The covariant basis vectors $\mathbf{Z}_{i}$, given by can be said to represent the rates of change of the position vector $\mathbf{R}$ with respect to each of the coordinates $Z^{i}$. As a reminder, the symbols $\mathbf{R}\left( Z\right)$ and $\mathbf{Z}_{i}\left( Z\right)$ represent the functions $\mathbf{R}\left( Z^{1},Z^{2} ,Z^{3}\right)$ and $\mathbf{Z}_{i}\left( Z^{1},Z^{2},Z^{3}\right)$, i.e. the position vector $\mathbf{R}$ and the covariant basis $\mathbf{Z}_{i}$ as functions of the coordinates $Z^{i}$.

In addition to the ubiquitous covariant basis $\mathbf{Z}_{i}$, we have, on a number of occasions, encountered the second-order system that represents the rates of change of each of the covariant basis vectors $\mathbf{Z}_{i}$ with respect to each of the coordinates $Z^{j}$. It is natural to use two subscripts to enumerate the elements of this system and we will therefore use the symbol $\mathbf{\Gamma}_{ij}$ to denote it, i.e. Note that the symbol $\mathbf{\Gamma}$ is bold to indicate that the elements of $\mathbf{\Gamma}_{ij}$ are vectors.

The system $\mathbf{\Gamma}_{ij}$ can also be thought of as the collection of the second-order derivatives of the position vector $\mathbf{R}$ with respect to the coordinates: In the future, we will consider higher-order derivatives. The Christoffel symbol, however, is about the second order.

Since partial derivatives commute, i.e. $\mathbf{\Gamma}_{ij}$ is symmetric, i.e. or

In affine coordinates, the vectors $\mathbf{\Gamma}_{ij}$ vanish since the covariant basis is the same at all points. In all other coordinate systems, the covariant basis varies from one point to another and we can therefore expect $\mathbf{\Gamma}_{ij}$ to be nonzero. In the $n$-dimensional space, the second-order system $\mathbf{\Gamma}_{ij}$ consists of $n^{2}$ vectors. However, because of the aforementioned symmetry, only $n\left( n+1\right) /2$ vectors can be distinct: $6$ in three dimensions, $3$ in two dimensions, and $1$ in one dimension.

Looking ahead, the Christoffel symbol $\Gamma_{ij}^{k}$ will be defined as the contravariant coefficients of $\mathbf{\Gamma}_{ij}$, i.e. Before we get there, however, let us visualize the vectors $\mathbf{\Gamma }_{ij}$ in polar coordinates. This will later enable us to establish the values of $\Gamma_{ij}^{k}$ in all the common coordinate systems.

Let us begin with the vector Recall that $\mathbf{Z}_{1}$ is the unit vector pointing in the radial direction away from the origin. Therefore, the vector $\mathbf{Z}_{1}$ remains constant as the coordinate $Z^{1}$, i.e. $r$, changes, as illustrated in the following figure: Therefore, we conclude that $\partial\mathbf{Z}_{1}/\partial Z^{1}$ vanishes, i.e.

We will now calculate both and confirm that the two vectors are equal. Let us begin with $\mathbf{\Gamma }_{12}$. As $Z^{2}$, i.e. $\theta$, increases, the outward unit vector $\mathbf{Z}_{1}$ rotates counterclockwise. We have studied this very vector-valued function on a number of occasions and know that its derivative is the unit vector counterclockwise orthogonal to $\mathbf{Z}_{1}$.

Let us now turn our attention to $\mathbf{\Gamma}_{21}$. Recall that $\mathbf{Z}_{2}$ is a vector of length $r$ that is counterclockwise orthogonal to $\mathbf{Z}_{1}$. As the coordinate $Z^{1}$, i.e. $r$, increases, the length of the $\mathbf{Z}_{2}$ increases and, in fact, equals $r$, while its direction remains constant. Thus, $\mathbf{\Gamma}_{21}$ is a unit vector in the same direction, i.e. counterclockwise orthogonal to $\mathbf{Z}_{1}$. As expected, the two calculations yielded the same result!

