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| {{Distinguish|Levi-Civita connection}}
| | Oscar is what my wife loves to call me and I totally dig that title. For years I've been operating as a payroll clerk. To collect cash is one of the things I adore most. My family members life in Minnesota and my family members enjoys it.<br><br>Stop by my web page [http://ece.modares.ac.ir/mnl/?q=node/1234839 ece.modares.ac.ir] |
| {{hatnote|See [[Ricci calculus]], [[Einstein notation]], and [[Raising and lowering indices]] for the [[index notation]] used in the article.}}
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| In [[mathematics]], particularly in [[linear algebra]], [[tensor analysis]], and [[differential geometry]], the '''Levi-Civita symbol''' represents a collection of numbers; defined from the [[parity of a permutation|sign of a permutation]] of the [[natural number]]s 1, 2, …, ''n'', for some positive integer ''n''. It is named after the [[Italian people|Italian]] [[mathematician]] and [[physicist]] [[Tullio Levi-Civita]]. Other names include the '''[[permutation]] symbol''', '''antisymmetric symbol''', or '''alternating symbol''', which refer to its [[antisymmetric tensor|antisymmetric]] property and definition in terms of permutations.
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| The standard letters to denote the Levi-Civita symbol are the [[Greek alphabet|Greek]] lower case [[epsilon]] ''ε'' or ''ϵ'', or less commonly the [[Latin alphabet|Latin]] lower case ''e''. [[Index notation]] allows one to display permutations in a way compatible with tensor analysis:
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| :<math>\varepsilon_{i_1 i_2 \cdots i_n}</math>
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| where ''each'' index ''i''<sub>1</sub>, ''i''<sub>2</sub>, …, ''i''<sub>''n''</sub> takes values 1, 2, …, ''n''. There are ''n<sup>n</sup>'' indexed values of <math>\varepsilon_{i_1i_2\cdots i_n}</math>, which can be arranged into an ''n''-dimensional array. The key definitive property of the symbol is ''total [[Antisymmetric tensor|antisymmetry]]'' in all the indices. When any two indices are interchanged, equal or not, the symbol is negated:
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| :<math>\varepsilon_{\cdots i_p \cdots i_q \cdots }=-\varepsilon_{\cdots i_q \cdots i_p \cdots } .</math>
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| If any two indices are equal, the symbol is zero. When all indices are unequal, we have:
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| :<math>\varepsilon_{i_1 i_2 \cdots i_n} = (-1)^p \varepsilon_{1 2 \cdots n} ,</math>
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| where ''p'' (called the parity of the permutation) is the number of interchanges of indices necessary to unscramble ''i''<sub>1</sub>, ''i''<sub>2</sub>, …, ''i''<sub>''n''</sub> into the order 1, 2, …, ''n'', and the factor (−1)<sup>''p''</sup> is called the [[Parity of a permutation|sign or signature]] of the permutation. The value ''ε''<sub>12…''n''</sub> must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose {{nowrap|1=''ε''<sub>12…''n''</sub> = +1}}, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
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| The term "''n''-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol ''n'' matches the [[dimension]]ality of the relevant [[vector space]] in question, which may be [[Euclidean space|Euclidean]] or [[non-Euclidean space|non-Euclidean]], pure space or [[spacetime]]. The values of the Levi-Civita symbol are independent of any [[metric tensor]] and [[coordinate system]].
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| The Levi-Civita symbol may be used to express the [[determinant]] of a square matrix and the [[cross product]] of two vectors in 3d Euclidean space.
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| ==Definition==
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| The common dimensionalities of the Levi-Civita symbol are in 3d and 4d, and to some extent 2d, so it is useful to see these definitions before the general one in any number of dimensions.
