# Levis 4d

See Ricci calculus, Einstein notation, and Raising and lowering indices for the index notation used in the article. In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2. ., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita.

Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms levis 4d permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e.

Index notation allows one to display permutations in a way compatible with tensor analysis: levis 4d i 1 i 2 … i n {\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}} where each index i 1, i 2. ., i n takes levis 4d 1, 2.

., n. There are n n indexed values of ε i levis 4d i 2. i n, which can be arranged into an n-dimensional array. The key defining property of the symbol is total antisymmetry in the indices. When any two indices are interchanged, equal or not, the symbol is negated: ε … i p … i q … = − ε … i q … i p …. {\displaystyle \varepsilon _{\dots i_{p}\dots i_{q}\dots }=-\varepsilon _{\dots i_{q}\dots i_{p}\dots }.} If any two indices are equal, the symbol is zero.

When all indices are unequal, we have: ε i 1 i 2 … i n = ( − 1 ) p ε 1 2 … n{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}=(-1)^{p}\varepsilon _{1\,2\,\dots n},} where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i 1, i 2. ., i n into the order 1, 2. ., n, and the factor (−1) p is called the sign or signature of the permutation. The value ε 1 2 . n must be defined, else the particular values of the symbol for all permutations are indeterminate.

Most authors choose ε 1 2 . n = +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. The term " n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, R 3 {\displaystyle \mathbb {R} ^{3}} or Minkowski space.

The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.

The Levi-Civita symbol allows the determinant of a square matrix, and the cross product levis 4d two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation. Contents • 1 Definition • 1.1 Two dimensions • 1.2 Three dimensions • 1.3 Four dimensions • 1.4 Generalization to n dimensions • 2 Properties • 2.1 Two dimensions • 2.2 Three dimensions • 2.2.1 Index and symbol values • 2.2.2 Product • 2.3 n dimensions • 2.3.1 Index and symbol values • 2.3.2 Product • 2.4 Proofs • 3 Applications and examples • 3.1 Determinants • 3.2 Vector cross product • 3.2.1 Cross product (two vectors) • 3.2.2 Triple scalar product (three vectors) • 3.2.3 Curl (one vector field) • 4 Tensor density • 5 Levi-Civita tensors • 5.1 Example: Minkowski space • 6 See also • 7 Notes • 8 References • 9 External links Definition [ edit ] The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining levis 4d general case.

Two dimensions [ edit ] In two dimensions, the Levi-Civita symbol is defined by: ε i j = { + 1 if ( ij ) = ( 12 ) − 1 if ( ij ) = ( 21 ) 0 if i = j {\displaystyle \varepsilon _{ij}={\begin{cases}+1&{\text{if }}(i,j)=(1,2)\\-1&{\text{if }}(i,j)=(2,1)\\\;\;\,0&{\text{if }}i=j\end{cases}}} The values can be arranged into a 2 × 2 antisymmetric matrix: ( ε levis 4d ε 12 ε 21 ε 22 ) = ( 0 1 − 1 0 ) {\displaystyle {\begin{pmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{21}&\varepsilon _{22}\end{pmatrix}}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}} Use of the two-dimensional symbol is relatively uncommon, although in certain specialized topics like supersymmetry [1] and twistor theory [2] it appears in the context of 2- spinors.

The three- and higher-dimensional Levi-Civita symbols are used more commonly. Three dimensions [ edit ] For the indices ( i, j, k) in ε ijk, the values 1, 2, 3 occurring in the cyclic order (1, 2, 3) correspond levis 4d ε = +1, while occurring in the reverse cyclic order correspond to ε = −1, otherwise ε = 0.

In three dimensions, the Levi-Civita symbol is defined by: [3] ε i j k = { + 1 if ( ijk ) is ( 123 )( 231 )or ( 312 )− 1 if ( ijk ) is ( 321 )( 132 )or ( 213 )0 if i = jor j = kor k = i {\displaystyle \varepsilon _{ijk}={\begin{cases}+1&{\text{if }}(i,j,k){\text{ is }}(1,2,3),(2,3,1),{\text{ or }}(3,1,2),\\-1&{\text{if }}(i,j,k){\text{ is }}(3,2,1),(1,3,2),{\text{ or }}(2,1,3),\\\;\;\,0&{\text{if }}i=j,{\text{ or }}j=k,{\text{ or }}k=i\end{cases}}} That is, ε ijk is 1 if ( i, j, k) is an even permutation of (1, 2, 3), −1 if it is an odd permutation, and 0 if any index is repeated.

