Although the permutation systems $e_{ijk}$eijk and $e^{ijk}$eijk are indeed objects of immense beauty and utility, they are not tensors. I hope that at this advanced point in our narrative, we no longer have to extol the benefits of the tensor framework and may consider the tensor property to be of self-evident value. In this Chapter, we will introduce the *Levi-Civita symbols* $\varepsilon_{ijk}$εijk and $\varepsilon^{ijk}$εijk, named after Tullio Levi-Civita, which are the tensor versions, so to speak, of the permutation systems. Thanks to their tensor property, the Levi-Civita symbols open new avenues for the creation of invariant operations, such as the *cross product*, and various differential operators, such as the *curl*. Although the cross product has already been described in Chapter 3 from a geometric perspective, its *component space* representation is most effectively expressed with the help of the Levi-Civita symbols. Meanwhile, the curl will be discussed in Chapter 18.

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⋯inεi1⋯in=Zei1⋯inand(17.1)=Zei1⋯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

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

in three dimensions,

$\begin{aligned}\varepsilon_{ij} & =\sqrt{Z}e_{ij}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.5\right)\\\varepsilon^{ij} & =\frac{1}{\sqrt{Z}}e^{ij}\ \ \ \ \ \ \ \ \ \ \left(17.6\right)\end{aligned}$εijεij=Zeijand(17.5)=Z1eij(17.6)

in two dimensions and

$\begin{aligned}\varepsilon_{i} & =\sqrt{Z}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.7\right)\\\varepsilon^{i} & =\frac{1}{\sqrt{Z}}\ \ \ \ \ \ \ \ \ \ \left(17.8\right)\end{aligned}$εiεi=Zand(17.7)=Z1(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.

Our initial order of business is to investigate the tensor property of the Levi-Civita symbols $\varepsilon_{ijk}$εijk and $\varepsilon^{ijk}$εijk. To this end, we will first explore the transformation rules for the permutation systems $e_{ijk}$eijk and $e^{ijk}$eijk under a change of coordinates and subsequently extend those rules to the Levi-Civita symbols. We will discover that the Levi-Civita symbols are indeed tensors, albeit with a significant qualification.

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′=eijkJi′iJj′jJk′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

$e_{ijk}A_{r}^{i}A_{s}^{j}A_{t}^{k}=\det A~e_{rst}. \tag{16.113}$eijkAriAsjAtk=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,

$e_{ijk}J_{i^{\prime}}^{i}J_{j^{\prime}}^{j}J_{k^{\prime}}^{k}=\det J~e_{i^{\prime}j^{\prime}k^{\prime}}.\tag{17.10}$eijkJi′iJj′jJk′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−1JeijkJi′iJj′jJk′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)

Observe from the two identities above that the transformation rules for the permutation systems $e_{ijk}$eijk and $e^{ijk}$eijk deviate from the definition of a tensor only by the presence of $\det J$detJ. Let us then use this near miss as a rationale for introducing the concept of a *relative tensor*.

### 17.3.1Definition and elementary properties

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′=detMJTjiJii′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.

It is left as an exercise to show that the collection of relative tensors of a given weight are closed under addition and multiplication by numbers. In other words, the sum of two relative tensors of weight $M$M as well as a product of a relative tensor of weight $M$M and a number are also relative tensors of weight $M$M. Thus, the collection of relative tensors of a given weight is closed under linear combinations. Furthermore, the product of a relative tensor of weight $M$M and a relative tensor of weight $N$N is a relative tensor of weight $M+N$M+N. Finally, a contraction of a relative tensor of weight $M$M is also a relative tensor of weight $M$M.

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−1JeijkJi′iJj′jJk′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 $e_{ijk}e^{rst}$eijkerst, 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.

Having establishing the (relative) tensor property of the permutation systems, we find ourselves halfway towards doing the same for the Levi-Civita symbols. In order to complete this task, we must investigate the tensor property of the volume element $\sqrt{Z}$Z and will therefore turn our attention now to determinants.

### 17.3.2The relative tensor property of determinants

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!1eijkerstAirAjsAkt.(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!1eijkerstAirAjsAkt,(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!1δrstijkAirAjsAkt,(16.117)

is also an absolute invariant.

### 17.3.3The relative tensor property of the volume element $\sqrt{Z}$Z

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*.

### 17.3.4Why $\sqrt{Z}$Z in the Levi-Civita symbols?

Recall that, much like the volume element $\sqrt{Z}$Z, the permutation symbol $e^{ijk}$eijk is also a relative tensor of weight $1$1. Thus, *dividing* $e^{ijk}$eijk by $\sqrt{Z}$Z yields an absolute tensor -- albeit, only with respect to orientation-preserving coordinate transformations. Meanwhile, $e_{ijk}$eijk is a relative tensor of weight $-1$−1. Thus, *multiplying* $e_{ijk}$eijk by $\sqrt{Z}$Z also yields an absolute tensor -- once again, only with respect to orientation-preserving coordinate changes. Of course, this is *precisely* how the Levi-Civita symbols are constructed, but you may ask -- why must we use $\sqrt{Z}$Z in order to balance the permutations systems and not *another* relative invariant of weight $1$1 that is not limited 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.

Does the resulting object $U$U offer a better alternative to $\sqrt{Z}$Z? It has one indisputable advantage over $\sqrt{Z}$Z: it is an unqualified relative invariant of weight $1$1. However, since the construction of $U$U requires us to a single out one coordinate system -- i.e. the Cartesian coordinates of a particular orientation where $U=1$U=1 -- it is a tensor of the synthetic variety, as described in Section 14.13, and, as such, its uses are significantly limited. Thus, on balance, the volume element $\sqrt{Z}$Z is the superior choice.

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⋯inεi1⋯in=Zei1⋯inand(17.1)=Zei1⋯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⋯jni1⋯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⋯jni1⋯in=ei1⋯inej1⋯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⋯jni1⋯in=εi1⋯inεj1⋯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⋯jni1⋯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

$A_{r}^{i}A_{s}^{j}-A_{r}^{j}A_{s}^{i}=\det A~\delta_{rs}^{ij} \tag{16.149}$AriAsj−ArjAsi=detAδrsij(16.149)

derived in Section 16.10, where $\det A$detA denotes the determinant of $A_{j}^{i}$Aji. Importantly, if $A_{j}^{i}$Aji is a tensor, then all elements in this identity, including $\det A$detA, are tensors. The same cannot be said, however, for a covariant tensor $A_{ij}$Aij which satisfies the identity

$A_{ir}A_{js}-A_{is}A_{jr}=\det A~e_{ij}e_{rs}, \tag{16.152}$AirAjs−AisAjr=detAeijers,(16.152)

where $\det A$detA denotes the determinant of $A_{ij}$Aij, or for a contravariant tensor $A^{ij}$Aij which satisfies the identity

$A^{ir}A^{js}-A^{is}A^{jr}=\det A~e^{ij}e^{rs}, \tag{16.153}$AirAjs−AisAjr=detAeijers,(16.153)

where $\det A$detA denotes the determinant of $A^{ij}$Aij. Thus, the symbol $A$A denotes three different objects in the three preceding identities. We will now show that, with the help of the Levi-Civita symbols, these identities can be fully tensorized while the symbol $A$A can be assigned a unique meaning.