Notice that in describing the vector $\mathbf{\Gamma}_{21}=\mathbf{\Gamma }_{12}$ as the unit vector in the direction counterclockwise orthogonal to $\mathbf{Z}_{1}$, we used pure geometric terms. However, since we now have the covariant basis $\mathbf{Z}_{i}$ at our disposal, we are also able to describe the same vector analytically by giving the expressions for its contravariant components. Since $\mathbf{\Gamma}_{21}=\mathbf{\Gamma} _{12}$ points in the same direction as $\mathbf{Z}_{2}$ and has length $1$ (while the length of $\mathbf{Z}_{2}$ is $r$), we have

Finally, we turn our attention to As the coordinate $Z^{2}$, i.e. $\theta$, increases, $\mathbf{Z}_{2}$ rotates counterclockwise while maintaining its length of $r$. This is once again an evolution we are quite familiar with and can readily conclude that $\mathbf{\Gamma}_{22}$ is a vector of length $r$ that is counterclockwise orthogonal to $\mathbf{Z}_{2}$.

In order to give the analytical expressions for the contravariant components of $\mathbf{\Gamma}_{22}$, as we did for $\mathbf{\Gamma}_{12}=\mathbf{\Gamma }_{21}$, note that $\mathbf{\Gamma}_{22}$ points in the direction opposite to $\mathbf{Z}_{1}$. Since the length of $\mathbf{\Gamma}_{22}$ is $r$ (while the length of $\mathbf{Z}_{1}$ is $1$), we have

Let us summarize our findings by arranging the analytical expressions for the vectors $\mathbf{\Gamma}_{ij}$ in a $2\times2$ matrix:

In the preceding Section, we established the analytical expressions for the vectors in polar coordinates. More specifically, we determined the contravariant components of $\mathbf{\Gamma}_{ij}$, i.e. components with respect to the covariant basis $\mathbf{Z}_{i}$. When a single vector $\mathbf{U}$ is decomposed with respect to the covariant basis, the resulting contravariant components $U^{i}$ form a first-order system. When each vector in a first-order system $\mathbf{U}_{i}$ is decomposed with respect to the covariant basis, the resulting components $U_{i}^{j}$, where form a second-order system. When each vector in a second-order system $\mathbf{U}_{ij}$ is decomposed with respect to the covariant basis, the contravariant components $U_{ij}^{k}$, where form a third-order system.

The Christoffel symbol $\Gamma_{ij}^{k}$ is precisely that for the vectors $\mathbf{\Gamma}_{ij}$, i.e. the third-order system that consists of contravariant components of $\mathbf{\Gamma}_{ij}$. In other words, the Christoffel symbol $\Gamma_{ij}^{k}$ is defined by the identity Since $\mathbf{\Gamma}_{ij}$ is symmetric, i.e. the Christoffel symbol $\Gamma_{ij}^{k}$ is symmetric in its subscripts:

In the $n$-dimensional space, $\Gamma_{ij}^{k}$ has $n^{3}$ elements. However, because of the above symmetry, only $n^{2}\left( n+1\right) /2$ entries can be distinct, i.e. $18$ in three dimensions, $6$ in two dimensions, and $1$ in one dimension. In special coordinate systems, the Christoffel symbol $\Gamma_{ij}^{k}$ typically has only a few nonzero elements, so much so that it is usually more efficient to describe $\Gamma_{ij}^{k}$ by listing its few nonzero elements. For example, the Christoffel symbol in polar coordinates can be summarized as follows:

We have mentioned previously that, in order to avoid potential ambiguities, it is often necessary to agree on the order of indices. For the Christoffel symbol $\Gamma_{ij}^{k}$, two different conventions are found in the available texts. According to one, the superscript is treated as the first index and, according to the other, as the last. We will follow Hermann Weyl's Space Time Matter, and treat the superscript as the first index and the subscripts as second and third. We could denote the Christoffel symbol by the symbol $\Gamma_{\cdot ij}^{k}$ with a dot placeholder to make this choice explicit. However, we will prefer to rely on the stated convention rather than the placeholder.

