The covariant basis $\mathbf{Z}_{i}$ and its companion contravariant basis $\mathbf{Z}^{i}$ are equally well-suited for the decomposition of vectors at a given point. Let us first consider the components of vectors with respect to the covariant basis $\mathbf{Z}_{i}$. These are known as the contravariant components for reasons that will become obvious shortly.

Denote by $U^{1}$, $U^{2}$, and $U^{3}$, or, collectively, $U^{i}$, the components of a vector $\mathbf{U}$ with respect to the covariant basis $\mathbf{Z}_{i}$, i.e. With the help of the summation convention, we write The summation convention compelled us to use a superscript to enumerate the components and therefore to use the term contravariant. As always, this choice will prove to be an accurate predictor of the manner in which $U^{i}$ transforms under a coordinates transformation.

Since the covariant basis $\mathbf{Z}_{i}$ varies from one point to another, one and the same vector decomposed with respect to bases at different points in the Euclidean space will yield different contravariant components $U^{i}$. Later on, we will provide an analytical method for determining whether two sets of components at two different points in space represent equal vectors.

With the introduction of contravariant coordinates, note that the almost trivial identity can now be interpreted in two different ways. On the one hand, it can be seen merely as an instance of the index-renaming property of the Kronecker delta in a contraction. Alternatively, the expression on the right can be seen as a linear combination for $\mathbf{Z}_{i}$ in terms of the basis vectors $\mathbf{Z}_{1}$, $\mathbf{Z}_{2}$, and $\mathbf{Z}_{3}$. Thus, the elements of the Kronecker delta can also be interpreted as the collection of the contravariant components of the elements of a basis with respect to itself.

When a vector $\mathbf{U}$ is decomposed with respect to the contravariant basis $\mathbf{Z}^{i}$, i.e. the resulting coefficients $U_{i}$ are known as the covariant components.

Naturally, the contravariant components $U^{i}$ and the covariant components $U_{i}$ are closely related and one nearly can guess the relationship between them simply from the placement of the indices. Recall that the covariant and the contravariant bases are related by the identity Substituting this relationship into the expansion $\mathbf{U}=U^{j} \mathbf{Z}_{j}$, we find Thus, since $\mathbf{U}$ is also given by $\mathbf{U}=U_{i}\mathbf{Z}^{i}$, we have Equating the coefficients, we arrive at the relationship between $U_{i}$ and $U^{i}$, i.e. Since the metric tensor is symmetric, i.e. $Z_{ij}=Z_{ji}$, we can write this identity in the more pleasing way Deriving the inverse relationship in similar fashion is left as an exercise. Alternatively, this relationship can be derived from by inverting the metric tensor, as described in Section 9.5.2.

Finally, we note that the relationship between the contravariant and covariant components of a vector, captured by the identities and is exactly the same as the relationship between the covariant and contravariant bases. These relationships are examples of index juggling which is the subject of Chapter 11. Note that objects related by index juggling are, in fact, so closely related that we will tend to think of them as two different manifestations of the same object.

The equations and define the components but do not indicate how to calculate them. Depending on the situation, a practical calculation can be accomplished in a number of ways. In a pure geometric context illustrated in the two-dimensional figure below, one can construct a parallelogram with sides parallel to $\mathbf{Z}_{1}$ and $\mathbf{Z}_{2}$ and a vertex at the tip of $\mathbf{U}$. Then the signed ratios of the lengths of the sides of the parallelogram to the lengths of the corresponding covariant basis vectors $\mathbf{Z}_{i}$ are the contravariant components $U^{i}$.

An alternative approach is based on decomposition by the dot product. First discussed in Section 2.4, it become our most frequently used approach thanks to its algebraic nature. We will revisit this method later in this Chapter where it will find a particularly elegant expression in the tensor notation.

Note that the two methods have a crucial feature in common: each works in any coordinate systems and each can be described exclusively in terms of quantities available in the context of the chosen coordinate system. This shared aspect of the two approaches is essential to the concept of a variant to which we now turn.

The covariant and contravariant bases $\mathbf{Z}_{i}$ and $\mathbf{Z}^{i}$, the contravariant and covariant components $U^{i}$ and $U_{i}$, and all other objects introduced in the previous Chapter are examples of variants. We will now explain the meaning of this important term.

