Consider a two-dimensional surface embedded in a three-dimensional space and referred to surface coordinates \(S^{\alpha}\). Furthermore, introduce an arbitrary coordinate system \(Z^{i}\) in the ambient space.

With the help of the two coordinate systems, the surface can be described parametrically, i.e. by specifying the values of the ambient coordinates as functions of surface coordinates. For a two-dimensional surface in a three-dimensional space, a parametric specification requires three functions of two variables. These functions are called the equations of the surface and are denoted by the same symbols \(Z^{i}\) as the ambient coordinates, i.e. As usual, these equations can be captured by a single indicial identity, which, by collapsing the function arguments, can be written in an even more compact form Not surprisingly, the one operation that can be applied to the equation of the surface, i.e. differentiating with respect to \(S^{\alpha}\) resulting in the system will give rise to an important object that will occupy much of our attention in the rest of our narrative.

Before proceeding with our analysis, however, let us consider a few classical examples of two-dimensional surfaces and one-dimensional planar curves.

Refer a sphere of radius \(R\) to the surface spherical coordinates \(\theta,\varphi\) described in the previous Chapter. If the ambient space is referred to Cartesian coordinates \(x,y,z\), aligned with the surface coordinates in the usual way, then the equations of the surface read

When the ambient space is referred to spherical coordinates \(r,\theta _{1},\varphi_{1}\), the equations of the surface appear in a particularly simple form The simplicity of these expressions indicates that spherical coordinates in the ambient space are a natural choice for describing the surface of a sphere. This simple example also illustrates the profound dependence of the equation of a surface on the choice of coordinates.

Refer a cylinder of radius \(R\) to coordinates \(\theta,z\) illustrated in the following figure. If the ambient space is referred to Cartesian coordinates \(x,y,z_{1}\), the equations for the cylinder read If the ambient space is referred to cylindrical coordinates \(r,\theta _{1},z_{1}\), the equations become even simpler, i.e. Interestingly, note that the equations of a sphere in spherical coordinates are the same as the equations of a cylinder in cylindrical coordinates.

A surface of revolution is formed by rotating a planar curve around a straight line (within the same plane) known as the axis of rotation. The curve that is being rotated may be referred to as the profile of the shape. Notice that both a sphere and a cylinder, as well as the upcoming torus, are examples of shapes of revolution.

It is natural to choose the angle of rotation \(\theta\) to be one of the two surface coordinates. The other coordinate can be any parameter \(\gamma\). If the ambient space is referred to Cartesian coordinates, then the shape of revolution about the \(z\)-axis is given by where \(H\left( \gamma\right) \) is the location of the point along the \(z\)-axis and \(G\left( \gamma\right) \) is the distance away from the \(z\)-axis. If the ambient space is instead referred to cylindrical coordinates \(r,\theta_{1},z_{1}\), we once again obtain equations of utmost simplicity, i.e. where \(H\left( \gamma\right) \) is the location of the point along the \(z\)-axis and \(G\left( \gamma\right) \) is the distance away from the \(z\)-axis. Finally, note that it is very common to choose in which case \(\gamma\) corresponds exactly to the location of the point along the axis of rotation.

A torus is the shape of revolution of a circle of radius \(r\) whose center is the distance \(R\) from from the rotation axis. This simple profile corresponds to the functions where we decided to use the letter \(\varphi\) instead of \(\gamma\) because of its interpretation as an angle. Thus, the equations of the torus read where both \(\theta\) and \(\varphi\) vary from \(0\) to \(2\pi\).

A planar curve is a one-dimensional "surface" embedded in a two-dimensional Euclidean plane which we will typically refer to either Cartesian or polar coordinates.

The most common type of curve that we encounter in our introductory Calculus courses is the graph of a function \(f\left( x\right) \). For example, the curve corresponding to the function is shown in the following figure. Every curve that represents the graph of a function \(f\left( x\right) \) can also be represented parametrically by the equations

Of course, not every planar curve corresponds to the graph of a function. A quintessential curve that is not the graph of a function is a circle. The following figure illustrates a circle of radius \(R\) centered at a point with coordinates \(\left( x_{0},y_{0}\right) \). It is described by the equations where, once again, we used \(\theta\) instead of \(\gamma\) because of its interpretation as an angle.