Let us focus our attention on the identity

$A_{ir}A_{js}-A_{is}A_{jr}=\det A~e_{ij}e_{rs} \tag{16.152}$AirAjs−AisAjr=detAeijers(16.152)

for a covariant tensor $A_{ij}$Aij where, once again, $\det A$detA denotes its determinant. Since the permutation system $e_{ij}$eij is expressed in terms of the Levi-Civita symbol $\varepsilon_{ij}$εij by the equation

$e_{ij}=\frac{\varepsilon_{ij}}{\sqrt{Z}},\tag{17.20}$eij=Zεij,(17.20)

we find that

$A_{ir}A_{js}-A_{is}A_{jr}=\frac{\det A}{Z}\varepsilon_{ij}\varepsilon_{rs}. \tag{16.152}$AirAjs−AisAjr=ZdetAεijεrs.(16.152)

Since $A_{ir}A_{js}-A_{is}A_{jr}$AirAjs−AisAjr, $\varepsilon_{ij}$εij, and $\varepsilon_{rs}$εrs are tensors, the quantity

$\frac{\det A}{Z}\tag{17.21}$ZdetA(17.21)

is *also* a tensor by the quotient theorem described in Section 14.14. It is, in fact, the determinant of another tensor closely related to $A_{ij}$Aij. Since $1/Z$1/Z is the determinant of the contravariant metric tensor $Z^{ij}$Zij, then $\det\left( A\right) /Z$det(A)/Z is the determinant of $A_{ir}Z^{ij}$AirZij by the multiplicative property of determinants. In other words, $\det\left( A\right) /Z$det(A)/Z is the determinant of $A_{\cdot j}^{i}$A⋅ji.

In light of this insight, let us agree to denote by the symbol $A$A the determinant of $A_{\cdot j}^{i}$A⋅ji, regardless of whether the context is concerned with $A_{ij}$Aij, $A_{j}^{i}$Aji, or $A^{ij}$Aij. With the help of this convention, the three identities at the center of this Section can be written as

$\begin{aligned}A_{ir}A_{js}-A_{is}A_{jr} & =A~\varepsilon_{ij}\varepsilon_{rs}\ \ \ \ \ \ \ \ \ \ \left(17.22\right)\\A_{r}^{i}A_{s}^{j}-A_{r}^{j}A_{s}^{i} & =A~\delta_{rs}^{ij}\ \ \ \ \ \ \ \ \ \ \left(16.149\right)\\A^{ir}A^{js}-A^{is}A^{jr} & =A~\varepsilon^{ij}\varepsilon^{rs}.\ \ \ \ \ \ \ \ \ \ \left(17.23\right)\end{aligned}$AirAjs−AisAjrAriAsj−ArjAsiAirAjs−AisAjr=Aεijεrs(17.22)=Aδrsij(16.149)=Aεijεrs.(17.23)

This convention makes logical sense from the Tensor Calculus point of view. If we were to associate a determinant-like invariant with a covariant tensor $A_{ij}$Aij or contravariant tensor $A^{ij}$Aij, it would need to be the determinant of $A_{\cdot j}^{i}$A⋅ji since, among the three determinants, it is the only one that is an invariant.

Note that an interesting special case of the above formulas is found when $A_{ij}$Aij is the metric tensor $Z_{ij}$Zij. Then $A_{\cdot j}^{i}$A⋅ji is the Kronecker delta $\delta_{j}^{i}$δji and therefore $A=1$A=1. Thus we have

$\begin{aligned}Z_{ir}Z_{js}-Z_{is}Z_{jr} & =\varepsilon_{ij}\varepsilon_{rs}\ \ \ \ \ \ \ \ \ \ \left(17.24\right)\\\delta_{r}^{i}\delta_{s}^{j}-\delta_{r}^{j}\delta_{s}^{i} & =\delta _{rs}^{ij}\ \ \ \ \ \ \ \ \ \ \left(17.25\right)\\Z^{ir}Z^{js}-Z^{is}Z^{jr} & =\varepsilon^{ij}\varepsilon^{rs}\ \ \ \ \ \ \ \ \ \ \left(17.26\right)\end{aligned}$ZirZjs−ZisZjrδriδsj−δrjδsiZirZjs−ZisZjr=εijεrs(17.24)=δrsij(17.25)=εijεrs(17.26)

where we note that the middle equation is precisely the identity

$\delta_{rs}^{ij}=\delta_{r}^{i}\delta_{s}^{j}-\delta_{r}^{j}\delta_{s}^{i} \tag{16.58}$δrsij=δriδsj−δrjδsi(16.58)

discovered in the previous Chapter.

Finally, note that one of the most striking applications of the formulas discussed in this Section will be found in a future book in the context of the *Gauss equations* for two-dimensional surfaces.

In this Section, we will show that the metrinilic property of the covariant derivative extends to the Levi-Civita symbols. In the context of a Euclidean space, which, as we know, is characterized by the availability of Cartesian coordinates, the metrinilic property with respect to the Levi-Civita symbols is rather easily established by considering the combinations

$\nabla_{p}\varepsilon_{ijk}\text{\ \ \ \ and\ \ \ \ }\nabla_{p}\varepsilon ^{ijk}\tag{17.27}$∇pεijkand∇pεijk(17.27)

in any Cartesian coordinates, where the Levi-Civita symbols $\varepsilon _{ijk}$εijk and $\varepsilon^{ijk}$εijk have constant values, while the covariant derivative coincides with the partial derivative. Therefore, the combinations $\nabla_{p}\varepsilon_{ijk}$∇pεijk and $\nabla_{p}\varepsilon^{ijk}$∇pεijk vanish, i.e.

$\begin{aligned}\nabla_{p}\varepsilon_{ijk} & =0\ \ \ \ \ \ \ \ \ \ \left(17.28\right)\\\nabla_{p}\varepsilon^{ijk} & =0.\ \ \ \ \ \ \ \ \ \ \left(17.29\right)\end{aligned}$∇pεijk∇pεijk=0(17.28)=0.(17.29)

Meanwhile, since the Levi-Civita symbols are tensors, vanishing in one coordinate system implies vanishing in all coordinate systems, as we set out to show. The fact the $\varepsilon_{ijk}$εijk is a tensor only with respect to orientation-preserving coordinate transformations has no effect on this argument.

However, we would also like to provide another argument that does not rely on the Euclidean nature of the space, so that we are later able to extend the result to Riemannian spaces. Therefore, we will give an alternative demonstration based on a direct application of the definition of the covariant derivative to the Levi-Civita symbol.

Denote by $T_{pijk}$Tpijk the covariant derivative of the permutation system $e_{ijk}$eijk, i.e.