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| ===Two dimensions===
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| The [[two dimensions|two-dimensional]] Levi-Civita symbol is defined by:
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| :<math> \varepsilon_{ij} =
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| \begin{cases}
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| +1 & \text{if } (i,j) \text{ is } (1,2) \\
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| -1 & \text{if } (i,j) \text{ is } (2,1) \\
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| \;\;\,0 & \text{if }i=j
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| \end{cases} </math>
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| The values can be arranged into a 2 × 2 [[antisymmetric matrix]]:
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| :<math> \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} \\ \varepsilon_{21} & \varepsilon_{22} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}</math>
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| Use of the 2d symbol is relatively uncommon, although in certain specialized topics like [[supersymmetry]]<ref>{{cite book|title=Supersymmetry|author=P. Labelle|series=Demystified|publisher=McGraw-Hill|pages=57–58|year=2010|isbn=978-0-07-163641-4}}</ref> and [[twistor theory]]<ref>{{cite web|accessdate=03/09/2013|title=Twistor Primer|author=F. Hadrovich|url=http://users.ox.ac.uk/~tweb/00004/index.shtml}}</ref> it appears in the context of 2-[[spinors]]. The 3d and higher-dimensional Levi-Civita symbols are used more commonly.
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| ===Three dimensions===
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| [[File:Permutation indices 3d numerical.svg|thumb|210|For the indices (''i'', ''j'', ''k'') in ''ε<sub>ijk</sub>'', the values 1, 2, 3 occurring in the cyclic order (1,2,3) (yellow) correspond to ''ε'' = +1, while occurring in the reverse cyclic order (red) correspond to ''ε'' = −1, otherwise ''ε'' = 0.]]
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| In [[three dimensions]], the Levi-Civita symbol is defined as follows:<ref name="Tyldesley">{{cite book| author=J.R. Tyldesley| title=An introduction to Tensor Analysis: For Engineers and Applied Scientists| publisher=Longman| year=1973 | isbn=0-582-44355-5}}</ref>
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| :<math> \varepsilon_{ijk} =
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| \begin{cases}
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| +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (3,1,2) \text{ or } (2,3,1), \\
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| -1 & \text{if } (i,j,k) \text{ is } (1,3,2), (3,2,1) \text{ or } (2,1,3), \\
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| \;\;\,0 & \text{if }i=j \text{ or } j=k \text{ or } k=i
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| \end{cases} </math>
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| i.e. <math> \varepsilon_{ijk} </math> is 1 if (''i'', ''j'', ''k'') is an [[even and odd permutations|even permutation]] of (1,2,3), −1 if it is an [[odd permutation]], and 0 if any index is repeated. In three dimensions, and not higher, the [[cyclic permutation]]s of (1,2,3) are all even permutations, similarly the [[anticyclic permutation]]s are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of (1,2,3) and easily obtain all the even or odd permutations.
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| Analogous to 2d matrices, the values of the 3d Levi-Civita symbol can be arranged into a 3×3×3 array:
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| :[[File:Epsilontensor.svg|200px]]
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| where ''i'' is the depth, ''j'' the row and ''k'' the column.
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| Some examples:
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| :<math> \varepsilon_{\color{BrickRed}{1}\color{Violet}{3}\color{Orange}{2}} = -\varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}} = - 1</math>
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| :<math>\varepsilon_{\color{Violet}{3}\color{BrickRed}{1}\color{Orange}{2}} = -\varepsilon_{\color{Orange}{2}\color{BrickRed}{1}\color{Violet}{3}} = -(-\varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}}) = 1</math>
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| :<math>\varepsilon_{\color{Orange}{2}\color{Violet}{3}\color{BrickRed}{1}} = -\varepsilon_{\color{BrickRed}{1}\color{Violet}{3}\color{Orange}{2}} = -(-\varepsilon_{\color{BrickRed}{1}\color{Orange}{2}\color{Violet}{3}}) = 1</math>
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| :<math>\varepsilon_{\color{Orange}{2}\color{Violet}{3}\color{Orange}{2}} = -\varepsilon_{\color{Orange}{2}\color{Violet}{3}\color{Orange}{2}} = 0</math>
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| ===Four dimensions===
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| In [[Four-dimensional space|four dimensions]], the Levi-Civita symbol is defined as:
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| :<math>\varepsilon_{ijkl } =
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| \begin{cases}
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| +1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\
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| -1 & \text{if }(i,j,k,l) \text{ is an odd permutation of } (1,2,3,4) \\
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| 0 & \text{otherwise}
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| \end{cases}
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| </math>
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| These values can be arranged into a 4×4×4×4 array, although in 4d and higher this is difficult to draw.
