Levi-Civita symbol: Difference between revisions
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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 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> or minus one if the order is different, or zero if any two of the indices are the same. The Levi-Civita symbol also is used to denote the ''alternating tensor''<ref name=Sharma> | </ref> or minus one if the order is different, or zero if any two of the indices are the same. | ||
The Levi-Civita symbol also is used to denote the ''alternating tensor''<ref name=Sharma> | |||
{{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}} | {{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}} | ||
</ref> or the ''completely antisymmetric tensor'' with three indices in three dimensions. It is named after the Italian mathematician and physicist [[Tullio Levi-Civita]]. | </ref> or the ''completely antisymmetric tensor'' with three indices in three dimensions. It is named after the Italian mathematician and physicist [[Tullio Levi-Civita]]. 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> | ||
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> | |||
{{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}} | {{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}} | ||
Revision as of 10:02, 2 January 2011
The Levi-Civita symbol, usually denoted as εijk equals one if i,j,k = 1,2,3 or any permutation that keeps the same cyclic order,[1] or minus one if the order is different, or zero if any two of the indices are the same.
The Levi-Civita symbol also is used to denote the alternating tensor[2] or the completely antisymmetric tensor with three indices in three dimensions. It is named after the Italian mathematician and physicist Tullio Levi-Civita. 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 εij, εijk, εijkl.[3] Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating ε123 = εxyz = 1 in Cartesian coordinates.
Notes
- ↑ 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 Scoby McCurdy (1894). “Cyclic order”, An exercise book in algebra. D. C. Heath & Co., p. 59.
- ↑ Vinod K. Sharma (2009). “§9.2 Alternating tensor (or Levi-Civita symbol)”, Matrix Methods and Vector Spaces in Physics. Prentice-Hall of India Pvt.Ltd, p. 370. ISBN 8120338669.
- ↑ T. Padmanabhan (2010). Gravitation: Foundations and Frontiers. Cambridge University Press, p. 22. ISBN 0521882230.