For a more intricate example, consider the equations that describe the curve in the following figure.

Interestingly, many curves that can be represented by a graph of a function are still more easily represented parametrically. One example is a cycloid, which represents the trajectory of a point on a wheel rolling without slippage along a straight line. A cycloid that corresponds to a wheel of radius \(R\) is specified by the equations

A general planar curve in polar coordinates \(r,\theta\) is given by As we described in Chapter TBD of Introduction to Tensor Calculus, \(r\left( \gamma\right) \) can assume negative values and \(\theta\left( \gamma\right) \) can assume any value whatsoever. When \(r\left( \gamma\right) \) is negative, the point with coordinates \(\left( r,\theta\right) \) is plotted as \(\left( -r,\theta+\pi\right) \). To repeat an example from that Chapter, the curve given by the equation is shown in the following figure. For another example, the curve given by the equations has the following shape:

Polar coordinates have their own version of a graph, i.e. curves that may be described by specifying \(r\) as a function of \(\theta\) which is equivalent to the parametric equations For example, the following spiral corresponds to where \(\theta\) varies form \(-4\pi\) to \(4\pi\). It is also possible, but less common, to specify \(\theta\) as a function of \(r\).

This wraps up the description of the most common surfaces and curves that will be used throughout the rest of our narrative as illustration of the relevant concepts.

We now turn our attention to the shift tensor \(Z_{\alpha}^{i}\) -- a central object in the analysis of embedded surfaces that represents the tangent space. Furthermore, the shift tensor helps relate corresponding surface and ambient quantities. In fact, we will approach the shift tensor by exploring the relationship between the surface basis \(\mathbf{S}_{\alpha}\) and the ambient basis \(\mathbf{Z}_{i}\).

Recall that the ambient covariant basis \(\mathbf{Z}_{i}\) is given by the equation where \(\mathbf{R}\left( Z\right) \) is the position vector as a function of the ambient coordinates \(Z^{i}\). Similarly, the surface covariant basis \(\mathbf{S}_{\alpha}\) is given by the equation where \(\mathbf{R}\left( S\right) \) is the surface restriction of the position vector \(\mathbf{R}\) as a function of the surface coordinates \(S^{\alpha}\).

As usual, in order to relate the derivatives we must first relate the functions To find the correct relationship, imagine the position vector \(\mathbf{R}\) corresponding to a specific surface point \(P\) with surface coordinates \(S^{\alpha}\). The vector \(\mathbf{R}\) can be obtained in two alternative ways. The first is to evaluate the function \(\mathbf{R}\left( S\right) \) at the coordinates \(S^{\alpha}\). The second is to determine the ambient coordinates \(Z^{i}\) of the point \(P\) by evaluating the equations of the surface \(Z^{i}\left( S\right) \) at \(S^{\alpha}\) and then to evaluate the function \(\mathbf{R}\left( Z\right) \) at the resulting values of the coordinates \(Z^{i}\). This line of reasoning yields the identity

Differentiating this identity by the chain rule with respect to \(S^{\alpha}\), we have where we recognize two of the three elements, i.e. The third element, is the shift tensor! It is denoted by the symbol \(Z_{\alpha}^{i}\), i.e. By convention, the Latin index is considered first and the Greek index is considered second. Therefore, we do not need a placeholder to help us track the order of the indices. Meanwhile, the precise meaning of the word tensor, when applied to an object featuring both ambient and surface indices, will be described in the next Section.

With the help of the symbol \(Z_{\alpha}^{i}\), the relationship between the surface and the ambient bases reads For each \(\alpha\), this identity represents the vector \(\mathbf{S}_{\alpha}\) as a linear combination of the ambient basis vectors \(\mathbf{Z}_{i}\). Thus, \(Z_{\alpha}^{i}\) are the components of the surface basis element \(\mathbf{S}_{\alpha}\) with respect to the ambient basis \(\mathbf{Z}_{i}\). It is for this reason that the shift tensor is said to represent the tangent space.

