17. The Levi-Civita Symbol and the Cross Product (2025)

In $n$n dimensions, the Levi-Civita symbols $\varepsilon_{i_{1}\cdots i_{n}}$εi1​⋯in​​ and $\varepsilon^{i_{1}\cdots i_{n}}$εi1​⋯in​ are defined by the equations

$$\begin{aligned}\varepsilon_{i_{1}\cdots i_{n}} & =\sqrt{Z}e_{i_{1}\cdots i_{n}}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.1\right)\\\varepsilon^{i_{1}\cdots i_{n}} & =\frac{e^{i_{1}\cdots i_{n}}}{\sqrt{Z}},\ \ \ \ \ \ \ \ \ \ \left(17.2\right)\end{aligned}$$εi1​⋯in​​εi1​⋯in​​=Z​ei1​⋯in​​and(17.1)=Z​ei1​⋯in​​,(17.2)​

where $\sqrt{Z}$Z​ is the volume element first introduced in Chapter 9. In the natural Euclidean dimensions, the above definition reduces to

$$\begin{aligned}\varepsilon_{ijk} & =\sqrt{Z}e_{ijk}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.3\right)\\\varepsilon^{ijk} & =\frac{e^{ijk}}{\sqrt{Z}},\ \ \ \ \ \ \ \ \ \ \left(17.4\right)\end{aligned}$$εijk​εijk​=Z​eijk​and(17.3)=Z​eijk​,(17.4)​

in three dimensions,

$$\begin{aligned}\varepsilon_{ij} & =\sqrt{Z}e_{ij}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.5\right)\\\varepsilon^{ij} & =\frac{1}{\sqrt{Z}}e^{ij}\ \ \ \ \ \ \ \ \ \ \left(17.6\right)\end{aligned}$$εij​εij​=Z​eij​and(17.5)=Z​1​eij(17.6)​

in two dimensions and

$$\begin{aligned}\varepsilon_{i} & =\sqrt{Z}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.7\right)\\\varepsilon^{i} & =\frac{1}{\sqrt{Z}}\ \ \ \ \ \ \ \ \ \ \left(17.8\right)\end{aligned}$$εi​εi​=Z​and(17.7)=Z​1​(17.8)​

in one dimension. You may be surprised that the one-dimensional Levi-Civita symbols $\varepsilon_{i}$εi​ and $\varepsilon^{i}$εi remain remarkably useful objects, even though they do not exhibit the sign-alternating property associated with the permutation systems.

As we noted in the previous Chapter, the permutation systems $e_{ijk}$eijk​ and $e^{ijk}$eijk are not tensors. In other words, if, say, $e_{i^{\prime}j^{\prime }k^{\prime}}$ei′j′k′​, is defined in the primed coordinates in the exact same terms as $e_{ijk}$eijk​ in the unprimed coordinates, then there is no reason to expect that the two objects are related by the requisite identity

$$e_{i^{\prime}j^{\prime}k^{\prime}}=e_{ijk}J_{i^{\prime}}^{i}J_{j^{\prime}} ^{j}J_{k^{\prime}}^{k}, \tag{-}$$ei′j′k′​=eijk​Ji′i​Jj′j​Jk′k​,(-)

and, indeed, this identity does not hold.

However, the combination on the right is familiar to us from our study of determinants in Chapter 16. Recall that for a second-order system $A_{j}^{i}$Aji​, we have

$$e_{ijk}A_{r}^{i}A_{s}^{j}A_{t}^{k}=\det A~e_{rst}. \tag{16.113}$$eijk​Ari​Asj​Atk​=detAerst​.(16.113)

Thus, if

$$J\text{ is the matrix corresponding to the Jacobian }J_{i^{\prime}}^{i},\tag{17.9}$$JisthematrixcorrespondingtotheJacobianJi′i​,(17.9)

then, according to the preceding identity,

$$e_{ijk}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}=\det J~e_{i^{\prime}j^{\prime}k^{\prime}}.\tag{17.10}$$eijk​Ji′i​Jj′j​Jk′k​=detJei′j′k′​.(17.10)

Thus, the transformation rule for the subscripted permutation system $e_{ijk}$eijk​ reads

$$e_{i^{\prime}j^{\prime}k^{\prime}}=\det{}^{-1}J~e_{ijk}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}.\tag{17.11}$$ei′j′k′​=det−1Jeijk​Ji′i​Jj′j​Jk′k​.(17.11)

Correspondingly, the rule for the superscripted permutation symbol $e^{ijk}$eijk features the determinant of $J$J rather than its inverse, i.e.

$$e^{i^{\prime}j^{\prime}k^{\prime}}=\det J~e^{ijk}J_{i}^{i^{\prime}} J_{j}^{j^{\prime}}J_{k}^{k^{\prime}}.\tag{17.12}$$ei′j′k′=detJeijkJii′​Jjj′​Jkk′​.(17.12)

A variant $T_{j}^{i}$Tji​ with a representative collection of indices is called a relative tensor of weight $M$M if it transforms according to the rule

$$T_{j^{\prime}}^{i^{\prime}}=\det{}^{M}J~T_{j}^{i}J_{i}^{i^{\prime} }J_{j^{\prime}}^{j}.\tag{17.13}$$Tj′i′​=detMJTji​Jii′​Jj′j​.(17.13)

In particular, conventional tensors are relative tensors of weight $0$0, and are often referred to as absolute tensors to highlight their special place among relative tensors.