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| Some examples:
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| :<math>\varepsilon_{\color{BrickRed}{1}\color{RedViolet}{4}\color{Violet}{3}\color{Orange}{\color{Orange}{2}}} = -\varepsilon_{\color{BrickRed}{1}\color{Orange}{\color{Orange}{2}}\color{Violet}{3}\color{RedViolet}{4}} = - 1</math>
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| :<math>\varepsilon_{\color{Orange}{\color{Orange}{2}}\color{BrickRed}{1}\color{Violet}{3}\color{RedViolet}{4}} = -\varepsilon_{\color{BrickRed}{1}\color{Orange}{\color{Orange}{2}}\color{Violet}{3}\color{RedViolet}{4}} = -1</math>
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| :<math>\varepsilon_{\color{RedViolet}{4}\color{Violet}{3}\color{Orange}{\color{Orange}{2}}\color{BrickRed}{1}} = -\varepsilon_{\color{BrickRed}{1}\color{Violet}{3}\color{Orange}{\color{Orange}{2}}\color{RedViolet}{4}} = -(-\varepsilon_{\color{BrickRed}{1}\color{Orange}{\color{Orange}{2}}\color{Violet}{3}\color{RedViolet}{4}}) = 1</math>
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| :<math>\varepsilon_{\color{Violet}{3}\color{Orange}{\color{Orange}{2}}\color{RedViolet}{4}\color{Violet}{3}} = -\varepsilon_{\color{Violet}{3}\color{Orange}{\color{Orange}{2}}\color{RedViolet}{4}\color{Violet}{3}} = 0</math>
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| ===Generalization to ''n'' dimensions===
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| The Levi-Civita symbol can be generalized to [[n-dimensional space|''n'' dimensions]]:<ref name="Kay">{{cite book| author=D.C. Kay| title=Tensor Calculus| publisher=Schaum’s Outlines, McGraw Hill (USA)| year=1988 | isbn=0-07-033484-6}}</ref>
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| :<math>\varepsilon_{a_1 a_2 a_3 \ldots a_n} =
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| \begin{cases}
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| +1 & \text{if }(a_1 , a_2 , a_3 , \ldots , a_n) \text{ is an even permutation of } (1,2,3,\dots,n) \\
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| -1 & \text{if }(a_1 , a_2 , a_3 , \ldots , a_n) \text{ is an odd permutation of } (1,2,3,\dots,n) \\
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| 0 & \text{otherwise}
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| \end{cases}
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| </math>
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| Thus, it is the [[Even and odd permutations|sign of the permutation]] in the case of a permutation, and zero otherwise.
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| Using the [[Multiplication#Capital Pi notation|capital Pi notation]] <math>\prod</math> for ordinary multiplication of numbers, an explicit expression for the symbol is:
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| :<math> \begin{align}
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| \varepsilon_{a_1 a_2 a_3 \ldots a_n} & = \prod_{1\leq i < j \leq n} \sgn ( a_j-a_i ) \\
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| & = \sgn(a_2 - a_1)\sgn(a_3 - a_1)\ldots\sgn(a_n - a_1)\sgn(a_3 - a_2)\sgn(a_4 - a_2)\ldots\sgn(a_n - a_2)\ldots\sgn(a_n - a_{n-1})
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| \end{align}</math>
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| where the product is totally antisymmetric in all indices, and the [[sign function]] (denoted by "sgn") extracts the sign of each difference discarding the [[absolute value]]. The formula is true for all index values, and for any ''n'' (when ''n'' = 1 or 0, this is the [[empty product]]). However, it is seldom used in practice since interchanging indices is quicker.