The definition of $\mathbf{Z}_{i}$ reads The very fact that it contains an explicit reference to the coordinates $Z^{i}$ points to its dependence on the choice of the coordinate system. Indeed, this is so: the covariant basis, at a fixed point in space, varies from one coordinate system to another. Objects that have this property are called variants, subject to an additional requirement described in the next paragraph. Clearly, any combination of variants -- be it a sum, a product, or the result of differentiation -- is a variant in its own right. Note, however, that the term variant refers to the potential to vary from one system to another and is not a guarantee of variability. Some special combinations of variants, such as $U^{i}\mathbf{Z}_{i}$, have the same value in all coordinate systems. Such variants, which are the central objects of our study, are known as invariants and are described in the next Section.

Importantly, the concept of a variant carries an additional requirement: in order to be considered a variant an object must be constructed by the same algorithm in all coordinate systems. This definition is admittedly rather informal and we will therefore attempt to clarify it by illustrating why each of the objects introduced in the previous Chapter are indeed variants.

Let us start with the covariant basis. To confirm that it is a variant, let us translate its analytical definition into words, i.e. in order to construct the covariant basis $Z_{i}$, differentiate the position vector $\mathbf{R}$ with respect to the coordinate $Z^{i}$. This algorithm is the same in all coordinates and therefore $\mathbf{Z}_{i}$ is a variant. For the covariant metric tensor $Z_{ij}$, note that it is constructed in two steps: 1) construct the covariant basis $\mathbf{Z}_{i}$ and 2) calculate the pairwise dot products $\mathbf{Z}_{i}\cdot \mathbf{Z}_{j}$. Thus, $Z_{ij}$ is a variant. The contravariant metric tensor $Z^{ij}$ includes the additional step 3) find the matrix inverse of $Z_{ij}$. Thus, it is also a variant. Similarly, the contravariant basis $\mathbf{Z}^{i}$ needs a further step 4) evaluate the contraction $Z^{ij}\mathbf{Z}j$. Finally, the contravariant components $U^{i}$ of a vector $\mathbf{U}$ require two steps: 1) construct the covariant basis $\mathbf{Z}_{i}$ and 2) decompose $U$ with respect to $\mathbf{Z}_{i}$.

The dependence of a variant on the coordinate system may be likened to the dependence of length on the units of measurement. For example, the length of a segment may be described as $25.4$ centimeters in the metric system or as $10$ inches in the imperial system. Meanwhile, the length of a segment is, of course, a valid concept that exists in the absence of any system of measurements. Therefore, a specific value of length in a particular measurement system can be thought of as a manifestation of length in that system. Similarly, a variant may be thought of as an entity that exists outside of any coordinate system, while the specific set of values that represent it in a particular coordinate system may be thought of as its particular coordinate manifestation. Thus, we can identify a variant with the algorithm that produces it, such as differentiation of the position vector with respect to coordinates for the covariant basis.

A variant, having different values in different coordinate systems, is said to transform from one coordinate system to another. Of great interest to us will be the transformation rule, i.e. the equation that relates the values of a variant in the different coordinate systems. We will discover that some variants, which we will call tensors, are subject to a particularly transformation rules. As you may expect, these special variants will play an instrumental role in our subject.

Finally, note that at this point, we would be hard-pressed to think of an object that is not a variant. We will encounter our first non-variant in Chapter 13 when we introduce the Jacobian which requires the presence of two coordinate systems.

The covariant basis $\mathbf{Z}_{i}$ and the contravariant components $U^{i}$ of a vector $\mathbf{U}$ are variants. Thus, by definition, the contraction is also a variant. However, it is a very special kind of variant: it has the same value, namely the vector $\mathbf{U}$, in all coordinate systems. Such a variant is called an invariant.

Invariants, by virtue of their independence from the choice of coordinates, are the ultimate objects of our study. Above all, the purpose of Tensor Calculus is to study the natural world (or, at least, a geometric idealization of it). Thus, every meaningful analysis must ultimately yield objects that are independent of coordinates.

Of course, the statement that the combination $U^{i}\mathbf{Z}_{i}$ is independent of coordinates is nearly tautological since the components $U^{i}$ are constructed precisely in such a way as to produce $\mathbf{U}$ by the contraction $U^{i}\mathbf{Z}_{i}$. Going forward, we will encounter invariants that arise in far more nontrivial ways. Nevertheless, the key to invariance will be exceptionally simple: the expression must be comprised of tensors and have all indices contracted away.

