In $n$ dimensions, the Levi-Civita symbols $\varepsilon_{i_{1}\cdots i_{n}}$ and $\varepsilon^{i_{1}\cdots i_{n}}$ are defined by the equations where $\sqrt{Z}$ is the volume element first introduced in Chapter 9. In the natural Euclidean dimensions, the above definition reduces to in three dimensions, in two dimensions and in one dimension. You may be surprised that the one-dimensional Levi-Civita symbols $\varepsilon_{i}$ and $\varepsilon^{i}$ 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 $\varepsilon_{ijk}$ and $\varepsilon^{ijk}$. To this end, we will first explore the transformation rules for the permutation systems $e_{ijk}$ and $e^{ijk}$ 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 $e_{ijk}$ and $e^{ijk}$ are not tensors. In other words, if, say, $e_{i^{\prime}j^{\prime }k^{\prime}}$, is defined in the primed coordinates in the exact same terms as $e_{ijk}$ in the unprimed coordinates, then there is no reason to expect that the two objects are related by the requisite identity 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 $A_{j}^{i}$, we have Thus, if then, according to the preceding identity, Thus, the transformation rule for the subscripted permutation system $e_{ijk}$ reads Correspondingly, the rule for the superscripted permutation symbol $e^{ijk}$ features the determinant of $J$ rather than its inverse, i.e.

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

A variant $T_{j}^{i}$ with a representative collection of indices is called a relative tensor of weight $M$ if it transforms according to the rule In particular, conventional tensors are relative tensors of weight $0$, and are often referred to as absolute tensors to highlight their special place among relative tensors.

A relative tensor $T$ of order zero is called a relative invariant and transforms according to the rule We should recognize that the term relative invariant is an oxymoron since the values $T$ and $T^{\prime}$ 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 $M$ as well as a product of a relative tensor of weight $M$ and a number are also relative tensors of weight $M$. Thus, the collection of relative tensors of a given weight is closed under linear combinations. Furthermore, the product of a relative tensor of weight $M$ and a relative tensor of weight $N$ is a relative tensor of weight $M+N$. Finally, a contraction of a relative tensor of weight $M$ is also a relative tensor of weight $M$.

Returning to the permutation symbols, which transform according to the equations and we observe that $e_{ijk}$ is a relative covariant tensor of weight $-1$ while $e^{ijk}$ is a relative contravariant tensor of weight $1$. (Since $e_{ijk}$ and $e^{ijk}$ 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 $e_{ijk}e^{rst}$, which we recognize as the delta symbol $\delta_{ijk}^{rst}$, 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 $\sqrt{Z}$ and will therefore turn our attention now to determinants.

Suppose that $A_{ij}$ is an absolute covariant tensor. Recall that its determinant is given by the formula Then, by the multiplicative property of relative tensors stated in the previous Section (and demonstrated in Exercise 17.2), $\det A$ is a relative invariant of weight $2$. Meanwhile, the determinant of an absolute tensor $A^{ij}$, which is given by is a relative invariant of weight $-2$. Finally, recall that, since the delta system $\delta_{rst}^{ijk}$ is an absolute tensor, we concluded in Chapter 16 that the determinant of a mixed tensor $A_{j}^{i}$, i.e. is also an absolute invariant.

As soon as we calculated the volume element $\sqrt{Z}$ in Cartesian and polar coordinates in Chapter 9, it became apparent that $\sqrt{Z}$ is not an invariant. Indeed, in Cartesian coordinates, and 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 $Z$, being the determinant of the covariant metric tensor $Z_{ij}$, is a relative invariant of weight $2$. In other words, if $Z^{\prime}$ is the determinant of $Z_{i^{\prime}j^{\prime}}$ in the primed coordinates, then $Z$ and $Z^{\prime }$ are related by the equation

As tempting as it may be to take the square roots and conclude that we must remember that the determinant of the Jacobian $J$ may very well be negative. Thus, the correct relationship is Therefore, we are not able to conclude that the volume element $\sqrt{Z}$ is a relative invariant of weight $1$. In order to actually reach that conclusion, we must restrict our attention to coordinate changes for which $\det J\gt 0$ -- in other words, orientation-preserving coordinate transformations. Under this condition, the above identity does simplify to To use the terminology introduced in Section 14.15, $\sqrt{Z}$ is a relative invariant of weight $1$ only with respect to orientation-preserving coordinate transformations.

