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The Three - index version of the Levi - Civita symbol, introduced in Eq., Has values {matheq}{matheq}===+1,===−1, all other =0.{endmatheq}{endmatheq} suppose now that we have rank - 3 pseudotensor ijk, which in one particular Cartesian coordinate system is equal to i jk. Then, letting stand for matrix of coefficients in orthogonal transformation of {matheq}{matheq}{endmatheq}{endmatheq} we have in transform coordinate system {matheq}{matheq}=det,{endmatheq}{endmatheq} by definition of pseudotensor. All terms of pqr sum will vanish except those where pqr is permutation of 123, and when pqr is such permutation, sum will correspond to determinant of except that its rows will have been permuted from 123 to i jk. This means that pqr sum will have value i jk det, and {matheq}{matheq}==,{endmatheq}{endmatheq} where the final result depends on the fact that | det | = 1. If the reader is uncomfortable with the above analysis, result can be checked by enumeration of contributions of six permutations that correspond to nonzero values of {matheq}{matheq}{endmatheq}{endmatheq} equation not only shows that it is rank - 3 pseudotensor, but that it is also isotropic. In other words, it has the same components in all rotated Cartesian coordinate systems, and 1 times those component values in all Cartesian systems that are reached by improper rotations.

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In three dimensions, Levi - Civita symbol is defined by: {matheq}{\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}}}{endmatheq} that is, ijk is 1 if there is even permutation of, 1 if it is an odd permutation, and 0 if any index is repeat. In three dimensions only, cyclic permutations are all even permutations, similarly anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations and easily obtain all even or odd permutations. Analogous to 2 - dimensional matrices, values of 3 - dimensional Levi - Civita symbols can be arranged into 3 3 3 array: where I is depth, j is row and k is column.

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In three dimensions, when all i, j, k, m, n each take value 1, 2, and 3: {matheq}{\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}=\delta _{j}{}^{m}\delta _{k}{}^{n}-\delta _{j}{}^{n}\delta _{k}{}^{m}}{endmatheq} {matheq}{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}{endmatheq} {matheq}{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}{endmatheq} Levi - Civita symbol is related to the Kronecker delta. In three dimensions, relationship is given by the following equations: {matheq}{\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}}\ {matheq}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}}}{endmatheq} special case of this result is: {matheq}{\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}}{endmatheq} in Einstein notation, duplication of i index implies sum on i. Previous is then denoted ijk imn = jm kn jn km.

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If = and b = are vectors of 3, their cross product can be written as determinant: {matheq}{\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}}{endmatheq} {matheq}{\displaystyle (\mathbf {a\times b} )^{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}.}{endmatheq} in Einstein notation, summation symbols may be omit, and i component of their cross product equals {matheq}{\displaystyle (\mathbf {a\times b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}.}{endmatheq} {matheq}{\displaystyle (\mathbf {a\times b} )^{1}=a^{2}b^{3}-a^{3}b^{2}\,,}{endmatheq} then by cyclic permutations of 1 2 3 others can be derived immediately, without explicitly calculating them from above formulae: {matheq}{\displaystyle {\begin{aligned}(\mathbf {a\times b} )^{2}&=a^{3}b^{1}-a^{1}b^{3}\,,\ {matheq}\mathbf {a\times b} )^{3}&=a^{1}b^{2}-a^{2}b^{1}\,.\end{aligned}}}{endmatheq} from above expression for cross product, we have: {matheq}{\displaystyle \mathbf {a\times b} =-\mathbf {b\times a} }{endmatheq} if c = is third vector, then triple scalar product equals {matheq}{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=\varepsilon _{ijk}a^{i}b^{j}c^{k}.}{endmatheq} from this expression, it can be see that triple scalar product is antisymmetric when exchanging any pair of arguments. For example, {matheq}{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=-\mathbf {b} \cdot (\mathbf {a\times c} )}{endmatheq} if F = is a vector field defined on some open set of 3 as a function of position x =. Then i component of curl of F equals {matheq}{\displaystyle (\nabla \times \mathbf {F} )^{i}(\mathbf {x} )=\varepsilon ^{ijk}{\frac {\partial }{\partial x^{j}}}F_{k}(\mathbf {x} ),}{endmatheq} which follows from the cross product expression above, substituting components of gradient vector operator.

In any arbitrary curvilinear coordinate system and even in the absence of metric on manifold, Levi - Civita symbol as defined above may be considered to be tensor density field in two different ways. It may be regarded as contravariant tensor density of weight + 1 or as covariant tensor density of weight 1. In n dimensions using generalized Kronecker delta, {matheq}{\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}}}{endmatheq} notice that these are numerically identical. In particular, signs are the same.

The Permutation tensor, also called Levi - Civita tensor or isotropic tensor of rank 3, is a pseudotensor which is antisymmetric under interchange of any two slots. Recalling the definition of permutation symbol in terms of scalar triple product of Cartesian unit vectors, pseudotensor is generalization to an arbitrary basis defined by and, where is metric tensor. Is nonzero iff vectors are already independent. When viewed as a tensor, permutation symbol is sometimes known as the Levi - Civita tensor. The permutation tensor of rank four is important in general relativity, and has components defined as. Rank four permutation tensor satisfies identity

Mathematica supports several operations for combining or manipulating tensors. First, there is inner product; inner product of two tensors and B is contraction using the last index of and first index of B. It is therefore Tensor analog of matrix multiplication, and is written in Mathematica using dot operator. Also supported is Outer product Outer, also called direct product or Kronecker product, which is a product containing all indices of both factors. Contraction within the Tensor is supported by operation Tr, which can be used to form contraction on the first two indices of the Tensor. Dot and Tr Operations can be used to produce contractions of arbitrarily located Tensor indices by making use of operation Transpose, which supports arbitrary index permutations. We can permute indices to be contracted to necessary special positions, perform contractions, and then carry out any further index permutation that may be needed to reach proper index order. We will discuss Mathematica first. The Tensor of rank {matheq}{matheq}{endmatheq}{endmatheq} in 3 - D space corresponds to {matheq}{matheq}{endmatheq}{endmatheq} table of element values, with index values for each dimension ranging from 1 to 3. If Tensor elements are values of function {matheq}{matheq}{endmatheq}{endmatheq} of indices {matheq}{matheq}{endmatheq}{endmatheq} We can enter all elements by command of type illustrated here for Tensor of rank 4 with {matheq}{matheq}{endmatheq}{endmatheq} here {I1 3}, {I2 3}, {I3 3}, {I4 3} Indicate that the range of each {matheq}{matheq}{endmatheq}{endmatheq} is from 1 to 3. We see that output is nested list, with nesting depth equal to the rank of Tensor. Tensors with completely arbitrary elements can be entered using standard Mathematica list construction: Mathematica forces input into two - dimensional format if it is examined using MatrixForm. These are formats used for tensors of ranks 3 and 4: use of Mathematica commands referred to above is illustrated in the following comprehensive example. Example 8. 41 Tensor Operations in Mathematica Let's introduce two tensors for illustrative purposes: Let's also form rank - 3 Levi - Civita Tensor: We can verify that Levi is antisymmetric by examining the result of permuting its indices: second argument of Transpose gives order to which indices of the first argument are permute; for Levi, permutation We use is odd, so {matheq}{matheq}{endmatheq}{endmatheq} Levi. Reader can verify that even permutations leave Levi unchanged.

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