The combination $U^{i}\mathbf{Z}_{i}$ also offers an insight into the eventual meaning of the terms covariant and contravariant as applied to tensors, i.e. variants subject to a special transformation rule -- or, to be more precise, one of two rules that are the inverses of each other. One of the rules is called covariant and the other -- contravariant . Covariant tensors transform in the same way as the basis $\mathbf{Z}_{i}$ -- thus co. Contravariant tensors transform in the opposite way -- thus contra. When two objects that transform according to opposite rules are combined, the two transformations cancel each other and, as a result, the value of the combination in one coordinate system equals its value in the other coordinate system. In other words, the combination is an invariant. This, in a nutshell, is how invariants are produced in Tensor Calculus.

In Section 2.4, we considered the decomposition of a vector $\mathbf{U}$ with respect to a basis $\mathbf{b}_{1},\mathbf{b}_{2} ,\mathbf{b}_{3}$. Recall that the components of $\mathbf{U}$, which we at the time denoted by $U_{i}$, were given by the equations for an orthonormal, a.k.a. Cartesian, basis $\mathbf{b}_{i}$, for an orthogonal basis $\mathbf{b}_{i}$, and for an arbitrary basis $\mathbf{b}_{i}$.

We will now switch to the covariant basis $\mathbf{Z}_{i}$ and the contravariant components $U^{i}$. Note that replace $\mathbf{b}_{i}$ with $\mathbf{Z}_{i}$ and $U_{i}$ with $U^{i}$ in the above equations yields identities that are invalid from the tensor notation point of view, i.e. U^{i} & =mathbf{Z}_{i}cdotmathbf{U,}tag{$-$} U^{i} & =frac{mathbf{Z}_{i}cdotmathbf{U}}{mathbf{Z}_{i}cdot mathbf{Z}_{i}},text{ and }tag{$-$} left[ begin{array} {c} U^{1} U^{2} U^{3} end{array} right] & =left[ begin{array} {ccc} mathbf{Z}_{1}cdotmathbf{Z}_{1} & mathbf{Z}_{1}cdotmathbf{Z}_{2} & mathbf{Z}_{1}cdotmathbf{Z}_{3} mathbf{Z}_{2}cdotmathbf{Z}_{1} & mathbf{Z}_{2}cdotmathbf{Z}_{2} & mathbf{Z}_{2}cdotmathbf{Z}_{3} mathbf{Z}_{3}cdotmathbf{Z}_{1} & mathbf{Z}_{3}cdotmathbf{Z}_{2} & mathbf{Z}_{3}cdotmathbf{Z}_{3} end{array} right] ^{-1}left[ begin{array} {c} mathbf{Z}_{1}cdotmathbf{U} mathbf{Z}_{2}cdotmathbf{U} mathbf{Z}_{3}cdotmathbf{U} end{array} right] .tag{$-$} end{align} Recall from Section 9.5.4, that the $3\times3$ matrix above corresponds to the covariant metric tensor $Z_{ij}$. Therefore, its inverse corresponds to the contravariant metric tensor $Z^{ij}$.

Despite its invalid form, the equation U^{i}=mathbf{Z}_{i}cdotmathbf{U} tag{$-$} end{equation} is correct for an orthonormal basis $\mathbf{Z}_{i}$, although it is, of course, limited to that special case. This is observed for most equations that violate the rules of the tensor notation: they are typically correct only in a narrow range of special cases.

Crucially, the notational flaw in the above equation is easily fixed by replacing the covariant basis with the contravariant basis, i.e. In words: the contravariant component $U^{i}$ of the vector $\mathbf{U}$ is given by the dot product of $\mathbf{U}$ with the contravariant basis vector $\mathbf{Z}^{i}$. This equation remains valid for an orthonormal basis $\mathbf{Z}_{i}$ since, in that case, $\mathbf{Z}^{i}$ coincides with $\mathbf{Z}_{i}$. However, somewhat remarkably given its utmost simplicity, this equation is valid for an arbitrary basis $\mathbf{Z}_{i}$. In other words, it is valid in all coordinate systems.