Recall that the ambient components \(U^{i}\) of a vector \(\mathbf{U}\) are given by the dot product Therefore, In other words, the elements of the shift tensor are the pairwise dot products of the ambient and surface bases. By index juggling, we easily obtain the equivalent identities

Although we will save the bulk of our discussion of specific surfaces until the next Chapter, let us give the shift tensor for a sphere of radius \(R\) in both Cartesian and spherical ambient coordinates. As we stated earlier in this Chapter, for the ambient space is referred to Cartesian coordinates \(x,y,z\), the equations of the sphere read The shift tensor \(Z_{\alpha}^{i}\) then corresponds to the \(3\times2\) matrix Meanwhile, for the ambient space referred to spherical coordinates \(r,\theta_{1},\varphi_{1}\), the equations of the surface read the shift tensor \(Z_{\alpha}^{i}\) corresponds to the matrix

Once again, the equation tells us that the elements of the shift tensor are the components of the surface covariant basis \(\mathbf{S}_{\alpha}\) with respect to the ambient covariant basis \(\mathbf{Z}_{i}\). Thus, the columns of each \(3\times2\) matrix representing the shift tensor are the ambient components of the corresponding surface covariant basis vectors. For the surface coordinates \(\theta,\varphi\), the covariant basis \(\mathbf{S}_{\alpha}\) on the sphere is shown in the following figure and you should confirm that the columns of the matrices above correctly represent these vectors. In particular, when spherical coordinates \(r,\theta_{1},\varphi_{1}\) are chosen in the ambient space, the surface covariant basis \(\mathbf{S}_{\alpha}\) is represented by the last two vectors of the ambient basis \(\mathbf{Z}_{i}\), i.e. which corroborates the particularly simple form of the shift tensor.

The shift tensor \(Z_{\alpha}^{i}\) is our first example of a variant featuring both ambient and surface indices. Therefore, we must revisit the concept of a tensor as \(Z_{\alpha}^{i}\) forces us to contend with two simultaneous coordinate changes: one in the ambient space and one on the surface. This is the task to which we will now turn our attention.

From the identity it is clear that the elements of the shift tensor \(Z_{\alpha}^{i}\) change with both the ambient and surface coordinates. This invites us to expand the concept of a tensor to include transformations under simultaneous changes of ambient and surface coordinates. Before we give the general definition, let us examine how the shift tensor behaves in those circumstances.

Let \(Z^{i}\) and \(Z^{i^{\prime}}\) be the unprimed and primed ambient coordinates related by the functions At the same time, let \(S^{\alpha}\) and \(S^{\alpha^{\prime}}\) be the unprimed and primed surface coordinates related by the functions Also recall the definitions of the associated Jacobians, i.e. and

In the primed coordinates, the shift tensor \(Z_{\alpha^{\prime}}^{i^{\prime}}\) is given by the equation The ambient contravariant basis \(\mathbf{Z}^{i}\) depends only on the choice of the ambient coordinates and, being a tensor, transforms according to the rule Similarly, the surface covariant basis \(\mathbf{S}_{\alpha}\) depends only on the choice of the surface coordinates and, being a tensor, transforms according to the rule Therefore, for the shift tensor \(Z_{\alpha^{\prime}}^{i^{\prime}}\), we have In summary, In words, the shift tensor \(Z_{\alpha}^{i}\) exhibits the properties of a tensor with respect to changes of both ambient and surface coordinates.

Let us use this behavior as a blueprint for an expanded definition of a tensor. Consider a variant \(T_{j\beta}^{i\alpha}\) with a representative collection of ambient and surface indices. Then \(T_{j\beta}^{i\alpha}\) is an (absolute) tensor if its elements \(T_{j^{\prime}\beta^{\prime} }^{i^{\prime}\alpha^{\prime}}\) in the primed coordinate systems are related to \(T_{j\beta}^{i\alpha}\) by the equation According to this definition, the shift tensor is indeed a tensor. Furthermore, all previously defined surface and ambient variants that are tensors in the original sense remain tensors in the sense of the new, more general definition. Of course, for tensors in the ambient space, we must limit our attention to their surface restrictions in order for the new definition to apply.