Recall that, much like the volume element $\sqrt{Z}$, the permutation symbol $e^{ijk}$ is also a relative tensor of weight $1$. Thus, dividing $e^{ijk}$ by $\sqrt{Z}$ yields an absolute tensor -- albeit, only with respect to orientation-preserving coordinate transformations. Meanwhile, $e_{ijk}$ is a relative tensor of weight $-1$. Thus, multiplying $e_{ijk}$ by $\sqrt{Z}$ 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 $\sqrt{Z}$ in order to balance the permutations systems and not another relative invariant of weight $1$ that is not limited to orientation-preserving coordinate transformations?

Such an object proves difficult to come by. Indeed, suppose that $U$ is, in fact, such an object, i.e. a relative invariant of weight $1$ not subject to any qualifications. Since we have no further requirements of $U$, suppose that $U\equiv1$ in some Cartesian coordinate system $Z^{i}$. Then the values of $U$ are uniquely determined in all coordinate systems. Indeed, in any alternative coordinate system $Z^{i^{\prime}}$, $U^{\prime}$ is given by the equation where $J$ is the Jacobian of the coordinate transformation between $Z^{i}$ and $Z^{i^{\prime}}$. Since $U$ and $\sqrt{Z}$ coincide in the Cartesian coordinates $Z^{i}$ and transform by the similar rules we conclude that $U$ agrees with $\sqrt{Z}$ to within sign in all coordinates systems.

Does the resulting object $U$ offer a better alternative to $\sqrt{Z}$? It has one indisputable advantage over $\sqrt{Z}$: it is an unqualified relative invariant of weight $1$. However, since the construction of $U$ requires us to a single out one coordinate system -- i.e. the Cartesian coordinates of a particular orientation where $U=1$ -- 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 $\sqrt{Z}$ is the superior choice.

Recall that the general $n$-dimensional definition reads 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 $\mathbf{Z}_{i}$ and $\mathbf{Z}^{i}$, the metric tensors $Z_{ij}$ and $Z^{ij}$, and the volume element $\sqrt{Z}$.

Note that the complete delta system $\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}$ defined by the identity can be similarly expressed in terms of the Levi-Civita symbols, i.e. The appeal of this identity, compared to the original definition of $\delta_{j_{1}\cdots j_{n}}^{i_{1}\cdots i_{n}}$ 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 $A_{j}^{i}$ in two dimensions, recall the identity derived in Section 16.10, where $\det A$ denotes the determinant of $A_{j}^{i}$. Importantly, if $A_{j}^{i}$ is a tensor, then all elements in this identity, including $\det A$, are tensors. The same cannot be said, however, for a covariant tensor $A_{ij}$ which satisfies the identity where $\det A$ denotes the determinant of $A_{ij}$, or for a contravariant tensor $A^{ij}$ which satisfies the identity where $\det A$ denotes the determinant of $A^{ij}$. Thus, the symbol $A$ 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 $A$ can be assigned a unique meaning.

Let us focus our attention on the identity for a covariant tensor $A_{ij}$ where, once again, $\det A$ denotes its determinant. Since the permutation system $e_{ij}$ is expressed in terms of the Levi-Civita symbol $\varepsilon_{ij}$ by the equation we find that Since $A_{ir}A_{js}-A_{is}A_{jr}$, $\varepsilon_{ij}$, and $\varepsilon_{rs}$ are tensors, the quantity 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 $A_{ij}$. Since $1/Z$ is the determinant of the contravariant metric tensor $Z^{ij}$, then $\det\left( A\right) /Z$ is the determinant of $A_{ir}Z^{ij}$ by the multiplicative property of determinants. In other words, $\det\left( A\right) /Z$ is the determinant of $A_{\cdot j}^{i}$.