To observe why this is so, note that the equation in matrix terms reads Thus, it can be seen that the equation is simply the tensor form of the equation which is valid for an arbitrary basis $\mathbf{Z}_{i}$.

The remarkably simple equation can also be derived quite concisely by pure tensor means. Simply dot both sides of the equation with $\mathbf{Z}^{i}$, i.e. Since $\mathbf{Z}_{j}\cdot\mathbf{Z}^{i}=\delta_{j}^{i}$ and $U^{j}\delta _{j}^{i}=U^{i}$, we have or as we set out to show.

Note that the simplicity of the equation does not imply that the matrix inversion inherent in linear decomposition can be circumvented. Instead, this equation hides the inversion in the contravariant basis $\mathbf{Z}^{i}$ and, in doing so, organizes the decomposition algorithm with respect to an arbitrary basis in such a way that it appears as simple as for an orthonormal basis.

It is left as an exercise to show that the covariant components $U_{i}$ are given by the individual dot products with the elements $\mathbf{Z}_{i}$ of the covariant basis, i.e.

The remarkable compactness of these identities is a testament to the tensor framework. While economy of notation does not imply economy of computation, the expressive power of the tensor notation is undeniable as it represents the best of two worlds: tensor expressions are as simple as those associated with Cartesian coordinates yet are valid in all coordinate systems. This pattern is found throughout Tensor Calculus, and we are about to see it again in our discussion of the coordinate space representation of the dot product.

In Section 2.6, we established the expression for the dot product of two vectors in terms of their components. Specifically -- once again switching to the symbols $U^{i}$, $V^{i}$, and $\mathbf{Z}_{i}$ -- we showed that the dot product $\mathbf{U}\cdot\mathbf{V}$ is given by the matrix equation If the $3\times1$ matrices corresponding to $U^{i}$ and $V^{i}$ are denoted by $U$ and $V$, and $M$ once again denotes the matrix corresponding to $Z_{ij}$, then $\mathbf{U}\cdot\mathbf{V}$ is given by If the basis is orthonormal, and therefore $M$ is the identity matrix, then the dot product reduces to the classical form

You have probably already guessed that the tensor equivalent of $U^{T}MV$ is $Z_{ij}U^{i}U^{j}$. In fact, we have already essentially derived it in the tensor notation in Section 8.4. However, since the derivation fits on a single line, we are happy to repeat it here. Recall that In the second equation, rename the index $i$ into $j$, i.e. in preparation for using the two contractions in a single expression. Now, dot both sides of the equations above, i.e. and note the chain of identities Thus, we have arrived at the fundamental identity This crucial identity clarifies the meaning of the term metric in metric tensor: the metric tensor is used for evaluating dot products, and therefore lengths and angles, in the Euclidean space.

In terms of the underlying arithmetic operations, $Z_{ij}U^{i}V^{j}$ is equivalent to the matrix expression $U^{T}MV$ and it nearly matches its compactness. In addition, it is indifferent to the order of the multiplicative terms and lacks the need for the transpose. Furthermore, the expression $Z_{ij}U^{i}V^{j}$ is subject to further "compactization". Recall that the combination $Z_{ij}V^{j}$ yields the covariant component $V_{i}$. Therefore, the equation can be rewritten in the form or, by the same token, in the form Note that in the unpacked form, the last equation reads which clearly demonstrates that we have been able to achieve the simplicity of the classical Cartesian expression found in the equation from Chapter 2. Meanwhile, the equation is valid in all bases. Its simplicity speaks to the expressive power of the tensor framework.

Finally, note that the length of a vector $\mathbf{U}$ in component form is given by which can be expressed more compactly as

Although the equation along with its unpacked form is valid in all coordinate systems, it will be useful to document the explicit expressions in standard coordinate systems corresponding to the equation that expresses the dot product in terms of the contravariant coordinates only.

In Cartesian coordinates, the covariant metric tensor is represented by the identity matrix and therefore In other words,

In polar coordinates in a two-dimensional space, and, therefore, $\mathbf{U}\cdot\mathbf{V}$ is given by

In cylindrical coordinates, and, therefore, $\mathbf{U}\cdot\mathbf{V}$ is given by

Finally, in spherical coordinates, and, therefore, $\mathbf{U}\cdot\mathbf{V}$ is given by

With every geometric vector $\mathbf{U}$, we associate its contravariant components $U^{i}$. This association works in both directions: from the vector, we can determine the components, and from the components, we can reconstruct the vector. Furthermore, we can naturally associate the vector $\mathbf{U}=U^{i}\mathbf{Z}_{i}$ with any first-order system $U^{i}$ with scalar elements, regardless of how the system may have arisen.