In three dimensions only, the cyclic permutations of (1, 2, 3) are all even permutations, similarly the anticyclic permutations 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. Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a 3 × 3 × 3 array: where i is the depth ( blue: i = 1; red: i = 2; green: i = 3), j is the row and k is the column.

Some examples: ε 1 3 2 = − ε 1 2 3 = − 1 ε 3 1 2 = − ε 2 1 3 = − ( − ε 1 2 3 ) = 1 ε 2 3 1 = − ε 1 3 2 = − ( − ε 1 2 3 ) = 1 ε 2 3 2 = − ε 2 3 2 = 0 {\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}}&=-1\\\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\\\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\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}&=0\end{aligned}}} Four dimensions [ edit ] In four dimensions, the Levi-Civita symbol is defined by: ε i j k l = { + 1 if ( ijkl ) is an even permutation of ( 1234 ) − 1 if ( ijkl ) is an odd permutation of ( 1234 ) 0 otherwise {\displaystyle \varepsilon levis 4d }}(i,j,k,l){\text{ is an even permutation of }}(1,2,3,4)\\-1&{\text{if }}(i,j,k,l){\text{ is an odd permutation of }}(1,2,3,4)\\\;\;\,0&{\text{otherwise}}\end{cases}}} These values can be arranged into a 4 × 4 × 4 × 4 array, although in 4 dimensions and higher this is difficult to draw.

Some examples: ε 1 4 3 2 = − ε 1 2 3 4 = − 1 ε levis 4d 1 3 4 = − ε 1 2 3 4 = − 1 ε 4 3 2 1 = − ε 1 3 2 4 = − ( − ε 1 2 3 4 ) = 1 ε 3 2 4 3 = − ε 3 2 4 3 levis 4d 0 {\displaystyle {\begin{aligned}\varepsilon _{\color levis 4d {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\\\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\\\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\\\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\end{aligned}}} Generalization to n dimensions [ edit ] More generally, in levis 4d dimensions, the Levi-Civita symbol is defined by: [4] ε a 1 a 2 a 3 … a n = { + 1 if ( a 1a 2a 3…a n ) is an even permutation of ( 123…n ) − 1 if ( a 1a 2a 3…a n ) is an odd permutation of ( 123…n ) 0 otherwise {\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an even permutation of }}(1,2,3,\dots ,n)\\-1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an odd permutation of }}(1,2,3,\dots ,n)\\\;\;\,0&{\text{otherwise}}\end{cases}}} Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.

Using the capital pi notation Π for ordinary multiplication of numbers, an explicit expression for the symbol is: [ citation needed] ε a 1 a 2 a 3 … a n = ∏ 1 ≤ i < j ≤ n sgn ⁡ ( a j − a i ) = sgn ⁡ ( a 2 − a 1 ) sgn ⁡ ( a 3 − a 1 ) ⋯ sgn ⁡ ( a n − a 1 ) sgn ⁡ ( levis 4d 3 − a 2 ) sgn ⁡ ( a 4 − a 2 ) ⋯ sgn ⁡ ( a n − a 2 ) ⋯ sgn ⁡ ( a n − a n − 1 ) {\displaystyle {\begin{aligned}\varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}&=\prod _{1\leq i

The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O( n 2), whereas levis 4d sign can be computed from the parity of the permutation from its disjoint cycles in only O( n log( n)) cost. Properties [ edit ] A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor.

Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems related by orthogonal transformations. However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant −1, for example, a reflection in an levis 4d number of dimensions, it should acquire a minus sign if it were a tensor.

As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor. As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. [5] Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix. This implies that in coordinate frames different from levis 4d 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 levis 4d, the factor will be ±1 depending on whether the orientation of the frame is the same or not. [5] In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.

Summation levis 4d can be eliminated by using Einstein notation, where an index repeated between two or more terms indicates summation over that index. For example, ε i j k ε i m n ≡ ∑ i = 123 ε i j k ε i m n {\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}\equiv \sum _{i=1,2,3}\varepsilon _{ijk}\varepsilon ^{imn}}.

In the following examples, Einstein notation is used. Two dimensions [ edit ] In two dimensions, when all i, j, m, n each take the values 1 and 2: [3] ( 6) Product [ edit ] 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): [4] ε i j k ε l m n = - δ i l δ i m δ i n δ j l δ j m δ j n δ k l δ k m δ k n - = δ i l ( δ j m δ k n − δ j n δ k m ) − δ i m ( δ j levis 4d δ k n − δ j n δ k l ) + δ i n ( δ j l δ k m − δ j m δ k l ) .