In light of this insight, let us agree to denote by the symbol $A$ the determinant of $A_{\cdot j}^{i}$, regardless of whether the context is concerned with $A_{ij}$, $A_{j}^{i}$, or $A^{ij}$. With the help of this convention, the three identities at the center of this Section can be written as 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 $A_{ij}$ or contravariant tensor $A^{ij}$, it would need to be the determinant of $A_{\cdot j}^{i}$ 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 $A_{ij}$ is the metric tensor $Z_{ij}$. Then $A_{\cdot j}^{i}$ is the Kronecker delta $\delta_{j}^{i}$ and therefore $A=1$. Thus we have where we note that the middle equation is precisely the identity 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 in any Cartesian coordinates, where the Levi-Civita symbols $\varepsilon _{ijk}$ and $\varepsilon^{ijk}$ have constant values, while the covariant derivative coincides with the partial derivative. Therefore, the combinations $\nabla_{p}\varepsilon_{ijk}$ and $\nabla_{p}\varepsilon^{ijk}$ vanish, i.e. 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 $\varepsilon_{ijk}$ 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 $T_{pijk}$ the covariant derivative of the permutation system $e_{ijk}$, i.e. Since the permutation systems have constant values and therefore the partial derivative $\partial e_{ijk}/\partial Z^{p}$ vanishes, we have Observe that $T_{pijk}$ is skew-symmetric in the indices $i$, $j$, and $k$. Therefore, we need only to consider the elements $T_{p123}$ given by In each contraction on the right, there is only one nonzero term that corresponds to $m=1$ in the first contraction, $m=2$ in the second, and $m=3$ in the third, i.e. Factoring out $-e_{123}$, we find In other words, Thus, in general, Since the Christoffel symbol is symmetric in its subscripts and therefore we arrive at the following identity for the covariant derivative of the permutation systems:

Note that we encountered the combination $\Gamma_{mp}^{m}$ at the end of the last Chapter where we derived the identity 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 $\varepsilon_{ijk}$, we have Since and, as we just discovered, we find as we set out to show.

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

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

Fortunately, the two interpretations are equivalent and we, indeed, have In other words, the Levi-Civita symbols $\varepsilon_{ijk}$ and $\varepsilon ^{ijk}$, 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 Thus, the combination $\varepsilon_{rst}Z^{ir}Z^{js}Z^{kt}$ is given by Recall once again that where $\det A$ is the determinant of $A^{ij}$. This identity implies that since the determinant of the contravariant metric tensor $Z^{ij}$ is $1/Z$. Thus, Finally, since we arrive at the identity as we set out to show.

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

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.

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

The third condition implies that the cross product is anti-symmetric, i.e. 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. Indeed, $\mathbf{U}\times\mathbf{U}=-\mathbf{U}\times\mathbf{U}$ and must therefore be $\mathbf{0}$.

In Chapter 3, we demonstrated that the cross product satisfies the associative law and the distributive law but lacks associativity, i.e. Indeed, the vector on the left is orthogonal to $\mathbf{U}$ while the vector on the right is not necessarily so. However, the product $\mathbf{U} \times\left( \mathbf{V}\times\mathbf{W}\right)$ satisfies the identity which we will demonstrate below by working with the components of vectors. Meanwhile, note that it is not surprising that $\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right)$ is a linear combination of $\mathbf{V}$ and $\mathbf{W}$ since, being orthogonal to $\mathbf{V}\times\mathbf{W}$, it must lie in the plane spanned by $\mathbf{V}$ and $\mathbf{W}$.

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 $\mathbf{U}$. Specifically, the "cross product" $\times \mathbf{U}$ of a vector $\mathbf{U}$ is the vector $\mathbf{V}$ obtained from $\mathbf{U}$ by a counterclockwise $90^{\circ}$ rotation. Thus, the two-dimensional "cross product" is merely a synonym for a counterclockwise $90^{\circ}$ 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 $\mathbf{V}$ is orthogonal to $\mathbf{U}$, has the same length as $\mathbf{U}$, while the set $\mathbf{U}$, $\mathbf{V}$ is positively oriented. Furthermore, we will discover that the coordinate space representation of $\times\mathbf{U}$ can be given by a similar expression involving the two-dimensional Levi-Civita symbol $\varepsilon_{\alpha\beta}$.