Thus, there is a natural one-to-one correspondence between first-order systems and geometric vectors. In fact, this correspondence is so strong that we may go as far as referring to first-order systems as vectors. That is, we may write consider a vector $U^{i}$ instead of consider a vector with components $U^{i}$. In other words, in the context of Euclidean spaces, a vector and its components are so closely linked that the terms vector and components of a vector become interchangeable.

In order to demonstrate how a practical analysis may be conducted in the component space, let us analyze the motion of a material particle in a Euclidian space. Suppose that the Euclidean space is referred to an arbitrary coordinate system $Z^{i}$ and that the trajectory of the particle is given by the equations where $t$ represents time. The functions $Z^{i}\left( t\right)$ are referred to as the equations of the motion.

Let our goal is to express the components $V^{i}$ of the particle's velocity vector $\mathbf{V}$ and the components $A^{i}$ of the acceleration vector $\mathbf{A}$ in terms of the equations of the motion $Z^{i}\left( t\right)$. Note that at each point along the trajectory, the vectors $\mathbf{V}$ and $\mathbf{A}$ must be decomposed with respect to the covariant basis $\mathbf{Z}_{i}$ found at that point.

In this Section, we will show that the velocity components $V^{i}$ are given by the intuitive equation Meanwhile, the equation for the acceleration components $A^{i}$ will be postponed until later since it requires the introduction of the Christoffel symbol $\Gamma_{jk}^{i}$ which is the subject of Chapter 12.

We described the equation as intuitive since it can be easily seen to be true in an affine setting. Indeed, in affine coordinates, the position vector $\mathbf{R}$ along the trajectory is given by the function provided that $\mathbf{R}$ emanates from the origin of the coordinate system. In other words, the functions $Z^{i}\left( t\right)$ represent the components of $\mathbf{R}\left( t\right)$ with respect to the constant basis $\mathbf{i,j,k}$. Differentiating both sides with respect to $t$, we find that which leads to the conclusion that the velocity components $V^{i}$ are given by However, it is clear that the above derivation is limited to affine coordinates since it assumes that the coordinate basis does not vary with $t$. Thus, the fact that the equation remains valid in curvilinear coordinates is perhaps somewhat unexpected.

Let us now derive the equation in general curvilinear coordinates. Start with the definition of the velocity $\mathbf{V}$, i.e. where $\mathbf{R}\left( t\right)$ represents the values of the position vector along the trajectory. The function $R\left( t\right)$ can be formed by composing the function $\mathbf{R}\left( Z\right)$, i.e. the position vector a function of the coordinate $Z^{i}$, with the equations of the motion $Z^{i}\left( t\right)$. In other words, By the chain rule, we find Since the derivative $\partial\mathbf{R}\left( Z\right) /\partial Z^{i}$ yields the covariant basis $\mathbf{Z}_{i}$, we have Therefore, the derivatives are indeed the contravariant components $V^{i}$ of $\mathbf{V}$, i.e. as we set out to show.

Two surprising aspects of this identity that are worthy of note. First is the very fact that it is true despite the spatial variability of the covariant basis. As we show below, the analogous equations for the acceleration components $A^{i}$, i.e. A^{i}=frac{dV^{i}left( tright) }{dt} tag{$-$} end{equation} or, equivalently, A^{i}=frac{d^{2}Z^{i}left( tright) }{dt^{2}}, tag{$-$} end{equation} do not hold -- precisely due to the variability of the basis.

The second surprising aspect of the equation is the fact that it does not reference either the position vector or the covariant basis or any other geometric elements of the Euclidean space. Meanwhile, one and the same equations of motion describe completely different trajectories in different coordinate systems. For example, the equations describe a parabola in Cartesian coordinates and a spiral in polar coordinates. Thus, the two motions are characterized by completely different velocity vectors $\mathbf{V}\left( t\right)$. Nevertheless, the components $V^{i}$ of the velocity, i.e. are the same for both motions.