In the absence of the convention and the dot placeholder, the result of lowering the superscript, denoted, say, by $\Gamma_{rst}$, would be ambiguous since we would not know which of the indices is the lowered superscript. Thanks to the convention, on the other hand, we know that it is the index $r$. Nevertheless, even when the ordering of the indices is clarified by a convention, it is customary to separate the subscripts by a comma, as in

Some sources refer to $\Gamma_{k,ij}$ as the Christoffel symbol of the first kind and $\Gamma_{ij}^{k}$ as the Christoffel symbol of the second kind. However, we see no need to make this distinction since we treat systems related by index juggling as different manifestations of the same objects.

The equation where uniquely defines the Christoffel symbol $\Gamma_{ij}^{k}$ as the contravariant components of $\mathbf{\Gamma}_{ij}$. However, the equation does not provide an explicit analytical expression for $\Gamma _{ij}^{k}$. Of course, such an expression is readily available with the help of the dot product. In Chapter 10, we established that the contravariant components $U^{k}$ of a vector $\mathbf{U}$ are given by the dot product with the contravariant basis element $\mathbf{Z}^{k}$ with $\mathbf{U}$: Applying this formula to the vector we find This identity will feature prominently in many future analyses.

An application of the product rule to the dot product on the right of the above identity leads to another insightful expression for the Christoffel symbol $\Gamma_{ij}^{k}$. Using the product rule in the form $fg^{\prime }=\left( fg\right) ^{\prime}-f^{\prime}g$, we find Since the dot product $\mathbf{Z}^{k}\cdot\mathbf{Z}_{i}$ equals $\delta _{i}^{k}$, i.e. a system whose elements do not change from one point to another, the first term on the right vanishes. Switching the order of the vectors in the second term, we conclude that $\Gamma_{ij}^{k}$ is also given by the dot product This identity shows that the covariant components of the vectors $\partial\mathbf{Z}^{k}/\partial Z^{j}$ are given by $-\Gamma_{ij}^{k}$, i.e. Thus, the elements of the Christoffel symbol $\Gamma_{ij}^{k}$ are simultaneously the contravariant components of and minus the covariant components of

The introduction of the Christoffel symbol provides us with another opportunity to discuss a number of positive aspects of the tensor notation.

The Christoffel symbol $\Gamma_{ij}^{k}$ is a system of order three -- the highest order of any system we have considered so far. You may feel like the indicial complexity may be beginning to get out of control and that Cartan's proverbial orgies of indices are beginning to obscure the simple geometric picture. You may also think that identities, such as $\partial\mathbf{Z}_{i}/\partial Z^{j}=\Gamma_{ij}^{k}\mathbf{Z}_{k}$ are difficult to remember. In this Section, we will demonstrate that the complexity remains completely under control and that identities such as $\partial\mathbf{Z}_{i}/\partial Z^{j}=\Gamma_{ij}^{k}\mathbf{Z}_{k}$ are not only easy to remember but, in fact, practically write themselves!

Consider the derivative where we have purposefully changed the names of the indices in order to give ourselves a fresh start. We will now demonstrate how to recreate the expression on the right featuring the Christoffel symbol simply by following the logic of the tensor notation.

You will certainly remember that this expression involves the Christoffel symbol -- but in what combination? When you inspect the expression $\partial\mathbf{Z}_{l}/\partial Z^{m}$, you will notice that it has two subscripts: one from $\mathbf{Z}_{l}$ and the other from $Z^{m}$ being in the "denominator". Thus, these two subscripts will be matched up with the two subscripts of the Christoffel symbol. If the Christoffel symbol was not symmetric in its superscripts, you would need to remember whether you need $\Gamma_{lm}$ or $\Gamma_{ml}$. Fortunately, it is symmetric so we do not need to worry about the order of the subscripts.

Thus, we have so far The Christoffel symbol is missing its superscript. We have no choice but to place it there; let us name it $n$ and pair it up with a covariant basis $\mathbf{Z}_{n}$, i.e.

The Christoffel symbol $\Gamma_{ij}^{k}$ captures the rate of change of the covariant basis $\mathbf{Z}_{i}$ with respect to the coordinates $Z^{j}$. Since the covariant metric tensor $Z_{ij}$ is built up from the covariant basis $\mathbf{Z}_{l}$, we should be able to calculate its rate of change in terms of the Christoffel symbol.