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| ==Properties==
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| A [[tensor]] whose components in an [[orthonormal basis]] are given by the Levi-Civita symbol (a tensor of [[Covariance and contravariance of vectors|covariant]] rank ''n'') is sometimes called a '''permutation tensor'''. It is actually a [[pseudotensor]] because under an [[Orthogonal matrix|orthogonal transformation]] of [[jacobian matrix and determinant|jacobian determinant]] −1 (i.e., a [[rotation]] composed with a [[Reflection (mathematics)|reflection]]), it acquires a minus sign. As the Levi-Civita symbol is a pseudotensor, the result of taking a [[cross product]] is a [[pseudovector]], not a vector.<ref name="Riley et al">{{cite book| author=K.F. Riley, M.P. Hobson, S.J. Bence| title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref>
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| Under a general [[coordinate change]], the components of the permutation tensor are multiplied by the [[Jacobian matrix and determinant|jacobian]] of the [[transformation matrix]]. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.<ref name="Riley et al"/>
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| In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the [[Hodge dual]].
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| In a context of where tensor index notation is used to manipulate tensor components, the Levi-Civita symbol may be written with its indices as either subscripts or superscripts with no change in meaning, as might be convenient. Thus, one could write
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| :<math>\varepsilon^{ij\dots k} = \varepsilon_{ij\dots k} .</math> | |
| In these examples, superscripts should be considered equivalent with subscripts.
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| Summation symbols can be eliminated by using [[Einstein notation]], where an index repeated between two or more terms indicates summation over that index. For example
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| :<math>\varepsilon_{ijk} \varepsilon^{imn} \equiv \sum_{i=1,2,3} \varepsilon_{ijk} \varepsilon^{imn}</math>.
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| In the following examples, Einstein notation is used.
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| ===Two dimensions===
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| In [[two dimensions]], when all ''i'', ''j'', ''m'', ''n'' each take the values 1 and 2,<ref name="Tyldesley"/>
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| {{NumBlk|:|
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| <math>\varepsilon_{ij} \varepsilon^{mn} = \delta_i {}^m \delta_j {}^n - \delta_i {}^n \delta_j {}^m </math>
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| |{{EquationRef|1}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{ij} \varepsilon^{in} = \delta_j{}^n </math>
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| |{{EquationRef|2}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{ij} \varepsilon^{ij} = 2. </math>
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| |{{EquationRef|3}}}}
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| ===Three dimensions===
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| ;Index and symbol values:
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| In three dimensions, when all ''i'', ''j'', ''k'', ''m'', ''n'' each take values 1, 2, and 3:<ref name="Tyldesley"/>
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| {{NumBlk|:|
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| <math>\varepsilon_{ijk} \varepsilon^{imn}=\delta_j{}^{m}\delta_k{}^n - \delta_j{}^n\delta_k{}^m </math>
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| |{{EquationRef|4}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{jmn} \varepsilon^{imn}=2\delta^i_j </math>
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| |{{EquationRef|5}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{ijk} \varepsilon^{ijk}=6. </math>
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| |{{EquationRef|6}}}}
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| ;Product:
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| The Levi-Civita symbol is related to the [[Kronecker delta]]. In three dimensions, the relationship is given by the following equations (vertical lines denote the [[determinant]]):<ref name="Kay"/>
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| :<math>\begin{align}
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| \varepsilon_{ijk}\varepsilon_{lmn} & = \begin{vmatrix}
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| \delta_{il} & \delta_{im}& \delta_{in}\\
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| \delta_{jl} & \delta_{jm}& \delta_{jn}\\
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| \delta_{kl} & \delta_{km}& \delta_{kn}\\
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| \end{vmatrix}\\
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| & = \delta_{il}\left( \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\right) - \delta_{im}\left( \delta_{jl}\delta_{kn} - \delta_{jn}\delta_{kl} \right) + \delta_{in} \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \right).
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| \end{align}</math>
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| A special case of this result is ({{EquationNote|4}}):
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| :<math>
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| \sum_{i=1}^3 \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}
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| </math>
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| sometimes called the "[[Tensor contraction|contracted]] epsilon identity".