A relative tensor $T$T of order zero is called a relative invariant and transforms according to the rule

$$T^{\prime}=\det{}^{M}J~T.\tag{17.14}$$T′=detMJT.(17.14)

We should recognize that the term relative invariant is an oxymoron since the values $T$T and $T^{\prime}$T′ are generally distinct. In other words, a relative invariant is not invariant.

Returning to the permutation symbols, which transform according to the equations

$$e_{i^{\prime}j^{\prime}k^{\prime}}=\det{}^{-1}J~e_{ijk}J_{i^{\prime}} ^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k} \tag{17.11}$$ei′j′k′​=det−1Jeijk​Ji′i​Jj′j​Jk′k​(17.11)

and

$$e^{i^{\prime}j^{\prime}k^{\prime}}=\det J~e^{ijk}J_{i}^{i^{\prime}} J_{j}^{j^{\prime}}J_{k}^{k^{\prime}}, \tag{17.12}$$ei′j′k′=detJeijkJii′​Jjj′​Jkk′​,(17.12)

we observe that $e_{ijk}$eijk​ is a relative covariant tensor of weight $-1$−1 while $e^{ijk}$eijk is a relative contravariant tensor of weight $1$1. (Since $e_{ijk}$eijk​ and $e^{ijk}$eijk consist of the exact same values, it is interesting that one and the same object can be interpreted as tensors of different kinds.) Meanwhile, according to the product property of relative tensors, the combination $e_{ijk}e^{rst}$eijk​erst, which we recognize as the delta symbol $\delta_{ijk}^{rst}$δijkrst​, is an absolute tensor. Of course, this was already demonstrated in Chapter 16 by other means.

Suppose that $A_{ij}$Aij​ is an absolute covariant tensor. Recall that its determinant is given by the formula

$$\det A=\frac{1}{3!}e^{ijk}e^{rst}A_{ir}A_{js}A_{kt}. \tag{16.119}$$detA=3!1​eijkerstAir​Ajs​Akt​.(16.119)

Then, by the multiplicative property of relative tensors stated in the previous Section (and demonstrated in Exercise 17.2), $\det A$detA is a relative invariant of weight $2$2. Meanwhile, the determinant of an absolute tensor $A^{ij}$Aij, which is given by

$$\det A=\frac{1}{3!}e_{ijk}e_{rst}A^{ir}A^{js}A^{kt}, \tag{16.121}$$detA=3!1​eijk​erst​AirAjsAkt,(16.121)

is a relative invariant of weight $-2$−2. Finally, recall that, since the delta system $\delta_{rst}^{ijk}$δrstijk​ is an absolute tensor, we concluded in Chapter 16 that the determinant of a mixed tensor $A_{j}^{i}$Aji​, i.e.

$$\det A=\frac{1}{3!}\delta_{rst}^{ijk}A_{i}^{r}A_{j}^{s}A_{k}^{t}, \tag{16.117}$$detA=3!1​δrstijk​Air​Ajs​Akt​,(16.117)

is also an absolute invariant.

As soon as we calculated the volume element $\sqrt{Z}$Z​ in Cartesian and polar coordinates in Chapter 9, it became apparent that $\sqrt{Z}$Z​ is not an invariant. Indeed,

$$\sqrt{Z}=1 \tag{9.59}$$Z​=1(9.59)

in Cartesian coordinates, and

$$\sqrt{Z}=r \tag{9.60}$$Z​=r(9.60)

in polar coordinates. We will now be able to characterize this behavior in terms of the relative tensor property.