Recall the definition of the metric tensor $Z_{ij}$ Differentiating both sides, we find Each of the partial derivatives on the right side can be expressed in terms of the Christoffel symbol, i.e. Thus, Since $\mathbf{Z}_{l}\cdot\mathbf{Z}_{j}=Z_{jl}$ and $\mathbf{Z}_{i} \cdot\mathbf{Z}_{l}=Z_{il}$, This result has a slightly better form when the two terms are switched: Finally, absorbing the metric tensor into the Christoffel symbol, we find

The identity is not the least bit surprising. Since the metric tensor $Z_{ij}$ can be expressed in terms of the covariant basis $\mathbf{Z}_{i}$, we fully expected to find the expression for the derivatives of the former, i.e. $\partial Z_{ij}/\partial Z^{k}$, in terms of the derivatives of the latter, i.e. the Christoffel symbol. What may be at least somewhat surprising is that this relationship can be reversed: the Christoffel symbol can be expressed in terms of the derivatives of the metric tensor.{}

To remove the sense of surprise, let us think of the identity as representing equations in which the elements of the Christoffel symbol $\Gamma_{i,jk}$ are the unknowns, and let us do a simple count of equations and the unknowns. Since it has three free indices $i$, $j$, and $k$, the above equation represents $n^{3}$ equations in an $n$-dimensional space, which matches the number of the elements in a Christoffel symbol. The available symmetries reduce the system to $n^{2}\left( n+1\right) /2$ equations with $n^{2}\left( n+1\right) /2$ unknowns: $18$ in three dimensions, $6$ in two dimensions, and $1$ in one dimension. Thus, expressing the Christoffel symbol in terms of the derivatives of the metric tensor is simply a matter of solving a linear system of equations.

An effective approach to finding the solution relies on a creative manipulation of indices. From the identity obtain equivalent forms by cycling through the index names. For the first variation, rename $i\rightarrow j$, $j\rightarrow k$, $k\rightarrow i$. For the second variation, perform the very same renaming on the first variation. We end up with the following three equivalent identities:

The orgy of indices is on full display now! The Christoffel symbols in the above equations all have different combinations of indices. However, note that Sort all commuting indices in alphabetical order in every term of the system and notice that we are now able to isolate $\Gamma_{i,jk}$ by adding the first and the third equations and subtracting the second equation, i.e. Thus, Raising the subscript $i$ on both sides of the equation yields for the Christoffel symbol in the form $\Gamma_{jk}^{i}$, i.e. This result will prove crucial when we begin to develop Riemannian spaces and this identity will be adopted as the definition of the Christoffel symbol. Note that we, too, could have adopted this identity as the definition of the Christoffel symbol and, from it, derive all of the other identities involving the Christoffel symbol. This approach to defining the Christoffel symbol is known as intrinsic.

The elements of the Christoffel symbol $\Gamma_{ij}^{k}$ in all of the most common special coordinate systems can be derived by the approach outlined above in Section 12.2. In this Section, we document those values and leave their calculation as an exercise.

Since the covariant basis $\mathbf{Z}_{i}$ does not vary from one point to another in affine coordinates, the Christoffel symbol $Z_{ij}^{k}$ vanishes at all points:

The nonzero elements of the Christoffel symbol are the same as those for polar coordinates derived in Section 12.3:

In spherical coordinates, the nonzero elements of the Christoffel symbol are Deriving these expressions is left as an exercise.

Let us now illustrate how the Christoffel symbol enters analytical calculations. For this demonstration we will return to the analysis of a material particle moving along a trajectory given by the equation where the Christoffel symbol will appear in the expression for acceleration.