For a specific example, consider a particle moving along a helix in a three-dimensional Euclidean space. If the Euclidean space is referred to cylindrical coordinates $Z^{1},Z^{2},Z^{3}=r,\theta,z$ oriented in a natural way with respect to the helix, then the equations of the motion read where $R$ is the radius of the helix, $\omega$ is the angular velocity of the circular motion, and $2\pi a/\omega$ is the span of the helix. Therefore, the components of the velocity, given by the equation have the following constant values Of course, even though the components of the velocity vector are constant, the velocity vector itself is variable due to the variability of the accompanying basis.

The magnitude $V$ of the velocity can be calculated by the formula Along the helix, this formula reads and therefore Thus, It is left as an exercise to the repeat this calculation in Cartesian coordinates.

As we have already stated, the components $A^{i}$ of the acceleration vector $\mathbf{A}$ do not equal the time derivative of the components $V^{i}$ of the velocity vector, i.e. If nothing else, this is clear from the example of uniform motion along a helix that we analyzed in the previous Section. Since the components $V^{i}$ of the velocity vector are constant, their derivatives vanish, i.e. Meanwhile, the acceleration of the particle moving along a helix is not zero and therefore $A^{i}\neq0$.

The underlying reason for the failure of the formula is, of course, the spatial variability of the covariant basis. In order to establish the correct relationship, let us once again start our analysis with a geometric object: the acceleration vector $\mathbf{A}$, defined as the derivative of the velocity vector $\mathbf{V} \left( t\right)$, i.e. Since $\mathbf{V}\left( t\right)$ is given by the equation where $\mathbf{Z}_{i}\left( t\right)$ is a function that represents the covariant basis $\mathbf{Z}_{i}$ along the trajectory, differentiating the above equation yields This is the crucial moment. The second term does not vanish since the derivative represents the (nonvanishing) rate of change of the basis vector $\mathbf{Z}_{i}$ along the trajectory. As a result, and consequently Therefore, in order to advance our analysis further, we must tackle the spatial variability of the covariant basis. This line of inquiry, which leads to the concept of the Christoffel symbol, is developed in Chapter 12.

The analysis similar to that of the motion of a material particle can be applied to abstract geometric curves. Consider a curve parameterized by a generic variable $\gamma$ rather than time $t$. In terms of the $\gamma$, the equations of the curve read Based on our foregoing discussion, we can conclude that the derivative represents the contravariant components of a tangent vector.

Suppose now that the curve is parameterized by arc length $s$, i.e. In Chapter 5, we established that the derivative produces a unit tangent $\mathbf{T}$. Thus, the contravariant components $T^{i}$ of the unit tangent $\mathbf{T}$ are given by the equation The fact that $\mathbf{T}$ is unit length is expressed in terms of its components by the equation or, equivalently,

In Section 5.2, we considered a curve given by the vector equation where $\gamma$ is an arbitrary parameter along the curve. We showed that the arc length $s$ of the segment between the points corresponding to values $\gamma_{0}$ and $\gamma_{1}$ of the parameter $\gamma$ is given by the integral On the one hand, this formula, which does not require coordinates in the ambient space, proved to be of great theoretical value. On the other hand, due to the lack of ambient coordinates, this formula cannot be used for practical calculations of the length of any concrete curve since it features geometric quantities rather than algebraic expressions. However, since we have introduced coordinates in the ambient space and are now able to work with the components of vectors, we can modify the above formula in order to make it suitable for practical calculations.

Introduce an arbitrary coordinate system $Z^{i}$ in the ambient Euclidean space and let the equations of the curve read As we demonstrated in the previous Section, the derivative $\mathbf{R} ^{\prime}\left( \gamma\right)$ is given by the equation In other words, are the contravariant components of $\mathbf{R}^{\prime}\left( \gamma\right)$. Since the dot product of a vector $\mathbf{U}$ with itself is given by the formula we have With the help of this identity, the equation becomes

Let us once again stress one of the most crucial aspects of this formula: it is valid in arbitrary coordinates $Z^{i}$ as well as for an arbitrary parameterization $\gamma$ of the curve, provided that $\gamma_{0}\leq\gamma_{1}$. The practical advantage of this formula over the coordinate-free equation is overwhelming and speaks to the immense utility of coordinate systems.