$T_{pijk}=\nabla_{p}e_{ijk}=\frac{\partial e_{ijk}}{\partial Z^{p}}-\Gamma _{pi}^{m}e_{mjk}-\Gamma_{pj}^{m}e_{imk}-\Gamma_{pk}^{m}e_{ijm}.\tag{17.30}$Tpijk=∇peijk=∂Zp∂eijk−Γpimemjk−Γpjmeimk−Γpkmeijm.(17.30)

Since the permutation systems have constant values and therefore the partial derivative $\partial e_{ijk}/\partial Z^{p}$∂eijk/∂Zp vanishes, we have

$T_{pijk}=-\Gamma_{pi}^{m}e_{mjk}-\Gamma_{pj}^{m}e_{imk}-\Gamma_{pk}^{m} e_{ijm}.\tag{17.31}$Tpijk=−Γpimemjk−Γpjmeimk−Γpkmeijm.(17.31)

Observe that $T_{pijk}$Tpijk is skew-symmetric in the indices $i$i, $j$j, and $k$k. Therefore, we need only to consider the elements $T_{p123}$Tp123 given by

$T_{p123}=-\Gamma_{p1}^{m}e_{m23}-\Gamma_{p2}^{m}e_{1m3}-\Gamma_{p3}^{m} e_{12m}.\tag{17.32}$Tp123=−Γp1mem23−Γp2me1m3−Γp3me12m.(17.32)

In each contraction on the right, there is only one nonzero term that corresponds to $m=1$m=1 in the first contraction, $m=2$m=2 in the second, and $m=3$m=3 in the third, i.e.

$T_{p123}=-\Gamma_{p1}^{1}e_{123}-\Gamma_{p2}^{2}e_{123}-\Gamma_{p3}^{3} e_{123}.\tag{17.33}$Tp123=−Γp11e123−Γp22e123−Γp33e123.(17.33)

Factoring out $-e_{123}$−e123, we find

$T_{p123}=-\left( \Gamma_{p1}^{1}+\Gamma_{p2}^{2}+\Gamma_{p3}^{3}\right) e_{123}.\tag{17.34}$Tp123=−(Γp11+Γp22+Γp33)e123.(17.34)

In other words,

$T_{p123}=-\Gamma_{pm}^{m}e_{123}.\tag{17.35}$Tp123=−Γpmme123.(17.35)

Thus, in general,

$\nabla_{p}e_{ijk}=-\Gamma_{pm}^{m}e_{ijk}.\tag{17.36}$∇peijk=−Γpmmeijk.(17.36)

Since the Christoffel symbol is symmetric in its subscripts and therefore

$\Gamma_{pm}^{m}=\Gamma_{mp}^{m},\tag{17.37}$Γpmm=Γmpm,(17.37)

we arrive at the following identity for the covariant derivative of the permutation systems:

$\nabla_{p}e_{ijk}=-\Gamma_{mp}^{m}e_{ijk}.\tag{17.38}$∇peijk=−Γmpmeijk.(17.38)

Note that we encountered the combination $\Gamma_{mp}^{m}$Γmpm at the end of the last Chapter where we derived the identity

$\frac{\partial\sqrt{Z}}{\partial Z^{p}}=\sqrt{Z}\Gamma_{mp}^{m}. \tag{16.181}$∂Zp∂Z=ZΓmpm.(16.181)

This identity is about to play a pivotal role in our calculation as we analyzed the covariant derivative of the Levi-Civita symbols.

By the product rule, for the covariant derivative of the Levi-Civita symbol $\varepsilon_{ijk}$εijk, we have

$\nabla_{p}\varepsilon_{ijk}=\nabla_{p}\left( \sqrt{Z}e_{ijk}\right) =\nabla_{p}\sqrt{Z}~e_{ijk}+\sqrt{Z}\nabla_{p}e_{ijk}.\tag{17.39}$∇pεijk=∇p(Zeijk)=∇pZeijk+Z∇peijk.(17.39)

Since

$\nabla_{p}\sqrt{Z}=\frac{\partial\sqrt{Z}}{\partial Z^{p}}=\sqrt{Z}\Gamma _{mp}^{m} \tag{16.181}$∇pZ=∂Zp∂Z=ZΓmpm(16.181)

and, as we just discovered,

$\nabla_{p}e_{ijk}=-\Gamma_{mp}^{m}e_{ijk}. \tag{17.38}$∇peijk=−Γmpmeijk.(17.38)

we find

$\nabla_{p}\varepsilon_{ijk}=\sqrt{Z}\Gamma_{mp}^{m}e_{ijk}-\sqrt{Z}\Gamma _{mp}^{m}e_{ijk}=0\tag{17.40}$∇pεijk=ZΓmpmeijk−ZΓmpmeijk=0(17.40)

as we set out to show.

In summary, the Levi-Civita symbols $\varepsilon_{ijk}$εijk and $\varepsilon ^{ijk}$εijk are subject to the metrinilic property of the covariant derivative along with its fellow metrics the coordinate bases $\mathbf{Z}_{i}$Zi and $\mathbf{Z}^{i}$Zi and the metric tensors $Z_{ij}$Zij and $Z^{ij}$Zij.

Since we introduced the symbols $\varepsilon_{ijk}$εijk and $\varepsilon^{ijk}$εijk independently in a context that allows index juggling, each symbol $\varepsilon^{ijk}$εijk is potentially ambiguous. Indeed, does the symbol, say, $\varepsilon^{ijk}$εijk represent the contravariant Levi-Civita symbol $\varepsilon^{ijk}$εijk, as we defined it, or the covariant Levi-Civita symbol $\varepsilon_{ijk}$εijk with each of the indices raised, i.e. the combination $\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}$εrstZirZjsZkt?

Fortunately, the two interpretations are equivalent and we, indeed, have

$\varepsilon^{ijk}=\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}.\tag{17.41}$εijk=εrstZirZjsZkt.(17.41)

In other words, the Levi-Civita symbols $\varepsilon_{ijk}$εijk and $\varepsilon ^{ijk}$εijk, as defined independently of each other, are, in fact, related by index juggling. Had this not been the case, this ambiguity would have continually required special attention when the Levi-Civita symbols and index juggling were present in the same analysis.

To show that the above relationship holds, recall that

$\varepsilon_{rst}=\sqrt{Z}e_{rst}. \tag{16.28}$εrst=Zerst.(16.28)

Thus, the combination $\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}$εrstZirZjsZkt is given by

$\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}=\sqrt{Z}e_{rst}Z^{ir}Z^{js}Z^{kt}.\tag{17.42}$εrstZirZjsZkt=ZerstZirZjsZkt.(17.42)

Recall once again that

$e_{rst}A^{ir}A^{js}A^{kt}=\det A~e^{ijk}, \tag{16.120}$erstAirAjsAkt=detAeijk,(16.120)

where $\det A$detA is the determinant of $A^{ij}$Aij. This identity implies that

$e_{rst}Z^{ir}Z^{js}Z^{kt}=Z^{-1}e^{ijk}\tag{17.43}$erstZirZjsZkt=Z−1eijk(17.43)

since the determinant of the contravariant metric tensor $Z^{ij}$Zij is $1/Z$1/Z. Thus,

$\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}=\sqrt{Z}e_{rst}Z^{ir}Z^{js}Z^{kt}=\sqrt {Z}Z^{-1}e^{ijk}=\frac{1}{\sqrt{Z}}e^{ijk}.\tag{17.44}$εrstZirZjsZkt=ZerstZirZjsZkt=ZZ−1eijk=Z1eijk.(17.44)

Finally, since

$\varepsilon^{ijk}=\frac{e^{ijk}}{\sqrt{Z}}, \tag{17.4}$εijk=Zeijk,(17.4)

we arrive at the identity

$\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}=\varepsilon^{ijk}, \tag{17.41}$εrstZirZjsZkt=εijk,(17.41)

as we set out to show.