In Chapter 10, we learned that the contravariant components $V^{i}\left( t\right)$ of the velocity vector $\mathbf{V}\left( t\right)$ are given by Let us remind ourselves of the simple derivation that yielded this identity. If the position vector function $\mathbf{R}\left( t\right)$ describes the trajectory of the moving particle, then $\mathbf{R}\left( t\right)$ is given by the composition of the function $\mathbf{R}\left( Z\right)$, i.e. $\mathbf{R}\left( Z^{1},Z^{2},Z^{3}\right)$, with the equations of the motion $Z^{i}\left( t\right)$: Since differentiating both sides of the preceding equation with respect to $t$ yields or Since $\mathbf{V}=V^{i}\mathbf{Z}_{i}$, we conclude that the components $V^{i}\left( t\right)$ are given by

Let us now turn our attention to the components $A^{i}$ of the acceleration vector $\mathbf{A}$: In affine coordinates, where the covariant basis $\mathbf{Z}_{i}$ does not vary from one point to another, deriving the expression for $A^{i}\left( t\right)$ is an entirely straightforward exercise. Differentiating the identity with respect to $t$ yields In other words Since $\mathbf{A}=A^{i}\mathbf{Z}_{i}$, the components $A^{i}$ in affine coordinates are obtained by simply differentiating the components of velocity with respect to time This corresponds perfectly to our Cartesian intuition developed by our physics textbooks.

In curvilinear coordinates, however, we must account for the spatial variability of $\mathbf{Z}_{i}$. In order to acknowledge that variability we must write the equation for $\mathbf{V}\left( t\right)$ as where the function $\mathbf{Z}_{i}\left( t\right)$ represents the covariant basis vectors $\mathbf{Z}_{i}$ along the particle's trajectory. Differentiating both sides of this identity with respect to $t$ yields Dropping the functional dependence of the terms, we write Due to the presence of the second term, the identity $A^{i}=dV^{i}/dt$ is invalid in curvilinear coordinates.

In order to analyze the system $d\mathbf{Z}_{i}\left( t\right) /dt$, recall that the covariant basis $\mathbf{Z}_{i}$ is defined in the broader ambient space. Thus, $\mathbf{Z}_{i}\left( t\right)$ can be constructed by composing the function $\mathbf{Z}_{i}\left( Z\right)$, i.e. $\mathbf{Z} _{i}\left( Z^{1},Z^{2},Z^{3}\right)$, with the equations of the motion $Z^{i}\left( t\right)$: An application of the chain rule yields The last term, $dZ^{j}/dt$ is, of course, the velocity component $V^{j}$, so More interesting, however, is the term $\partial\mathbf{Z}_{i}\left( Z\right) /\partial Z^{j}$ which we recognize to be precisely $\Gamma_{ij} ^{k}\mathbf{Z}_{k}$. Thus, we arrive at the identity Substituting this result into the identity we arrive at the following expression for acceleration:

Since our goal is to determine the acceleration component $A^{i}$, we must modify the expression on the right so it appears in the form of a contraction with the covariant basis $\mathbf{Z}_{i}$. Thus, the covariant basis in the second term must have the subscript $i$. In order to achieve this, cycle the names of the indices $i\rightarrow j\rightarrow k\rightarrow i$ to obtain and We can now factor out $\mathbf{Z}_{i}$: From this form, we conclude that the component $A^{i}$ is given by

This formula is an important milestone as it succeeds in expressing the component $A^{i}$ of the acceleration in terms of objects that are available in the coordinate space. After all, recall that the identity tells us that the Christoffel symbol is available in the coordinate space once the metric tensor field has been calculated. Therefore, the equation demonstrates that the analysis of a moving material particle can be conducted -- at least up to the second derivative -- strictly in the coordinate space without a need for continual reference to the covariant basis $\mathbf{Z}_{i}$. The term $\Gamma_{jk}^{i}V^{j}V^{k}$ may be thought of as the correction for the spatial variability of the covariant basis.

In Chapter 10, we calculated the components of the velocity vector of a material particle moving around a circle of radius $R$ with angular velocity $\omega$ in polar coordinates $\left( r,\theta\right)$. We will now repeat that calculation and then take it a step further by calculating the components of the acceleration.