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| In [[Einstein notation]], the duplication of the ''i'' index implies the sum on ''i''. The previous is then denoted: <math> \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\,.</math>
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| :<math>
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| \sum_{i=1}^3 \sum_{j=1}^3 \varepsilon_{ijk}\varepsilon_{ijn} = 2\delta_{kn}
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| </math>
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| ===''n'' dimensions===
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| ;Index and symbol values:
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| In ''n'' dimensions, when all ''i''<sub>1</sub>,...,''i<sub>n</sub>'', ''j''<sub>1</sub>,...,''j''<sub>n</sub> take values 1, 2,..., ''n'':
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| {{NumBlk|:|
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| <math>\varepsilon_{i_1 \dots i_n} \varepsilon^{j_1 \dots j_n} = n! \delta_{[ i_1}{}^{j_1} \dots \delta_{i_n ]}{}^{j_n} = \delta^{j_1 \dots j_n}_{i_1 \dots i_n} </math>
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| |{{EquationRef|7}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{i_1 \dots i_k~i_{k+1}\dots i_n} \varepsilon^{i_1 \dots i_k~j_{k+1}\dots j_n}= k!(n-k)!~\delta_{[ i_{k+1}}{}^{j_{k+1}} \dots \delta_{i_n ]}{}^{j_n} = k!~\delta^{j_{k+1} \dots j_n}_{i_{k+1} \dots i_n} </math>
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| |{{EquationRef|8}}}}
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| {{NumBlk|:|
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| <math>\varepsilon_{i_1 \dots i_n}\varepsilon^{i_1 \dots i_n} = n! </math>
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| |{{EquationRef|9}}}}
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| where the exclamation mark (!) denotes the [[factorial]], and δ{{su|p=α…|b=β…}} is the [[generalized Kronecker delta]]. For any ''n'', the property
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| :<math>
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| \sum_{i,j,k,\dots=1}^n \varepsilon_{ijk\dots}\varepsilon_{ijk\dots} = n!
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| </math>
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| follows from the facts that
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| *every permutation is either even or odd,
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| *(+1)<sup>2</sup> = (−1)<sup>2</sup> = 1, and
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| *the number of permutations of any ''n''-element set number is exactly ''n''!.
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| ;Product:
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| In general, for ''n'' dimensions, one can write the product of two Levi-Civita symbols as:
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| :<math> \varepsilon_{i_1 i_2 \dots i_n} \varepsilon_{j_1 j_2 \dots j_n} = \begin{vmatrix}
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| \delta_{i_1 j_1} & \delta_{i_1 j_2} & \dots & \delta_{i_1 j_n} \\
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| \delta_{i_2 j_1} & \delta_{i_2 j_2} & \dots & \delta_{i_2 j_n} \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \delta_{i_n j_1} & \delta_{i_n j_2} & \dots & \delta_{i_n j_n} \\
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| \end{vmatrix} </math>.
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| ===Proofs===
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| For ({{EquationNote|1}}), both sides are [[antisymmetric]] with respect of ''ij'' and ''mn''. We therefore only need to consider the case ''i'' ≠ ''j'' and ''m'' ≠ ''n''. By substitution, we see that the equation holds for <math>\varepsilon_{12} \varepsilon^{12}</math>, i.e., for ''i'' = ''m'' = 1 and ''j'' = ''n'' = 2. (Both sides are then one). Since the equation is antisymmetric in ''ij'' and ''mn'', any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of ''ij'' and ''mn''.
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| Using ({{EquationNote|1}}), we have for ({{EquationNote|2}})
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| :<math> \varepsilon_{ij} \varepsilon^{in} = \delta_i{}^i \delta_j{}^n - \delta_i{}^n \delta_j{}^i = 2 \delta_j{}^n - \delta_j{}^n = \delta_j{}^n \,.</math>
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| Here we used the [[Einstein summation convention]] with ''i'' going from 1 to 2. Next, ({{EquationNote|3}}) follows similarly from ({{EquationNote|2}}).
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| To establish ({{EquationNote|5}}), notice that both sides vanish when ''i'' ≠ ''j''. Indeed, if ''i'' ≠ ''j'', then one can not choose ''m'' and ''n'' such that both permutation symbols on the left are nonzero. Then, with ''i'' ≠ ''j'' fixed, there are only two ways to choose ''m'' and ''n'' from the remaining two indices. For any such indices, we have
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| :<math>\varepsilon_{jmn} \varepsilon^{imn} = (\varepsilon^{imn})^2 = 1</math>
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| (no summation), and the result follows.