Importantly, unlike the Levi-Civita symbols, the permutation symbols $e_{ijk}$eijk and $e^{ijk}$eijk *do* suffer from the ambiguity related to index juggling, as $e^{ijk}$eijk does *not* equal $e_{ijk}$eijk with raised indices. Indeed, as we just saw

$e_{rst}Z^{ir}Z^{js}Z^{kt}=Z^{-1}e^{ijk}, \tag{17.43}$erstZirZjsZkt=Z−1eijk,(17.43)

and therefore,

$e_{rst}Z^{ir}Z^{js}Z^{kt}\neq e^{ijk}.\tag{17.45}$erstZirZjsZkt=eijk.(17.45)

We must acknowledge that this is a flaw in our notational system. However, one can live with this flaw since the need for juggling the indices of a permutation system almost never arises.

### 17.8.1A brief review of the cross product

The tensor property of the Levi-Civita symbols opens the door to the coordinate space expression for the cross product. However, let us begin by reviewing the geometric definition of the cross product given in Chapter 3.

(3.29)

In three dimensions, consider a pair of linearly independent vectors $\mathbf{U}$U and $\mathbf{V}$V that form an angle $\gamma$γ. Then their cross product $\mathbf{W}$W, denoted by $\mathbf{U}\times\mathbf{V}$U×V, is determined by the following three conditions. First, $\mathbf{W}$W is orthogonal to both $\mathbf{U}$U and $\mathbf{V}$V -- in other words, $\mathbf{W}$W lies along the unique straight line orthogonal to the plane spanned by $\mathbf{U}$U and $\mathbf{V}$V. Second, the length of $\mathbf{W}$W is the product of the length of $\mathbf{U}$U, the length of $\mathbf{V}$V, and the sine of the angle $\gamma$γ, i.e.

$\operatorname{len}\mathbf{W}=\operatorname{len}\mathbf{U}\operatorname{len} \mathbf{V}\sin\gamma. \tag{3.28}$lenW=lenUlenVsinγ.(3.28)

Finally, between the two opposite vectors that satisfy the first two conditions, $\mathbf{W}$W is selected in such a way that the set $\mathbf{U}$U, $\mathbf{V}$V, $\mathbf{W}$W is positively oriented.

The third condition implies that the cross product is *anti-symmetric*, i.e.

$\mathbf{U}\times\mathbf{V}=-\mathbf{V}\times\mathbf{U,} \tag{3.30}$U×V=−V×U,(3.30)

which tips us off to the connection between the cross product and skew-symmetric systems. From the anti-symmetric property, it also follows that the cross product of a vector with itself is zero, i.e.

$\mathbf{U}\times\mathbf{U}=\mathbf{0}. \tag{3.31}$U×U=0.(3.31)

Indeed, $\mathbf{U}\times\mathbf{U}=-\mathbf{U}\times\mathbf{U}$U×U=−U×U and must therefore be $\mathbf{0}$0.

In Chapter 3, we demonstrated that the cross product satisfies the associative law

$\alpha\left( \mathbf{U}\times\mathbf{V}\right) =\left( \alpha \mathbf{U}\right) \times\mathbf{V} \tag{3.33}$α(U×V)=(αU)×V(3.33)

and the distributive law

$\mathbf{U}\times\left( \mathbf{V}_{1}+\mathbf{V}_{2}\right) =\mathbf{U} \times\mathbf{V}_{1}+\mathbf{U}\times\mathbf{V}_{2}\mathbf{,} \tag{3.34}$U×(V1+V2)=U×V1+U×V2,(3.34)

but lacks associativity, i.e.

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) \neq\left( \mathbf{U}\times\mathbf{V}\right) \times\mathbf{W.} \tag{3.32}$U×(V×W)=(U×V)×W.(3.32)

Indeed, the vector on the left is orthogonal to $\mathbf{U}$U while the vector on the right is not necessarily so. However, the product $\mathbf{U} \times\left( \mathbf{V}\times\mathbf{W}\right)$U×(V×W) satisfies the identity

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) =\left( \mathbf{U}\cdot\mathbf{W}\right) \mathbf{V}-\left( \mathbf{U}\cdot \mathbf{V}\right) \mathbf{W} \tag{17.72}$U×(V×W)=(U⋅W)V−(U⋅V)W(17.72)

which we will demonstrate below by working with the components of vectors. Meanwhile, note that it is not surprising that $\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right)$U×(V×W) is a linear combination of $\mathbf{V}$V and $\mathbf{W}$W since, being orthogonal to $\mathbf{V}\times\mathbf{W}$V×W, it must lie in the plane spanned by $\mathbf{V}$V and $\mathbf{W}$W.

### 17.8.2The cross product in two dimensions

In two dimensions, there exists an operation analogous to the cross product that, unlike the conventional three-dimensional cross product, applies to a single vector $\mathbf{U}$U. Specifically, the "cross product" $\times \mathbf{U}$×U of a vector $\mathbf{U}$U is the vector $\mathbf{V}$V obtained from $\mathbf{U}$U by a counterclockwise $90^{\circ}$90∘ rotation.

(17.46)

Thus, the two-dimensional "cross product" is merely a synonym for a counterclockwise $90^{\circ}$90∘ rotation. Nevertheless, it is helpful to put this operation on an equal footing with the conventional cross product. Indeed, the close parallel is clear: the resulting vector $\mathbf{V}$V is orthogonal to $\mathbf{U}$U, has the same length as $\mathbf{U}$U, while the set $\mathbf{U}$U, $\mathbf{V}$V is positively oriented. Furthermore, we will discover that the coordinate space representation of $\times\mathbf{U}$×U can be given by a similar expression involving the two-dimensional Levi-Civita symbol $\varepsilon_{\alpha\beta}$εαβ.

### 17.8.3The coordinate space representation of the cross product

Let us now turn our attention to the coordinate space representation of the cross product. There exists a fundamental connection between skew-symmetry (an algebraic concept) and orthogonality (a geometric concept). For example, note that two Cartesian vectors $\left( a,b\right)$(a,b) and $\left( b,-a\right)$(b,−a) are orthogonal. In other words, in order to find a vector orthogonal to $\left( a,b\right)$(a,b), we must switch the components and multiply one of them by $-1$−1. This operation is equivalent to multiplication by the skew-symmetric matrix

$\left[ \begin{array} {rr} 0 & 1\\ -1 & 0 \end{array} \right]\tag{17.47}$[0−110](17.47)

which, as we know, is a representation of the two-dimensional permutation system $e_{ij}$eij.

