triple scalar product

Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.

Definition

In three dimensions, the Levi-Civita symbol is defined as follows:

varepsilon_{ijk} =
begin{cases} +1 & mbox{if } (i,j,k) mbox{ is } (1,2,3), (3,1,2) mbox{ or } (2,3,1), -1 & mbox{if } (i,j,k) mbox{ is } (3,2,1), (1,3,2) mbox{ or } (2,1,3), 0 & mbox{otherwise: }i=j mbox{ or } j=k mbox{ or } k=i, end{cases}

i.e. varepsilon_{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.

An alternative notation for the three dimensional Levi-Civita symbol without case differentiation is

varepsilon_{ijk} = -[(i-j)^2%3][(i-k)^2%3][(j-k)^2%3][(j-(i%3)-frac{1}{2})^2-frac{5}{4}]

with % representing the modulo operator and i,j,k in lbrace1,2,3rbrace .

For example, in linear algebra, the determinant of a 3×3 matrix A can be written

det A = sum_{i,j,k=1}^3 varepsilon_{ijk} a_{1i} a_{2j} a_{3k}

(and similarly for a square matrix of general size, see below)

and the cross product of two vectors can be written as a determinant:

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,j,k

1}^3 varepsilon_{ijk} mathbf{e_i} a_j b_k or more simply:
mathbf{a times b} = mathbf{c}, c_i = sum_{j,k=1}^3 varepsilon_{ijk} a_j b_k.

According to the Einstein notation, the summation symbol may be omitted.

The tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the permutation tensor. It is actually a pseudotensor because under an orthogonal transformation of jacobian determinant −1 (i.e., a rotation composed with a reflection), it acquires a minus sign. Because the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.

Note that under a general coordinate change, the components of the permutation tensor get multiplied by the 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.

Relation to Kronecker delta

The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations:

varepsilon_{ijk}varepsilon_{lmn} = det begin{bmatrix} delta_{il} & delta_{im}& delta_{in} delta_{jl} & delta_{jm}& delta_{jn} delta_{kl} & delta_{km}& delta_{kn} end{bmatrix}
= 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) ,
sum_{i=1}^3 varepsilon_{ijk}varepsilon_{imn} = delta_{jm}delta_{kn} - delta_{jn}delta_{km} ("contracted epsilon identity") (In Einstein notation, the duplication of the i index implies the sum on i. The previous is then noted: varepsilon_{ijk}varepsilon_{imn} = delta_{jm}delta_{kn} - delta_{jn}delta_{km})

sum_{i,j=1}^3 varepsilon_{ijk}varepsilon_{ijn} = 2delta_{kn}

Generalization to n dimensions

The Levi-Civita symbol can be generalized to higher dimensions:
varepsilon_{ijkldots} =
left{ begin{matrix} +1 & mbox{if }(i,j,k,l,dots) mbox{ is an even permutation of } (1,2,3,4,dots) -1 & mbox{if }(i,j,k,l,dots) mbox{ is an odd permutation of } (1,2,3,4,dots) 0 & mbox{if any two labels are the same} end{matrix} right.

Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.

Furthermore, for any n the property

sum_{i,j,k,dots=1}^n varepsilon_{ijkdots}varepsilon_{ijkdots} = n! follows from the facts that (a) every permutation is either even or odd, (b) (+1)2 = (-1)2 = 1, and (c) the permutations of any n-element set number exactly n!.

In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
In general n dimensions one can write the product of two Levi-Civita symbols as:

varepsilon_{ijkdots}varepsilon_{mnldots} = det begin{bmatrix}
delta_{im} & delta_{in} & delta_{il} & dots delta_{jm} & delta_{jn} & delta_{jl} & dots delta_{km} & delta_{kn} & delta_{kl} & dots vdots & vdots & vdots end{bmatrix} .

Now we can contract m indices. This will add a factor of m! to the determinant and we need to omit the relevant Kronecker delta.

Properties

(in these examples, superscripts should be considered equivalent with subscripts)

1. When n=2, we have for all i,j,m,n in {1,2},

varepsilon_{ij} varepsilon^{mn} = delta_i^m delta_j^n - delta_i^n delta_j^m, (1)
varepsilon_{ij} varepsilon^{in} = delta_j^n, (2)
varepsilon_{ij} varepsilon^{ij}=2. (3)

2. When n=3, we have for all i,j,k,m,n in {1,2,3},

varepsilon_{jmn} varepsilon^{imn}=2delta^i_j, (4)
varepsilon_{ijk} varepsilon^{ijk}=6, (5)
varepsilon_{ijk} varepsilon^{imn}=delta^{m}_jdelta^{n}_k - delta^{n}_jdelta^{m}_k. (6)

Proofs

For equation 1, both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case ineq j and mneq n. By substitution, we see that the equation holds for varepsilon_{12} varepsilon^{12}, 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. Using equation 1, we have for equation 2 varepsilon_{ij}varepsilon^{in} = delta_i^i delta_j^n - delta^n_i delta^i_j

= 2 delta_j^n - delta^n_j

= delta_j^n.

Here we used the Einstein summation convention with i going from 1 to 2. Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when ineq j. Indeed, if ineq 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 varepsilon_{jmn} varepsilon^{imn} = (varepsilon^{imn})^2 = 1 (no summation), and the result follows. Property (5) follows since 3!=6 and for any distinct indices i,j,k in {1,2,3}, we have varepsilon_{ijk} varepsilon^{ijk}=1 (no summation).

Examples

1. The determinant of an ntimes n matrix A=(a_{ij}) can be written as

det A = varepsilon_{i_1cdots i_n} a_{1i_1} cdots a_{ni_n},

where each i_l should be summed over 1,ldots, n.

Equivalently, it may be written as

det A = frac{1}{n!} varepsilon_{i_1cdots i_n} varepsilon_{j_1cdots j_n} a_{i_1 j_1} cdots a_{i_n j_n},

where now each i_l and each j_l should be summed over 1,ldots, n.

2. If A=(A^1, A^2, A^3) and B=(B^1, B^2, B^3) are vectors in R^3 (represented in some right hand oriented orthonormal basis), then the ith component of their cross product equals

(Atimes B)^i = varepsilon^{ijk} A^j B^k.

For instance, the first component of Atimes B is A^2 B^3-A^3 B^2. From the above expression for the cross product, it is clear that Atimes B = -Btimes A. Further, if C=(C^1, C^2, C^3) is a vector like A and B, then the triple scalar product equals

Acdot(Btimes C) = varepsilon^{ijk} A^i B^j C^k.

From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any adjacent arguments. For example, Acdot(Btimes C)= -Bcdot(Atimes C).

3. Suppose F=(F^1, F^2, F^3) is a vector field defined on some open set of R^3 with Cartesian coordinates x=(x^1, x^2, x^3). Then the ith component of the curl of F equals

(nabla times F)^i(x) = varepsilon^{ijk}frac{partial}{partial x^j} F^k(x).

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, for an n x n matrix, M,

M_{[ab]} = frac{1}{2}varepsilon_{abc} varepsilon^{cde} M_{de } = , frac{1}{2}(M_{ab} - M_{ba})

and for a rank 3 tensor T,

T_{[abc]} = , frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).

Tensor density

The Levi-Civita symbol may be considered to be a tensor density 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 four dimensions,
epsilon^{alpha beta gamma delta} = epsilon_{alpha beta gamma delta} ,.

See also

References

  • Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (See section 3.5 for a review of tensors in general relativity).


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