{INSERTKEYS} {\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\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).\end{aligned}}} A special case of this result is ( 4): ∑ i = 1 3 ε i j k ε i m n = δ j m δ k n − δ j n δ k m {\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}} sometimes called the " contracted epsilon identity".

In Einstein notation, the duplication of the i index implies the sum on i. The previous is then denoted ε ijkε imn = δ jmδ kn − δ jnδ km. ∑ i = 1 3 ∑ j = 1 3 ε i j k ε i j n = 2 δ k n {\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}} n dimensions [ edit ] Index and symbol values [ edit ] In n dimensions, when all i 1, ..., i n, j 1, ..., j n take values 1, 2, ..., n: ε i 1 … i k i k + 1 … i n ε i 1 … i k j k + 1 … j n = δ i 1 … i k i k + 1 … i n i 1 … i k j k + 1 … j n = k !

δ i k + 1 … i n j k + 1 … j n {\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=\delta _{i_{1}\ldots i_{k}~i_{k+1}\ldots i_{n}}^{i_{1}\dots i_{k}~j_{k+1}\ldots j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}} ( 9) where the exclamation mark ( !) denotes the factorial, and δ α... β... is the generalized Kronecker delta. For any n, the property ∑ i , j , k , ⋯ = 1 n ε i j k … ε i j k … = n ! {\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!} follows from the facts that • every permutation is either even or odd, • (+1) 2 = (−1) 2 = 1, and • the number of permutations of any n-element set number is exactly n!.

Product [ edit ] In general, for n dimensions, one can write the product of two Levi-Civita symbols as: ε i 1 i 2 … i n ε j 1 j 2 … j n = - δ i 1 j 1 δ i 1 j 2 … δ i 1 j n δ i 2 j 1 δ i 2 j 2 … δ i 2 j n ⋮ ⋮ ⋱ ⋮ δ i n j 1 δ i n j 2 … δ i n j n - .

{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}\varepsilon _{j_{1}j_{2}\dots j_{n}}={\begin{vmatrix}\delta _{i_{1}j_{1}}&\delta _{i_{1}j_{2}}&\dots &\delta _{i_{1}j_{n}}\\\delta _{i_{2}j_{1}}&\delta _{i_{2}j_{2}}&\dots &\delta _{i_{2}j_{n}}\\\vdots &\vdots &\ddots &\vdots \\\delta _{i_{n}j_{1}}&\delta _{i_{n}j_{2}}&\dots &\delta _{i_{n}j_{n}}\\\end{vmatrix}}.} Proofs [ edit ] For ( 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 ε 12 ε 12, that is, 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. Using ( 1), we have for ( 2) ε i j ε i n = δ i i δ j n − δ i n δ j i = 2 δ j n − δ j n = δ j n . {\displaystyle \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}\,.} Here we used the Einstein summation convention with i going from 1 to 2.

Next, ( 3) follows similarly from ( 2). To establish ( 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 ε j m n ε i m n = ( ε i m n ) 2 = 1 {\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=\left(\varepsilon ^{imn}\right)^{2}=1} (no summation), and the result follows. Then ( 6) follows since 3! = 6 and for any distinct indices i, j, k taking values 1, 2, 3, we have ε i j k ε i j k = 1 {\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=1} (no summation, distinct i, j, k) Applications and examples [ edit ] Determinants [ edit ] In linear algebra, the determinant of a 3 × 3 square matrix A = [ a ij] can be written [6] det ( A ) = ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 3 ε i j k a 1 i a 2 j a 3 k {\displaystyle \det(\mathbf {A} )=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{1i}a_{2j}a_{3k}} Similarly the determinant of an n × n matrix A = [ a ij] can be written as [5] det ( A ) = ε i 1 … i n a 1 i 1 … a n i n , {\displaystyle \det(\mathbf {A} )=\varepsilon _{i_{1}\dots i_{n}}a_{1i_{1}}\dots a_{ni_{n}},} where each i r should be summed over 1, ..., n, or equivalently: det ( A ) = 1 n !