With the help of the Levi-Civita symbol, the idea of orthogonality-by-skew-symmetry generalizes to any dimension. For an example in three dimensions, suppose that $U^{i}$Ui are the components of a vector $\mathbf{U}$U. Note that

$\varepsilon_{ijk}U^{i}U^{j}=0\tag{17.48}$εijkUiUj=0(17.48)

since a double contraction of a skew-symmetric system, $\varepsilon_{ijk}$εijk, with a symmetric system, $U^{i}U^{j}$UiUj, is zero. In a manner of speaking, "dot product" of $\varepsilon_{ijk}U^{i}$εijkUi and $U^{j}$Uj is zero. Thus, the combination $\varepsilon_{ijk}U^{i}$εijkUi -- no matter how the free index $k$k is eventually engaged -- produces an object "orthogonal" to $U^{j}$Uj. Similarly, in $n$n dimensions, the object $\varepsilon_{i_{1}\cdots i_{n}}U^{i_{1}}$εi1⋯inUi1 is "orthogonal" to $U^{i_{2}}$Ui2 no matter how the remaining indices $i_{3} ,\cdots,i_{n}$i3,⋯,in are engaged.

Let us now return to three dimensions and exploit this idea by considering the vector $\mathbf{W}$W with components

$W_{k}=\varepsilon_{ijk}U^{i}V^{j}.\tag{17.49}$Wk=εijkUiVj.(17.49)

In other words,

$\mathbf{W}=\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}.\tag{17.50}$W=εijkUiVjZk.(17.50)

Note that since $\varepsilon_{ijk}=\sqrt{Z}e_{ijk}$εijk=Zeijk, the expression on the right

$\mathbf{W}=\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}\tag{17.50}$W=εijkUiVjZk(17.50)

can be elegantly captured by the formula

$\mathbf{W}=\sqrt{Z}\left\vert \begin{array} {ccc} \mathbf{Z}^{1} & \mathbf{Z}^{2} & \mathbf{Z}^{3}\\ U^{1} & U^{2} & U^{3}\\ V^{1} & V^{2} & V^{3} \end{array} \right\vert\tag{17.51}$W=Z∣∣Z1U1V1Z2U2V2Z3U3V3∣∣(17.51)

which provides a practical recipe for evaluating $\mathbf{W}$W.

By design, $\mathbf{W}$W is orthogonal to both $\mathbf{U}$U and $\mathbf{V}$V. For a formal confirmation, note that

$\mathbf{W}\cdot\mathbf{U}=W_{k}U^{k}=\varepsilon_{ijk}U^{i}V^{j}U^{k}=0\tag{17.52}$W⋅U=WkUk=εijkUiVjUk=0(17.52)

and

$\mathbf{W}\cdot\mathbf{V}=W_{k}V^{k}=\varepsilon_{ijk}U^{i}V^{j}V^{k}=0\tag{17.53}$W⋅V=WkVk=εijkUiVjVk=0(17.53)

by the fact that a double-contraction of a skew-symmetric system and a symmetric system vanishes, whereas $U^{i}U^{k}$UiUk is a symmetric system in the first equation while $V^{j}V^{k}$VjVk is a symmetric system in the second. Thus, in one fell swoop, the skew-symmetry of the Levi-Civita symbol delivered a vector that satisfies the orthogonality requirement in the definition of the cross product.

Importantly, we should not lose sight of the fact that the tensor property of the Levi-Civita symbol was critical to our construction. Had the Levi-Civita symbol not been a tensor, our entire effort would have stopped dead in its tracks since we would have failed to produce an invariant, and therefore geometrically meaningful, expression. We should, however, reiterate the caveat that the Levi-Civita symbol is a tensor only with respect to orientation-preserving coordinate transformations. This issue will be addressed at the end of this Section.

Let us now turn our attention to the length of $\mathbf{W}$W and confirm that, in fact,

$\operatorname{len}\mathbf{W}=\operatorname{len}\mathbf{U}\operatorname{len} \mathbf{V}\sin\gamma. \tag{3.28}$lenW=lenUlenVsinγ.(3.28)

To show this, recall that the length of $\mathbf{W}$W is given by the equation

$\operatorname{len}^{2}\mathbf{W}=W_{k}W^{k}.\tag{17.54}$len2W=WkWk.(17.54)

In order to obtain $W^{k}$Wk, raise the subscript $k$k in the identity

$W_{k}=\varepsilon_{ijk}U^{i}V^{j} \tag{17.49}$Wk=εijkUiVj(17.49)

and introduce new letters $r$r and $s$s for the dummy indices in anticipation of the upcoming contraction, i.e.

$W^{k}=\varepsilon_{rs}^{\cdot\cdot k}U^{r}V^{s}\tag{17.55}$Wk=εrs⋅⋅kUrVs(17.55)

or, equivalently,

$W^{k}=\varepsilon^{rsk}U_{r}V_{s}\tag{17.56}$Wk=εrskUrVs(17.56)

Combining the expressions for $W_{k}$Wk and $W^{k}$Wk, we have

$W_{k}W^{k}=\varepsilon_{ijk}U^{i}V^{j}\varepsilon^{rsk}U_{r}V_{s}.\tag{17.57}$WkWk=εijkUiVjεrskUrVs.(17.57)

Since

$\varepsilon_{ijk}\varepsilon^{rsk}=\delta_{ijk}^{rsk} \tag{17.19}$εijkεrsk=δijkrsk(17.19)

and

$\delta_{ijk}^{rsk}=\delta_{ij}^{rs}=\delta_{i}^{r}\delta_{j}^{s}-\delta _{i}^{s}\delta_{j}^{r}, \tag{16.59}$δijkrsk=δijrs=δirδjs−δisδjr,(16.59)

we find that

$\varepsilon_{ijk}\varepsilon^{rsk}=\delta_{ijk}^{rsk}=\delta_{ij}^{rs} =\delta_{i}^{r}\delta_{j}^{s}-\delta_{i}^{s}\delta_{j}^{r}.\tag{17.58}$εijkεrsk=δijkrsk=δijrs=δirδjs−δisδjr.(17.58)

Substituting this relationship into the equation

$W_{k}W^{k}=\varepsilon_{ijk}U^{i}V^{j}\varepsilon^{rsk}U_{r}V_{s}, \tag{17.57}$WkWk=εijkUiVjεrskUrVs,(17.57)

we are able to eliminate the Levi-Civita symbols, i.e.