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| Then ({{EquationNote|6}}) follows since 3! = 6 and for any distinct indices ''i'', ''j'', ''k'' taking values 1, 2, 3, we have
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| :<math>\varepsilon_{ijk} \varepsilon^{ijk}=1</math> (no summation, distinct ''i'', ''j'', ''k'' ).
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| ==Applications and examples==
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| ===Determinants===
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| In [[linear algebra]], the [[determinant]] of a 3 × 3 [[square matrix]] '''A''' = (''a<sub>ij</sub>'') can be written<ref>{{cite book|edition=4th| author=S. Lipcshutz, M. Lipson| title=Linear Algebra| publisher=Schaum’s Outlines, McGraw Hill (USA)| year=2009 | isbn=978-0-07-154352-1}}</ref>
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| :<math>\det(\mathbf{A}) = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} a_{1i} a_{2j} a_{3k}</math>
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| Similarly the determinant of an ''n'' × ''n'' matrix '''A''' = (''a<sub>ij</sub>'') can be written as<ref name="Riley et al"/>
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| :<math> \det(\mathbf{A}) = \varepsilon_{i_1\cdots i_n} a_{1i_1} \cdots a_{ni_n},</math>
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| where each ''i<sub>r</sub>'' should be summed over 1,..., ''n'', or equivalently:
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| :<math> \det(\mathbf{A}) = \frac{1}{n!} \varepsilon_{i_1\cdots i_n} \varepsilon_{j_1\cdots j_n} a_{i_1 j_1} \cdots a_{i_n j_n},</math>
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| where now each ''i<sub>r</sub>'' and each ''j<sub>r</sub>'' should be summed over 1,.., ''n''. More generally, we have the identity<ref name="Riley et al"/>
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| :<math>\sum_{i_1,i_2,\dots}\varepsilon_{i_1\cdots i_n} a_{i_1 \, j_1} \cdots a_{i_n \, j_n} = \det(\mathbf{A}) \varepsilon_{j_1\cdots j_n}</math>
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| ===Vector cross product===
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| {{Main|cross product}}
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| ====Cross product (two vectors)====
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| If '''a''' = (''a''<sup>1</sup>, ''a''<sup>2</sup>, ''a''<sup>3</sup>) and '''b''' = (''b''<sup>1</sup>, ''b''<sup>2</sup>, ''b''<sup>3</sup>) are [[Vector (geometry)|vector]]s in <math>\mathbb{R}^3</math> (represented in some [[right-handed coordinate system]] using an [[orthonormal basis]]), their [[cross product]] can be written as a determinant:<ref name="Riley et al"/>
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| :<math>
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| \mathbf{a \times b} =
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| \begin{vmatrix}
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| \mathbf{e_1} & \mathbf{e_2} & \mathbf{e_3} \\
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| a^1 & a^2 & a^3 \\
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| b^1 & b^2 & b^3 \\
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| \end{vmatrix}
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| = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} \mathbf{e}_i a^j b^k
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| </math>
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| hence also using the Levi-Civita symbol, and more simply:
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| :<math>
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| (\mathbf{a \times b})_i = \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} a^j b^k.
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| </math>
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| In [[Einstein notation]], the summation symbols may be omitted, and the ''i''th component of their cross product equals<ref name="Kay"/>
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| :<math> (\mathbf{a\times b})_i = \varepsilon_{ijk} a^j b^k.</math>
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| The first component is
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| :<math>(\mathbf{a\times b})_1 = a^2 b^3-a^3 b^2\,,</math>
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| then by cyclic permutations of 1, 2, 3 the others can be derived immediately, without explicitly calculating them from the above formulae:
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| :<math>(\mathbf{a\times b})_2 = a^3 b^1-a^1 b^3\,,</math>
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| :<math>(\mathbf{a\times b})_3 = a^1 b^2-a^2 b^1\,.</math>
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| ====Triple scalar product (three vectors)====
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| From the above expression for the cross product, we have:
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| :<math>\mathbf{a\times b} = -\mathbf{b\times a}</math>.