{/INSERTKEYS}

ε i 1 … i n ε j 1 … j n a i 1 j 1 … a i n j n{\displaystyle \det(\mathbf {A} )={\frac {1}{n!}}\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}a_{i_{1}j_{1}}\dots a_{i_{n}j_{n}},} where now each i r and each j r should be summed over 1. ., n. More generally, we have the identity [5] ∑ i 1i 2… ε i 1 … i n a i 1 j 1 … a i n j n = det ( A ) ε j 1 … j n {\displaystyle \sum _{i_{1},i_{2},\dots }\varepsilon _{i_{1}\dots i_{n}}a_{i_{1}\,j_{1}}\dots a_{i_{n}\,j_{n}}=\det(\mathbf {A} )\varepsilon _{j_{1}\dots j_{n}}} Vector cross product [ edit ] Main article: cross product Levis 4d product (two vectors) [ edit ] Let ( e 1e 2levis 4d 3 ) {\displaystyle (\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} )} a positively oriented orthonormal basis of a vector space.

If ( a 1, a 2, a 3) and ( b 1, b 2, b 3) are the coordinates of the vectors a and b in this basis, levis 4d their cross product can be written as a determinant: [5] a × b = - e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 - = ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 3 ε i j k e i a j b k {\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\\end{vmatrix}}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}\mathbf {e} _{i}a^{j}b^{k}} hence also using the Levi-Civita symbol, and more simply: ( a levis 4d b ) i = ∑ j = 1 3 ∑ k = 1 3 ε i j k a j b k.

{\displaystyle (\mathbf {a\times b} )^{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}.} In Einstein notation, the summation symbols may be omitted, and the ith component of their cross product equals [4] ( a × b ) i = ε i j k a j b k.

{\displaystyle (\mathbf {a\times b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}.} The first component is ( a × b ) 1 = a 2 b 3 − a 3 b 2{\displaystyle (\mathbf {a\times b} )^{1}=a^{2}b^{3}-a^{3}b^{2}\,} then by cyclic permutations of 1, 2, 3 the others can be derived immediately, without explicitly calculating them from the above formulae: ( a × b ) 2 = a 3 b 1 − a 1 b 3( a × b ) 3 = a 1 b 2 − a 2 b 1.

{\displaystyle {\begin{aligned}(\mathbf {a\times b} )^{2}&=a^{3}b^{1}-a^{1}b^{3}\,\\(\mathbf {a\times b} )^{3}&=a^{1}b^{2}-a^{2}b^{1}\.\end{aligned}}} Triple levis 4d product (three vectors) [ edit ] From the above expression for the cross product, we have: a × b = − b × a {\displaystyle \mathbf {a\times b} =-\mathbf {b\times a} }.

If c = ( c 1, c 2, c 3) is a third vector, then the triple scalar product equals a ⋅ ( b × c ) = ε i j k a i b j c levis 4d. {\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=\varepsilon _{ijk}a^{i}b^{j}c^{k}.} From this expression, it can be levis 4d that the triple scalar product is antisymmetric when exchanging any pair of arguments.

For example, a ⋅ ( b × c ) = − levis 4d ⋅ ( a × c ) {\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=-\mathbf {b} \cdot (\mathbf {a\times c} )}. Curl (one vector field) [ edit ] If F = ( F 1, F 2, F 3) is a vector field defined on some open set of R 3 {\displaystyle \mathbb {R} ^{3}} as a function of position x = ( x 1, x 2, x 3) (using Cartesian coordinates).

Then the ith component of the curl of F equals [4] ( ∇ × F ) i ( x ) = ε i j k ∂ ∂ x j F k ( x ){\displaystyle (\nabla \times \mathbf {F} )^{i}(\mathbf {x} )=\varepsilon _{ijk}{\frac {\partial }{\partial x^{j}}}F^{k}(\mathbf {x} ),} which follows from the cross product expression above, substituting components of the gradient vector operator (nabla).

Tensor density [ edit ] In any arbitrary curvilinear coordinate system and even in the absence of a metric on levis 4d 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 contravariant tensor density of weight +1 or as a covariant tensor density of weight −1. In n dimensions using the generalized Kronecker delta, [7] [8] ε μ 1 … μ n = δ 1 … n μ 1 … μ n ε ν 1 … ν n = δ ν 1 … ν n 1 … n.

{\displaystyle {\begin{aligned}\varepsilon ^{\mu _{1}\dots \mu _{n}}&=\delta _{\,1\,\dots \,n}^{\mu _{1}\dots \mu _{n}}\,\\\varepsilon _{\nu _{1}\dots \nu _{n}}&=\delta _{\nu _{1}\dots \nu _{n}}^{\,1\,\dots \,n}\.\end{aligned}}} Notice that these are numerically identical. In particular, the sign is the same. Levi-Civita tensors [ edit ] On a pseudo-Riemannian manifold, one may define a coordinate-invariant covariant tensor field whose coordinate representation agrees with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation.