For a concrete example, let us calculate the arc length of a complete loop of the spiral in the following figure. The spiral can be described geometrically as a locus of points whose distance from a particular point $O$ equals the angle, in radians, to a reference ray $l$. Notice that the distance from $O$ to the point where the spiral intersects the reference line $l$ for the first time is $2\pi$. Evaluating the length of this curve in a coordinate-free setting would present a formidable challenge. With the use of coordinates, however, this task becomes a routine exercise in ordinary Calculus. Let us perform the required analysis in two alternative ambient coordinate systems: Cartesian coordinates $x,y$ and polar coordinates $r,\theta$.

First, introduce a Cartesian coordinate system $x,y$ that lines up with the reference line $l$ as in the following figure. The equations of the curve that describe the spiral in these coordinates read Since, in Cartesian coordinates, the metric tensor $Z_{ij}$ at all points corresponds to the identity matrix, we have The derivatives $x^{\prime}\left( \gamma\right)$ and $y^{\prime}\left( \gamma\right)$ are given by and therefore the integrand is given by Thus the arc length $s$ is given by the ordinary integral A routine evaluation of the integral yields the final answer

Now introduce the polar coordinate system $r,\theta$, where the origin of the coordinate system coincides with the point $O$ and the polar axis coincides with the ray $l$. In these coordinates the equations of the curve take on a particularly simple form, i.e. Recall that the metric tensor $Z_{ij}$ corresponds to the matrix Along a general curve given by the equations $r=r\left( \gamma\right)$ and $\theta=\theta\left( \gamma\right)$ spiral, $Z_{ij}\left( \gamma\right)$ corresponds to the matrix Thus, for a general curve in polar coordinates, the integrand is given by For our specific curve, $Z_{ij}\left( \gamma\right)$ corresponds to Since derivatives $r^{\prime}\left( \gamma\right)$ and $\theta^{\prime }\left( \gamma\right)$ are given by the integrand is given by Not surprisingly, we obtained the exact same expression for the integrand since it equals the length of the vector $R^{\prime}\left( \gamma\right)$ and does not depend on the choice of the ambient coordinates. Thus, the rest of the analysis can proceed as before.

This simple examples illustrates the great advantage of the coordinate space approach as it enables us to convert geometric expressions into arithmetic ones which can then be evaluated by the robust techniques of ordinary Calculus.

The equation demonstrates that, in combination with the equations of the curve, the metric tensor gives us the ability to calculate the length of any curve. In other words, we do not need any additional geometric details regarding either the curve or the coordinate system.

Crucially, this connection between the metric tensor and the lengths of curves also works in the opposite direction. Namely, an ability to measure the lengths of curves can be used to calculate the metric tensor. Thus, any quantity that can be calculated solely from the metric tensor (and its derivatives) can, in fact, be calculated from the very narrow ability to measure the lengths of curves. This insight will have important implications in the study of Riemannian spaces which we will undertake in Chapter 20.

Let us begin by calculating the entry $Z_{11}$ at a fixed point $A$ with coordinates $\left( A^{1},A^{2},A^{3}\right)$. Consider the coordinate line corresponding to the coordinate $Z^{1}$ illustrated below in a two-dimensional figure. The equations of this coordinate line read For a positive $\gamma$, denote the length of this curve from the point $A$ to the point with coordinate $\gamma$ by $L\left( \gamma\right)$. Importantly, the assumption that we are able to measure the lengths of curves by some means (perhaps by taking measurements with the help of some mechanical device) implies that the function $L\left( \gamma\right)$ is available -- meaning that we can evaluate its values and the values of its derivatives.

Since $Z^{1}\left( \gamma\right)$ is the only function in the equations of the curve that varies with $\gamma$, the only surviving term in the sum is Therefore, $L\left( \gamma\right)$ is given by the integral By the Fundamental Theorem of Calculus, the derivative $L^{\prime}\left( \gamma\right)$ equals the integrand at $t=\gamma$, i.e. Thus, the element $Z_{11}$ of the metric tensor at the point $A$ is given by the formula Thus, we have demonstrated a way to calculate $Z_{11}$ from the lengths of curves. The same approach can be used to evaluate other "diagonal" elements $Z_{22}$ and $Z_{33}$ of $Z_{ij}$.