Based on the findings of the previous Section, the object $Z$Z, being the determinant of the covariant metric tensor $Z_{ij}$Zij​, is a relative invariant of weight $2$2. In other words, if $Z^{\prime}$Z′ is the determinant of $Z_{i^{\prime}j^{\prime}}$Zi′j′​ in the primed coordinates, then $Z$Z and $Z^{\prime }$Z′ are related by the equation

$$Z^{\prime}=\det{}^{2}J~Z.\tag{17.15}$$Z′=det2JZ.(17.15)

As tempting as it may be to take the square roots and conclude that

$$\sqrt{Z^{\prime}}=\det J~\sqrt{Z}, \tag{-}$$Z′​=detJZ​,(-)

we must remember that the determinant of the Jacobian $J$J may very well be negative. Thus, the correct relationship is

$$\sqrt{Z^{\prime}}=\left\vert \det J\right\vert \sqrt{Z}.\tag{17.16}$$Z′​=∣detJ∣Z​.(17.16)

Therefore, we are not able to conclude that the volume element $\sqrt{Z}$Z​ is a relative invariant of weight $1$1. In order to actually reach that conclusion, we must restrict our attention to coordinate changes for which $\det J\gt 0$detJ>0 -- in other words, orientation-preserving coordinate transformations. Under this condition, the above identity does simplify to

$$\sqrt{Z^{\prime}}=\det J~\sqrt{Z}. \tag{-}$$Z′​=detJZ​.(-)

To use the terminology introduced in Section 14.15, $\sqrt{Z}$Z​ is a relative invariant of weight $1$1 only with respect to orientation-preserving coordinate transformations.

Such an object proves difficult to come by. Indeed, suppose that $U$U is, in fact, such an object, i.e. a relative invariant of weight $1$1 not subject to any qualifications. Since we have no further requirements of $U$U, suppose that $U\equiv1$U≡1 in some Cartesian coordinate system $Z^{i}$Zi. Then the values of $U$U are uniquely determined in all coordinate systems. Indeed, in any alternative coordinate system $Z^{i^{\prime}}$Zi′, $U^{\prime}$U′ is given by the equation

$$U^{\prime}=\det J~U,\tag{17.17}$$U′=detJU,(17.17)

where $J$J is the Jacobian of the coordinate transformation between $Z^{i}$Zi and $Z^{i^{\prime}}$Zi′. Since $U$U and $\sqrt{Z}$Z​ coincide in the Cartesian coordinates $Z^{i}$Zi and transform by the similar rules

$$U^{\prime}=\det J~U\text{ \ \ \ \ and \ \ \ \ }\sqrt{Z^{\prime}}=\left\vert \det J\right\vert \sqrt{Z},\tag{17.18}$$U′=detJUandZ′​=∣detJ∣Z​,(17.18)

we conclude that $U$U agrees with $\sqrt{Z}$Z​ to within sign in all coordinates systems.

Recall that the general $n$n-dimensional definition reads

$$\begin{aligned}\varepsilon_{i_{1}\cdots i_{n}} & =\sqrt{Z}e_{i_{1}\cdots i_{n}}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.1\right)\\\varepsilon^{i_{1}\cdots i_{n}} & =\frac{e^{i_{1}\cdots i_{n}}}{\sqrt{Z}}. \ \ \ \ \ \ \ \ \ \ \left(17.2\right)\end{aligned}$$εi1​⋯in​​εi1​⋯in​​=Z​ei1​⋯in​​and(17.1)=Z​ei1​⋯in​​.(17.2)​

By construction, the Levi-Civita symbols are absolute tensors with respect to orientation-preserving coordinate transformations. They are often thought of as members of the metrics family, alongside the covariant and the contravariant bases $\mathbf{Z}_{i}$Zi​ and $\mathbf{Z}^{i}$Zi, the metric tensors $Z_{ij}$Zij​ and $Z^{ij}$Zij, and the volume element $\sqrt{Z}$Z​.

Note that the complete delta system $\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}$δj1​⋯jn​i1​⋯in​​ defined by the identity

$$\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}=e^{i_{1}\cdots i_{n}} e_{j_{1}\cdots j_{n}}. \tag{16.37}$$δj1​⋯jn​i1​⋯in​​=ei1​⋯in​ej1​⋯jn​​.(16.37)

can be similarly expressed in terms of the Levi-Civita symbols, i.e.

$$\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}=\varepsilon^{i_{1}\cdots i_{n} }\varepsilon_{j_{1}\cdots j_{n}}.\tag{17.19}$$δj1​⋯jn​i1​⋯in​​=εi1​⋯in​εj1​⋯jn​​.(17.19)

The appeal of this identity, compared to the original definition of $\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}$δj1​⋯jn​i1​⋯in​​ in terms of the permutation systems, is that all of its constituent elements are tensors. However, while the Levi-Civita symbols are relative tensors only with respect to orientation-preserving coordinate changes, the delta system is an unqualified absolute tensor.

For a second-order system $A_{j}^{i}$Aji​ in two dimensions, recall the identity

$$A_{r}^{i}A_{s}^{j}-A_{r}^{j}A_{s}^{i}=\det A~\delta_{rs}^{ij} \tag{16.149}$$Ari​Asj​−Arj​Asi​=detAδrsij​(16.149)