$W_{k}W^{k}=\left( \delta_{i}^{r}\delta_{j}^{s}-\delta_{i}^{s}\delta_{j} ^{r}\right) U^{i}V^{j}U_{r}V_{s}.\tag{17.59}$WkWk=(δirδjs−δisδjr)UiVjUrVs.(17.59)

Upon multiplying out and absorbing the Kronecker deltas, we arrive at the final expression for $W_{k}W^{k}$WkWk in terms of the components of $\mathbf{U}$U and $\mathbf{V}$V:

$W_{k}W^{k}=U^{i}V^{j}U_{i}V_{j}-U^{i}V^{j}U_{j}V_{i}.\tag{17.60}$WkWk=UiVjUiVj−UiVjUjVi.(17.60)

Let us give the geometric interpretation to each familiar combination in this identity. Namely,

$\begin{aligned}W_{k}W^{k} & =\mathbf{W}\cdot\mathbf{W}=\operatorname{len}^{2} \mathbf{W}\text{,}\ \ \ \ \ \ \ \ \ \ \left(17.61\right)\\U_{i}U^{i} & =\mathbf{U\cdot U}=\operatorname{len}^{2}\mathbf{U}\text{,}\ \ \ \ \ \ \ \ \ \ \left(17.62\right)\\V_{j}V^{j} & =\mathbf{V}\cdot\mathbf{V}=\operatorname{len}^{2} \mathbf{V}\text{, and}\ \ \ \ \ \ \ \ \ \ \left(17.63\right)\\U_{j}V^{j} & =U^{i}V_{i}=\mathbf{U}\cdot\mathbf{V}=\operatorname{len} \mathbf{U}\operatorname{len}\mathbf{V}\cos\gamma.\ \ \ \ \ \ \ \ \ \ \left(17.64\right)\end{aligned}$WkWkUiUiVjVjUjVj=W⋅W=len2W,(17.61)=U⋅U=len2U,(17.62)=V⋅V=len2V,and(17.63)=UiVi=U⋅V=lenUlenVcosγ.(17.64)

Thus, in geometric terms, the preceding equation reads

$\operatorname{len}^{2}\mathbf{W}=\operatorname{len}^{2}\mathbf{U} \operatorname{len}^{2}\mathbf{V}-\operatorname{len}^{2}\mathbf{U} \operatorname{len}^{2}\mathbf{V}\cos^{2}\gamma,\tag{17.65}$len2W=len2Ulen2V−len2Ulen2Vcos2γ,(17.65)

which, thanks to the trigonometric identity $1-\cos^{2}\gamma=\sin^{2}\gamma$1−cos2γ=sin2γ, reduces to

$\operatorname{len}^{2}\mathbf{W}=\operatorname{len}^{2}\mathbf{U} \operatorname{len}^{2}\mathbf{V}\sin^{2}\gamma.\tag{17.66}$len2W=len2Ulen2Vsin2γ.(17.66)

Taking the square root of both sides, we arrive at the desired equation

$\operatorname{len}\mathbf{W}=\operatorname{len}\mathbf{U}\operatorname{len} \mathbf{V}\sin\gamma. \tag{3.28}$lenW=lenUlenVsinγ.(3.28)

Thus, the length condition in the definition of the cross product is confirmed.

To test whether the orientation condition is satisfied, me must determine whether $\mathbf{U}$U, $\mathbf{V}$V, and $\mathbf{W}$W form a positively oriented set. Recall from Chapter 16, that the sign of the combination $e_{ijk}U^{i}V^{j}W^{k}$eijkUiVjWk, and therefore that of $\varepsilon_{ijk}U^{i} V^{j}W^{k}$εijkUiVjWk, tells us whether the orientation of $\mathbf{U},\mathbf{V} ,\mathbf{W}$U,V,W is the same as that of the basis $\mathbf{Z}_{i}$Zi. Since

$\varepsilon_{ijk}U^{i}V^{j}W^{k}=W_{k}W^{k}\gt 0,\tag{17.67}$εijkUiVjWk=WkWk>0,(17.67)

we conclude that the orientation $\mathbf{U},\mathbf{V},\mathbf{W}$U,V,W is indeed the *same* as the orientation as the basis $\mathbf{Z}_{i}$Zi. Thus, the combination

$\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}\tag{17.68}$εijkUiVjZk(17.68)

equals $\mathbf{U}\times\mathbf{V}$U×V only when the coordinate system is positively oriented. In a negatively oriented coordinate system, we must reverse the sign, i.e.

$\mathbf{W}=-\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}.\tag{17.69}$W=−εijkUiVjZk.(17.69)

Of course, we should have a priori known that the equation

$\mathbf{W}=\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k} \tag{17.50}$W=εijkUiVjZk(17.50)

contains an inconsistency, since $\mathbf{W}$W is an invariant while $\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}$εijkUiVjZk is an invariant only with respect to orientation-preserving coordinate systems due to the corresponding feature of the Levi-Civita symbol $\varepsilon_{ijk}$εijk.

Finally, note that the coordinate space representation for the two-dimensional cross product $\mathbf{V}=\times\mathbf{U}$V=×U reads

$\mathbf{V}=\varepsilon_{ij}U^{i}\mathbf{Z}^{j}\tag{17.70}$V=εijUiZj(17.70)

in right-handed coordinate systems and

$\mathbf{V}=-\varepsilon_{ij}U^{i}\mathbf{Z}^{j}\tag{17.71}$V=−εijUiZj(17.71)

in left-handed coordinate systems. Proving these identities is left as an exercise.

In this Section, we will demonstrate the formula

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) =\left( \mathbf{U}\cdot\mathbf{W}\right) \mathbf{V}-\left( \mathbf{U}\cdot \mathbf{V}\right) \mathbf{W}\tag{17.72}$U×(V×W)=(U⋅W)V−(U⋅V)W(17.72)

which gives an expression for the double cross product $\mathbf{U} \times\left( \mathbf{V}\times\mathbf{W}\right)$U×(V×W) in terms of the dot product and linear combinations. This demonstration, for which we will give full details, is an excellent exercise in the tensor technique. The derivations of other cross product identities, which use many of the same elements of the tensor technique, will be left for exercises.

Denote the cross product $\mathbf{V}\times\mathbf{W}$V×W by $\mathbf{S}$S, i.e.