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 $\left( a,b\right)$ and $\left( b,-a\right)$ are orthogonal. In other words, in order to find a vector orthogonal to $\left( a,b\right)$, we must switch the components and multiply one of them by $-1$. This operation is equivalent to multiplication by the skew-symmetric matrix which, as we know, is a representation of the two-dimensional permutation system $e_{ij}$.

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

Let us now return to three dimensions and exploit this idea by considering the vector $\mathbf{W}$ with components In other words, Note that since $\varepsilon_{ijk}=\sqrt{Z}e_{ijk}$, the expression on the right can be elegantly captured by the formula which provides a practical recipe for evaluating $\mathbf{W}$.

By design, $\mathbf{W}$ is orthogonal to both $\mathbf{U}$ and $\mathbf{V}$. For a formal confirmation, note that and by the fact that a double-contraction of a skew-symmetric system and a symmetric system vanishes, whereas $U^{i}U^{k}$ is a symmetric system in the first equation while $V^{j}V^{k}$ 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 $\mathbf{W}$ and confirm that, in fact, To show this, recall that the length of $\mathbf{W}$ is given by the equation In order to obtain $W^{k}$, raise the subscript $k$ in the identity and introduce new letters $r$ and $s$ for the dummy indices in anticipation of the upcoming contraction, i.e. or, equivalently, Combining the expressions for $W_{k}$ and $W^{k}$, we have Since and we find that Substituting this relationship into the equation we are able to eliminate the Levi-Civita symbols, i.e. Upon multiplying out and absorbing the Kronecker deltas, we arrive at the final expression for $W_{k}W^{k}$ in terms of the components of $\mathbf{U}$ and $\mathbf{V}$: Let us give the geometric interpretation to each familiar combination in this identity. Namely, Thus, in geometric terms, the preceding equation reads which, thanks to the trigonometric identity $1-\cos^{2}\gamma=\sin^{2}\gamma$, reduces to Taking the square root of both sides, we arrive at the desired equation 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 $\mathbf{U}$, $\mathbf{V}$, and $\mathbf{W}$ form a positively oriented set. Recall from Chapter 16, that the sign of the combination $e_{ijk}U^{i}V^{j}W^{k}$, and therefore that of $\varepsilon_{ijk}U^{i} V^{j}W^{k}$, tells us whether the orientation of $\mathbf{U},\mathbf{V} ,\mathbf{W}$ is the same as that of the basis $\mathbf{Z}_{i}$. Since we conclude that the orientation $\mathbf{U},\mathbf{V},\mathbf{W}$ is indeed the same as the orientation as the basis $\mathbf{Z}_{i}$. Thus, the combination equals $\mathbf{U}\times\mathbf{V}$ only when the coordinate system is positively oriented. In a negatively oriented coordinate system, we must reverse the sign, i.e. Of course, we should have a priori known that the equation contains an inconsistency, since $\mathbf{W}$ is an invariant while $\varepsilon_{ijk}U^{i}V^{j}\mathbf{Z}^{k}$ is an invariant only with respect to orientation-preserving coordinate systems due to the corresponding feature of the Levi-Civita symbol $\varepsilon_{ijk}$.

Finally, note that the coordinate space representation for the two-dimensional cross product $\mathbf{V}=\times\mathbf{U}$ reads in right-handed coordinate systems and in left-handed coordinate systems. Proving these identities is left as an exercise.

In this Section, we will demonstrate the formula which gives an expression for the double cross product $\mathbf{U} \times\left( \mathbf{V}\times\mathbf{W}\right)$ 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 $\mathbf{V}\times\mathbf{W}$ by $\mathbf{S}$, i.e. Then the components of the cross product $\mathbf{U}\times\left( \mathbf{V}\times\mathbf{W}\right)$, i.e. $\mathbf{U}\times\mathbf{S}$, are given by the question Since we have which is the final expression for $\mathbf{U}\times\left( \mathbf{V} \times\mathbf{W}\right)$ in terms of the components of the vectors $\mathbf{U}$, $\mathbf{V}$, and $\mathbf{W}$. In order to return to geometric quantities, note that $U^{r}W_{r}=\mathbf{U}\cdot\mathbf{W}$ and $U^{r} V_{r}=\mathbf{U}\cdot\mathbf{V}$. Thus, we arrive at the equation as we set out to do.