Alternating tensor (or Levi-Civita symbol) |isbn=8120338669 |publisher=Prentice-Hall of India Pvt.Ltd |year=2009}}
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| </ref> or the ''completely antisymmetric tensor'' with ''n'' indices in ''n'' dimensions. The completely antisymmetric tensor with ''n'' indices in ''n''-dimensions has only one independent component, and is denoted in two, three and four dimensions as ε<sub>ij</sub>, ε<sub>ijk</sub>, ε<sub>ijkl</sub>.<ref name=Padmanabhan>
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| {{cite book |title=Gravitation: Foundations and Frontiers |author=T. Padmanabhan |url=http://books.google.com/books?id=BSfe2MjbQ3gC&pg=PA22 |pages=p. 22 |isbn=0521882230 |publisher=Cambridge University Press |year=2010}}
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| </ref> Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating ε<sub>123</sub> = ε<sub>xyz</sub> = 1 in [[Cartesian coordinates]].
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| ==Notes==
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| <references/>
| |
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.
Both the symbol and the tensor are named after the Italian mathematician and physicist Tullio Levi-Civita.