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| If '''c''' = (''c''<sup>1</sup>, ''c''<sup>2</sup>, ''c''<sup>3</sup>) is another vector, then the [[triple scalar product]] equals
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| :<math> \mathbf{a}\cdot(\mathbf{b\times c}) = \varepsilon_{ijk} a^i b^j c^k.</math>
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| From this expression, it can be seen that the [[triple scalar product]] is antisymmetric when exchanging any pair of arguments. For example,
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| :<math>\mathbf{a}\cdot(\mathbf{b\times c})= -\mathbf{b}\cdot(\mathbf{a\times c})</math>.
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| ====Curl (one vector field)====
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| If '''F''' = (''F''<sup>1</sup>, ''F''<sup>2</sup>, ''F''<sup>3</sup>) is a vector field defined on some [[open set]] of <math>\mathbb{R}^3</math> as a [[function (mathematics)|function]] of [[position vector|position]] '''x''' = (''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>) (using [[Cartesian coordinates]]). Then the ''i''th component of the [[curl (mathematics)|curl]] of '''F''' equals<ref name="Kay"/>
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| :<math> (\nabla \times \mathbf{F})^i(\mathbf{x}) = \varepsilon^{ijk}\frac{\partial}{\partial x^j} F^k(\mathbf{x}),</math>
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| which follows from the cross product expression above, substituting components of the [[gradient]] vector [[linear operator|operator]] (nabla).
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| ==Tensor density==
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| In any arbitrary [[curvilinear coordinate system]] and even in the absence of a [[metric tensor|metric]] on the [[manifold]], the Levi-Civita symbol as defined above may be considered to be a [[tensor density]] field in two different ways. It may be regarded as a [[Covariance and contravariance of vectors|contravariant]] tensor density of weight +1 or as a [[Covariance and contravariance of vectors|covariant]] tensor density of weight −1. In ''n'' dimensions using the [[generalized Kronecker delta]],<ref>{{cite book |author=David Lovelock, Hanno Rund |title=Tensors, Differential Forms, and Variational Principles |publisher=Courier Dover Publications |year=1989 |isbn=0-486-65840-6 |page=113}}</ref>
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| :<math>\varepsilon^{\mu_1 \cdots \mu_n} = \delta^{\mu_1 \cdots \mu_n}_{\,1 \,\cdots \,n} \,</math>
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| :<math>\varepsilon_{\nu_1 \cdots \nu_n} = \delta^{\,1 \,\cdots \,n}_{\nu_1 \cdots \nu_n} \,.</math>
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| Notice that these are numerically identical. In particular, the sign is the same.
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| ===Absolute tensor===
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| In the presence of a [[metric tensor]] field, one may define absolute covariant and contravariant tensor fields that agree with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to that metric. These absolute tensor fields should not be confused with each other, nor should they be confused with the tensor density fields mentioned above. One of these absolute tensor fields may be converted to the other by raising or lowering the indices with the metric as is usual, but a minus sign is needed if the [[metric signature]] contains an odd number of negatives. For example, in [[Minkowski space]] (the four dimensional [[spacetime]] of [[special relativity]])
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| :<math>E^{\alpha \beta \gamma \delta} E^{\rho \sigma \mu \nu} = - g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} \delta^{\rho \sigma \mu \nu}_{\zeta \eta \theta \iota} \,</math>
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| :<math>E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} = - g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \,</math>
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| :<math>E^{\alpha \beta \gamma \delta} = - g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} E_{\zeta \eta \theta \iota} \,.</math>
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| Notice the minus signs.
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| ==See also==
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| *[[Symmetric tensor]]
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| *[[Antisymmetric tensor]]
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| *[[Kronecker delta]]
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| *[[List of permutation topics]]
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| ==References==
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| {{reflist}}
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| * {{cite book |page=p.85–86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
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| ==External links==
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| {{PlanetMath attribution|id=4116|title=Levi-Civita permutation symbol}}
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| * [http://mathworld.wolfram.com/PermutationTensor.html Permutation Tensor] - mathworld.wolfram
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| {{tensors}}
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| [[Category:Linear algebra]]
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| [[Category:Tensors]]
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| [[Category:Permutations]]
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| [[Category:Articles containing proofs]]
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