In polar coordinates $Z^{1}=r$ and $Z^{2}=\theta$, the equations of the motion read Since the components $V^{i}$ of velocity correspond to As we have just derived, the components $A^{i}$ of acceleration are given by Since the velocity components $V^{i}\left( t\right)$ are constants, the only surviving term is $\Gamma_{jk}^{i}V^{j}V^{k}$: Let us fully unpack this identity as follows: Since $V^{1}=0$ and the only nonvanishing elements of the Christoffel symbol are the only surviving term is $\Gamma_{22}^{1}V^{2}V^{2}$ in $A^{1}$ and we find: which completes our calculation of the components $A^{i}$. We must emphasize that the entire analysis was performed strictly in the coordinate space.

Stepping back into the geometric space, the acceleration vector $\mathbf{A}$ can be obtained by contracting $A^{i}$ with the covariant basis $\mathbf{Z} _{i}$. The result is the vector that has a length of $\omega^{2}R$ and points directly towards the center of the circle. This is a familiar result from elementary Physics textbooks.

Let us take our analysis a step further and determine the components $J^{i}$ of the jolt $\mathbf{J}\left( t\right)$, defined as the derivative of acceleration, i.e. It is left as an exercise to show that the components $J^{i}$ are given by the identity

Thanks to the symmetry $\Gamma_{jk}^{i}=\Gamma_{kj}^{i}$ of the Christoffel symbol, the identity can be rewritten in the form This form is particularly interesting since it invites the introduction of a new differential operator which we will call the $\delta/\delta t$-derivative, pronounced the delta-by-delta-$t$ derivative.

The $\delta/\delta t$-derivative applies to first-order time-dependent variants with superscripts defined along the trajectory of the particle. For such a variant $U^{i}\left( t\right)$ and the trajectory given by the equations of the motion define where, as a reminder, $V^{i}$ are the componets of velocity along the trajectory given by the equation Note the novel structural aspect of this operator, where applying the $\delta/\delta t$-derivative to any element of the system $U^{i}$ involves all other elements of the system since the term $V^{j}\Gamma_{jk}^{i}U^{k}$ engages all of them in a contraction.

With the help of the $\delta/\delta t$-derivative, we can rewrite the equations and In the more compact forms and In general, for a vector $\mathbf{U}\left( t\right)$ with components $U^{i}\left( t\right)$, i.e. the components of the derivative $\mathbf{U}^{\prime}\left( t\right)$ are given by $\delta U^{i}\left( t\right) /\delta t$, i.e.

The $\delta/\delta t$-derivative can be extended to first-order variants with covariant indices and, in fact, to variants of arbitrary order and arbitrary indicial signatures. The resulting full-fledged $\delta/\delta t$-derivative has an array of remarkable properties that make it an operator of great utility. However, we will not discuss the development of this operator here for fear that that would steal the thunder of the covariant derivative which we will introduce right after we establish the concept of a tensor in Chapter 14.

If a vector $\mathbf{U}\left( t\right)$ is constant along the trajectory of a particle, i.e. then its components satisfy or, in expanded form, This identity gives us a coordinate-space criterion for determining whether two vectors $\mathbf{A}$ and $\mathbf{B}$ at different points in space are equal. In order to apply the criterion, one needs to connect the two points by a smooth trajectory and then interpret the above identity as a system of ordinary differential equations. If $t_{1}$ corresponds to the point where $\mathbf{A}$ is found and $t_{2}$ corresponds to the point where $\mathbf{B}$ is found, then we can solve this system of equations from $t_{1}$ to $t_{2}$ with the components of $\mathbf{A}$ as the initial condition. If the resulting components at $t_{2}$ coincide with the components of $\mathbf{B}$, then $\mathbf{A}$ and $\mathbf{B}$ are equal.

Exercise 12.1Show that the sum is symmetric in $i$ and $j$.

Exercise 12.2Derive the expression for the derivative of the contravariant metric tensor in terms of the Christoffel symbol.

Exercise 12.3Derive the values of the Christoffel symbol in each of the special coordinate systems described in Section 12.6.

Exercise 12.4Demonstrate that the components $J^{i}$ of jolt are given by the equation

Exercise 12.5Show that

Exercise 12.6Confirm that the Riemann-Christoffel identity which will be demonstrated in Chapter 15, holds in polar coordinates.