This tensor should not be confused with the tensor density field mentioned above. The presentation in this section closely follows Carroll 2004.

The covariant Levi-Civita tensor (also known as the Riemannian volume form) in any coordinate system that matches the selected orientation is E a 1 … a n = - det [ g a b ] - ε a 1 … a n{\displaystyle E_{a_{1}\dots a_{n}}={\sqrt {\left-\det[g_{ab}]\right-}}\,\varepsilon _{a_{1}\dots a_{n}}\,} where g ab is the representation of the metric in that coordinate system. We can similarly consider a contravariant Levi-Civita tensor by raising the indices with the metric as usual, E a 1 … a n = E b 1 … b n ∏ i = 1 n g a i b i{\displaystyle E^{a_{1}\dots a_{n}}=E_{b_{1}\dots b_{n}}\prod _{i=1}^{n}g^{a_{i}b_{i}}\,} but notice that if the metric signature contains an odd number of negatives q, then the sign of the components of this tensor differ from the standard Levi-Civita symbol: E a 1 … a n = sgn ⁡ ( det [ g a b ] ) - det levis 4d g a b ] - ε a 1 … a n{\displaystyle E^{a_{1}\dots a_{n}}={\frac {\operatorname {sgn} \left(\det[g_{ab}]\right)}{\sqrt {\left-\det[g_{ab}]\right-}}}\,\varepsilon ^{a_{1}\dots a_{n}},} where sgn(det[g ab]) = (−1) q, and ε a 1 … a n {\displaystyle \varepsilon ^{a_{1}\dots a_{n}}} is the usual Levi-Civita symbol discussed in the rest of this article.

More explicitly, when the tensor and basis orientation are chosen such that E 01 … n = + - det [ g a b ] - {\textstyle Levis 4d n}=+{\sqrt {\left-\det[g_{ab}]\right-}}}we have that E 01 … n = sgn ⁡ ( det [ g a b ] ) - det [ g a b ] - {\displaystyle E^{01\dots n}={\frac {\operatorname {sgn}(\det[g_{ab}])}{\sqrt {\left-\det[g_{ab}]\right-}}}}.

From this we can infer the identity, E μ 1 … μ p α 1 … α n − p E μ 1 levis 4d μ p β 1 … β n − p = ( − 1 ) q p ! δ β 1 … β n − p α 1 … α n − p{\displaystyle E^{\mu _{1}\dots \mu _{p}\alpha _{1}\dots \alpha _{n-p}}E_{\mu _{1}\dots \mu _{p}\beta _{1}\dots \beta _{n-p}}=(-1)^{q}p!\delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}\,} where δ β 1 … β n − p α 1 … α n − p = ( n − p ) !

δ β 1 [ α levis 4d … δ β n − p α n − p ] {\displaystyle \delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}=(n-p)!\delta _{\beta _{1}}^{\lbrack \alpha _{1}}\dots \delta _{\beta _{n-p}}^{\alpha _{n-p}\rbrack }} is the generalized Kronecker delta.

Example: Minkowski space [ edit ] In Minkowski space (the four-dimensional spacetime of special relativity), the covariant Levi-Civita tensor is E α β γ δ = ± - det [ g μ ν ] - ε α β γ δ{\displaystyle E_{\alpha \beta \gamma \delta }=\pm {\sqrt {\left-\det[g_{\mu \nu }]\right-}}\,\varepsilon _{\alpha \beta \gamma \delta }\,} where the sign depends on the orientation of the basis.

The contravariant Levi-Civita tensor is E α β γ δ = g α ζ g β η g γ θ g δ ι E ζ η θ ι. {\displaystyle E^{\alpha \beta \gamma \delta }=g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }E_{\zeta \eta \theta \iota }\.} The following are examples of the general identity above specialized to Minkowski space (with the negative sign arising from the odd number of negatives in the signature of the metric tensor in either sign convention): E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ levis 4d δ levis 4d δ ρ σ μ ν ζ η θ ι E α β γ δ E ρ σ μ ν = − g α ζ g β η g γ θ g δ ι δ ζ η θ ι ρ σ μ ν E α β γ δ E α β γ δ = − 24 E α β γ δ E ρ β γ δ = − 6 δ ρ α E α β γ δ E ρ σ γ δ = − 2 δ ρ σ α β E α β γ δ E ρ σ θ δ = − δ ρ σ θ α β γ.