$S^{k}=\varepsilon^{ijk}V_{i}W_{j}.\tag{17.73}$Sk=εijkViWj.(17.73)

Then the components of the cross product $\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right)$U×(V×W), i.e. $\mathbf{U}\times\mathbf{S}$U×S, are given by the question

$\varepsilon_{rst}U^{r}S^{s}=\varepsilon_{rst}U^{r}\varepsilon^{ijs}V_{i}W_{j}.\tag{17.74}$εrstUrSs=εrstUrεijsViWj.(17.74)

Since

$\varepsilon_{rst}\varepsilon^{ijs}=\delta_{rst}^{ijs}=-\delta_{rts} ^{ijs}=\delta_{r}^{j}\delta_{t}^{i}-\delta_{r}^{i}\delta_{t}^{j},\tag{17.75}$εrstεijs=δrstijs=−δrtsijs=δrjδti−δriδtj,(17.75)

we have

$\varepsilon_{rst}U^{r}\varepsilon^{ijs}V_{i}W_{j}=\left( \delta_{r}^{j} \delta_{t}^{i}-\delta_{r}^{i}\delta_{t}^{j}\right) U^{r}V_{i}W_{j}=U^{r} V_{t}W_{r}-U^{r}V_{r}W_{t},\tag{17.76}$εrstUrεijsViWj=(δrjδti−δriδtj)UrViWj=UrVtWr−UrVrWt,(17.76)

which is the final expression for $\mathbf{U}\times\left( \mathbf{V} \times\mathbf{W}\right)$U×(V×W) in terms of the components of the vectors $\mathbf{U}$U, $\mathbf{V}$V, and $\mathbf{W}$W. In order to return to geometric quantities, note that $U^{r}W_{r}=\mathbf{U}\cdot\mathbf{W}$UrWr=U⋅W and $U^{r} V_{r}=\mathbf{U}\cdot\mathbf{V}$UrVr=U⋅V. Thus, we arrive at the equation

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) =\left( \mathbf{U}\cdot\mathbf{W}\right) \mathbf{V}-\left( \mathbf{U}\cdot \mathbf{V}\right) \mathbf{W,} \tag{17.72}$U×(V×W)=(U⋅W)V−(U⋅V)W,(17.72)

as we set out to do.

Other cross product identities, whose derivations are left as exercises, include

$\begin{aligned}\left( \mathbf{U}\times\mathbf{V}\right) \times\left( \mathbf{U} \times\mathbf{W}\right) & =\left( \mathbf{U}\cdot\left( \mathbf{V} \times\mathbf{W}\right) \right) \mathbf{U}\ \ \ \ \ \ \ \ \ \ \left(17.77\right)\\\left( \mathbf{U}\times\mathbf{V}\right) \cdot\left( \mathbf{W} \times\mathbf{X}\right) & =\left( \mathbf{U}\cdot\mathbf{W}\right) \left( \mathbf{V}\cdot\mathbf{X}\right) -\left( \mathbf{U}\cdot \mathbf{X}\right) \left( \mathbf{V}\cdot\mathbf{W}\right)\ \ \ \ \ \ \ \ \ \ \left(17.78\right)\\\mathbf{U}\cdot\left( \mathbf{V}\times\mathbf{W}\right) & =\mathbf{V} \cdot\left( \mathbf{W}\times\mathbf{U}\right) =\mathbf{W}\cdot\left( \mathbf{U}\times\mathbf{V}\right)\ \ \ \ \ \ \ \ \ \ \left(17.79\right)\\\mathbf{0} & =\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) +\mathbf{V}\times\left( \mathbf{W}\times\mathbf{U}\right) +\mathbf{W} \times\left( \mathbf{U}\times\mathbf{V}\right) .\ \ \ \ \ \ \ \ \ \ \left(17.80\right)\end{aligned}$(U×V)×(U×W)(U×V)⋅(W×X)U⋅(V×W)0=(U⋅(V×W))U(17.77)=(U⋅W)(V⋅X)−(U⋅X)(V⋅W)(17.78)=V⋅(W×U)=W⋅(U×V)(17.79)=U×(V×W)+V×(W×U)+W×(U×V).(17.80)

In this Section, we will put forth an approach to restore the full tensor property to the volume element $\sqrt{Z}$Z and the Levi-Civita symbols $\varepsilon_{ijk}$εijk and $\varepsilon^{ijk}$εijk. Recall that the volume element $\sqrt{Z}$Z transforms according to the rule

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

where the presence of the absolute value limits its tensor property to orientation-preserving coordinate transformations. Correspondingly, the transformation rules for the Levi-Civita symbols $\varepsilon_{ijk}$εijk and $\varepsilon^{ijk}$εijk depend on the sign of $\det J$detJ, i.e.

$\begin{aligned}\varepsilon_{i^{\prime}j^{\prime}k^{\prime}} & =\operatorname{sign}\left( \det J\right) \varepsilon_{ijk}J_{i^{\prime}}^{i}J_{j^{\prime}} ^{j}J_{k^{\prime}}^{k}\text{ and}\ \ \ \ \ \ \ \ \ \ \left(17.81\right)\\\varepsilon^{i^{\prime}j^{\prime}k^{\prime}} & =\operatorname{sign}\left( \det J\right) \varepsilon^{ijk}J_{i}^{i^{\prime}}J_{j}^{j^{\prime}} J_{k}^{k^{\prime}}\ \ \ \ \ \ \ \ \ \ \left(17.82\right)\end{aligned}$εi′j′k′εi′j′k′=sign(detJ)εijkJi′iJj′jJk′kand(17.81)=sign(detJ)εijkJii′Jjj′Jkk′(17.82)

where

$\operatorname{sign}x=\left\{ \begin{array} {ll} \phantom{+} 1\text{,} & \text{if }x\gt 0\\ -1\text{,} & \text{if }x \lt 0. \end{array} \right.\tag{17.83}$signx={+1,−1,ifx>0ifx<0.(17.83)

As a result, the tensor property of the Levi-Civita symbols $\varepsilon _{ijk}$εijk and $\varepsilon^{ijk}$εijk is also restricted to orientation-preserving coordinate transformations, i.e. those for which $\operatorname{sign}\left( \det J\right) =1$sign(detJ)=1.

In a Euclidean space, the full tensor property of the volume element, and therefore of the Levi-Civita symbols, can be restored by leveraging the availability of the absolute sense of orientation. Namely, let the *orientation factor* $\Pi$Π be the "sign" of the coordinate orientation, i.e.

$\Pi=\left\{ \begin{array} {ll} \phantom{+} 1 & \text{for a right-handed coordinate system}\\ -1, & \text{for a left-handed coordinate system.} \end{array} \right.\tag{17.84}$Π={+1−1,foraright-handedcoordinatesystemforaleft-handedcoordinatesystem.(17.84)

With the help of $\Pi$Π, redefine the volume element to be the quantity

$\Pi\sqrt{Z}\tag{17.85}$ΠZ(17.85)

and redefine the Levi-Civita symbols to be

$\begin{aligned}\varepsilon^{ijk} & =\frac{e^{ijk}}{\Pi\sqrt{Z}}\ \ \ \ \ \ \ \ \ \ \left(17.86\right)\\\varepsilon_{ijk} & =\Pi\sqrt{Z}e_{ijk}.\ \ \ \ \ \ \ \ \ \ \left(17.87\right)\end{aligned}$εijkεijk=ΠZeijk(17.86)=ΠZeijk.(17.87)

With respect to these alternative definitions, the volume element is an unqualified relative invariant of weight $1$1, while the Levi-Civita symbols are absolute tensors. Furthermore, the formula

$\mathbf{W}=\varepsilon_{ijk}U^{k}V^{j}\mathbf{Z}^{k} \tag{17.50}$W=εijkUkVjZk(17.50)

becomes a universal expression for $\mathbf{U}\times\mathbf{V}$U×V.

However, despite the apparent utility of this approach, there is a compelling reason to continue using the conventional definitions. Namely, the orientation factor is possible only in Euclidean spaces where the concept of orientation is available in an absolute sense. Thus,we would not be able to generalize this approach to Riemannian spaces and if we were to adopt the new definitions in Euclidean spaces, we would create a discrepancy between the two types of spaces. Meanwhile, it is a fundamental tenet of our subject to treat Euclidean spaces strictly as a special case of Riemannian spaces.

Exercise 17.1A change of coordinates is called orthogonal if

$J^{T}J=I,\tag{17.88}$JTJ=I,(17.88)

where $J$J is the matrix corresponding to the Jacobian of the coordinate transformation. Show that any relative tensor is an absolute tensor with respect to orthogonal transformations. In particular, show that all relative tensors are absolute tensors if we limit ourselves to Cartesian coordinates.

Exercise 17.2Show that if $S_{jk}^{i}$Sjki is a relative tensor of weight $M$M and $T_{t}^{rs}$Ttrs is a relative tensor of weight $N$N, then $S_{jk}^{i}T_{t}^{rs}$SjkiTtrs is a relative tensor of weight $M+N$M+N. In particular, if $M=-N$M=−N, then $S_{jk}^{i}T_{t}^{rs}$SjkiTtrs is an absolute tensor.

Exercise 17.3Since $\delta_{rst}^{ijk}=e^{ijk}e_{rst}$δrstijk=eijkerst, show that the statement in the preceding exercise offers an alternative reason for the tensor property of $\delta_{rst}^{ijk}$δrstijk.

Exercise 17.4Show that the relative tensor property is reflexive, symmetric, and transitive in the sense described in Section 14.12.

Exercise 17.5Show that the result of contracting a relative tensor of weight $M$M is also a relative tensor of weight $M$M.

Exercise 17.6In three dimensions, show that the determinant of a relative tensor $a_{ij}$aij of weight $M$M is a relative invariant of weight $2+3M$2+3M. Show that in $n$n dimensions, this expression generalizes to $2+nM$2+nM.

Exercise 17.7Confirm the identity

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

for the transformation between Cartesian and polar coordinates in two dimensions, as well as the transformation between Cartesian and spherical coordinates in three dimensions.

Exercise 17.8Show that

$\varepsilon_{\cdot ik}^{i}=0.\tag{17.89}$ε⋅iki=0.(17.89)

Exercise 17.9Show that the determinants of $A_{\cdot j}^{i}$A⋅ji and $A_{i}^{\cdot j}$Ai⋅j are the same. Thus, the symbol $A$A introduced in Section 17.5 is well-defined regardless of which index of $A_{ij}$Aij is raised or which index of $A^{ij}$Aij is lowered.

Exercise 17.10In a one-dimensional space, show that $\varepsilon^{i}\mathbf{Z}_{i}$εiZi is a unit vector that points in the direction of $\mathbf{Z}_{1}$Z1. Conclude that it is therefore an invariant, but only with respect to orientation-preserving coordinate transformations.

Exercise 17.11By appealing directly to the definition of the covariant derivative, show that it is metrinilic with respect to the Levi-Civita symbol, i.e.

$\nabla_{p}\varepsilon^{ijk}=0. \tag{17.40}$∇pεijk=0.(17.40)

Exercise 17.12Use the metrinilic property of the Levi-Civita symbols to show the same for the delta systems, i.e.

$\nabla_{m}\delta_{rst}^{ijk},\ \ \nabla_{m}\delta_{rs}^{ij},\ \ \nabla _{m}\delta_{r}^{i}=0.\tag{17.90}$∇mδrstijk,∇mδrsij,∇mδri=0.(17.90)

Exercise 17.13In a three-dimensional space, show that

$\varepsilon_{ijk}=\mathbf{Z}_{i}\cdot\left( \mathbf{Z}_{j}\times \mathbf{Z}_{k}\right)\tag{17.91}$εijk=Zi⋅(Zj×Zk)(17.91)

in a right-handed coordinate system. Similarly, show that

$\mathbf{Z}^{i}=\frac{1}{2}\varepsilon^{ijk}\mathbf{Z}_{j}\times\mathbf{Z}_{k},\tag{17.92}$Zi=21εijkZj×Zk,(17.92)

subject to the same qualification.

Exercise 17.14Show that

$\mathbf{Z}^{i}\times\mathbf{Z}_{i}=\mathbf{0}.\tag{17.93}$Zi×Zi=0.(17.93)

Exercise 17.15Show that

$\left( \mathbf{U}\times\mathbf{V}\right) \times\left( \mathbf{U} \times\mathbf{W}\right) =\left( \mathbf{U}\cdot\left( \mathbf{V} \times\mathbf{W}\right) \right) \mathbf{U.} \tag{17.77}$(U×V)×(U×W)=(U⋅(V×W))U.(17.77)

Exercise 17.16Show that

$\left( \mathbf{U}\times\mathbf{V}\right) \cdot\left( \mathbf{W} \times\mathbf{X}\right) =\left( \mathbf{U}\cdot\mathbf{W}\right) \left( \mathbf{V}\cdot\mathbf{X}\right) -\left( \mathbf{U}\cdot\mathbf{X}\right) \left( \mathbf{V}\cdot\mathbf{W}\right) . \tag{17.78}$(U×V)⋅(W×X)=(U⋅W)(V⋅X)−(U⋅X)(V⋅W).(17.78)

Exercise 17.17Demonstrate that

$\mathbf{U}\cdot\left( \mathbf{V}\times\mathbf{W}\right) =\mathbf{V} \cdot\left( \mathbf{W}\times\mathbf{U}\right) =\mathbf{W}\cdot\left( \mathbf{U}\times\mathbf{V}\right) \tag{17.79}$U⋅(V×W)=V⋅(W×U)=W⋅(U×V)(17.79)

by showing that each product equals

$\varepsilon_{ijk}U^{i}V^{j}W^{k}.\tag{17.94}$εijkUiVjWk.(17.94)

Exercise 17.18Use the identity

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) =\left( \mathbf{U}\cdot\mathbf{W}\right) \mathbf{V}-\left( \mathbf{U}\cdot \mathbf{V}\right) \mathbf{W,} \tag{17.72}$U×(V×W)=(U⋅W)V−(U⋅V)W,(17.72)

to show that

$\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right) +\mathbf{V} \times\left( \mathbf{W}\times\mathbf{U}\right) +\mathbf{W}\times\left( \mathbf{U}\times\mathbf{V}\right) =\mathbf{0}. \tag{17.80}$U×(V×W)+V×(W×U)+W×(U×V)=0.(17.80)

Exercise 17.19For the two-dimensional cross product $\mathbf{V}=\times\mathbf{U}$V=×U, described in Section 17.8.2, show that

$\mathbf{V}=\varepsilon_{ij}U^{i}\mathbf{Z}^{j} \tag{17.70}$V=εijUiZj(17.70)

in a right-handed coordinate system and

$\mathbf{V}=-\varepsilon_{ij}U^{i}\mathbf{Z}^{j} \tag{17.71}$V=−εijUiZj(17.71)

in a left-handed coordinate system.