{\displaystyle {\begin{aligned}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 }\\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 }\\E^{\alpha \beta \gamma \delta }E_{\alpha \beta \gamma \delta }&=-24\\E^{\alpha \beta \gamma \delta }E_{\rho \beta \gamma \delta }&=-6\delta _{\rho }^{\alpha }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \gamma \delta }&=-2\delta _{\rho \sigma }^{\alpha \beta }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \theta \delta }&=-\delta _{\rho \sigma \theta }^{\alpha \beta \gamma }\.\end{aligned}}} See also [ edit ] • List of permutation topics • Symmetric tensor Notes [ edit ] • ^ Labelle, P.

(2010). Supersymmetry. Demystified. McGraw-Hill. pp. 57–58. ISBN 978-0-07-163641-4. • ^ Hadrovich, F. "Twistor Primer". Retrieved 2013-09-03. • ^ a b c Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists.

Longman. ISBN 0-582-44355-5. • ^ a b c d Kay, D. C. (1988). Tensor Calculus. Schaum's Outlines. McGraw Hill. ISBN 0-07-033484-6. • ^ a b c d e Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010). Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 978-0-521-86153-3. • ^ Lipcshutz, S.; Lipson, M. (2009).

Linear Algebra. Schaum's Outlines (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1. • ^ Murnaghan, F. D. (1925), "The generalized Kronecker symbol and its application to the theory of determinants", Amer. Math. Monthly, 32 (5): 233–241, doi: 10.2307/2299191, JSTOR 2299191 • ^ Lovelock, David; Rund, Hanno (1989). Tensors, Differential Forms, and Variational Principles.

Courier Dover Publications. p. 113. ISBN 0-486-65840-6. References [ edit ] • Misner, C.; Thorne, K. S.; Wheeler, J. A. (1973).

Gravitation. W. H. Freeman & Co. pp. 85–86, §3.5. ISBN 0-7167-0344-0. • Neuenschwander, D. E. (2015). Tensor Calculus for Physics. Johns Hopkins University Press. pp. 11, 29, 95. ISBN 978-1-4214-1565-9. • Carroll, Sean M. (2004), Spacetime and Geometry, Addison-Wesley, ISBN 0-8053-8732-3 External links [ edit ] This article incorporates material from Levi-Civita permutation levis 4d on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. • Weisstein, Eric W.

"Permutation Tensor". MathWorld. • affine connection • basis • Cartan formalism (physics) • connection form • covariance and contravariance of vectors • differential form • dimension • exterior form • fiber bundle • geodesic • Levi-Civita connection • linear map • manifold • matrix • multivector • pseudotensor • spinor • vector, vector space Notable tensors Edit links levis 4d This page was last edited on 27 April 2022, at 09:28 (UTC).

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none There is a general formula for the product of multidimensional Levi-Civita symbols: $$\epsilon_{i_1 i_2 i_3 \ldots i_n}\epsilon_{j_1 j_2 j_3 \ldots j_n}= \det A$$ where A is the matrix with elements $$(A)_{mn}=\delta_{i_m j_n}$$.

Using this you could push out an identity with a bit of work. A more direct way is to look at symmetry consider expressions of the form: $$\epsilon^{\mu i_1 \ldots i_n} \epsilon_{\nu i_1 \ldots i_n}$$. (In your case levis 4d. The Levi-Civita symbol is zero unless all the terms are different, and there are only n+1 different choices for the indicies; thus for any given choice of $$i_1,\ldots,i_n$$ there is only one choice of mu such that the first term doesn't vanish, and only one choice of nu such that the second term doesn't vanish.

Consequently the whole expression is proportional to $$\delta^{\mu}_{\nu}$$. Now to find the constant of proportionality just work with any case: to get a non-vanishing term we require all the indicies to be different. If we choose mu=nu, then we have n ways of choosing i_1, (n-1) ways of choosing i_2. ., 1 way of choosing i_n; and so we get exactly n! non-vanishing terms. Now clearly each term is either 0 or 1, and so we conclude $$\epsilon^{\mu i_1 \ldots i_n} \epsilon_{\nu i_1 \ldots i_n}=n! \delta^{\mu}_{\nu}$$. So $$\epsilon^{\mu \beta \gamma \delta} \epsilon_{\nu \beta \gamma \delta} = 3! \delta^{\mu}_{\nu}$$ Hope I convinced you!
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## Unboxing и обзор коллаборации Levis x Air Jordan 4

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