Finally, it is left as an exercise to demonstrate all of the common properties of tensors including the sum, product, and contraction properties, as well as the quotient theorem.

Consider a vector \(\mathbf{U}\) in the tangent plane at a point \(P\) on the surface. Like all vectors in the tangent plane, \(\mathbf{U}\) can be decomposed with respect to the surface covariant basis \(\mathbf{S}_{\alpha}\), leading to the contravariant components \(U^{\alpha}\), i.e. However, like any vector whatsoever, whether it lies in the tangent plane or not, \(\mathbf{U}\) can also be decomposed with respect to the ambient basis \(\mathbf{Z}_{i}\) resulting in the ambient components \(U^{i}\), i.e. The question is, how are \(U^{i}\) and \(U^{\alpha}\) related?

In order to answer this question, substitute the identity into the equation which yields This identity tells us that the combination \(U^{\alpha}Z_{\alpha}^{i}\) represents the ambient coordinates of \(\mathbf{U}\). In other words,

In this identity, the shift tensor plays a role that can be more accurately described as "upshifting": translating the surface components of a tangent vector into its ambient components. It is left as an exercise, which should be attempted after Section 3.8, to show that for a tangent vector \(\mathbf{U}\) with ambient components \(U^{i}\), the surface components \(U^{\alpha}\) can be recovered by contraction with the shift tensor on the ambient index, i.e.

A curve embedded in a surface can be specified by the dependence of the surface coordinates \(S^{\alpha}\) of the points on the curve on a parameter \(\gamma\), i.e. Our present goal is to express the length of the curve by an integral in terms of the functions \(S^{\alpha}\left( \gamma\right) \).

Recall from Chapter TBD of the Introduction to Tensor Calculus, that the length \(s\) of a segment of a curve given by the vector equation is expressed by the integral Since the vector \(\mathbf{R}^{\prime}\left( \gamma\right) \) is tangential to the curve it is also tangential to the surface in which the curve is embedded. Therefore, it can be expressed with respect to both ambient and surface covariant bases. As we learned in Chapter TBD of Introduction to Tensor Calculus, the contravariant components of \(\mathbf{R}^{\prime}\left( \gamma\right) \) are where are the ambient equations of the curve. This conclusion follows from differentiating the identity with respect to \(\gamma\), i.e. With the help of the components of \(\mathbf{R}^{\prime}\left( \gamma\right) \), the geometric integral can be converted into the arithmetic integral as we established in Chapter TBD of Introduction to Tensor Calculus.

Similarly, the surface components of \(\mathbf{R}^{\prime}\left( \gamma\right) \) are as can be shown by differentiating the identity with respect to \(\gamma\). We have Therefore, the length of the curve is also given by the formula Thus, as we might have expected, the length of a curve on a surface can be calculated by referring to its equation within the surface and the surface covariant metric tensor, while ignoring how either object is embedded in the larger ambient space.

The connection between the surface and the ambient basis, i.e. immediately leads to the connection between the surface and the ambient metric tensors. Indeed, since we have In summary, The right side of this identity may be described as having each of the subscripts of \(Z_{ij}\) operated on by the shift tensor.

The pivotal identity can be written in a number of other useful forms. Since the first alternative form reads If you prefer to see the index \(i\) in important identities, you may rewrite the same identity as

Finally, by raising the index \(\alpha\), we arrive at which is perhaps the most elegant form of the same identity. This identity is usually written with the two sides reversed, i.e. Interestingly, the identity that relates the surface and the ambient metric tensors \(S_{\alpha\beta}\) and \(Z_{ij}\) features neither object explicitly.

We ought to remark on the identity from the Linear Algebra point of view. Both \(Z_{i}^{\alpha}\) and \(Z_{\beta }^{i}\) correspond to \(3\times2\) matrices. If \(Z_{i}^{\alpha}\) corresponds to a matrix \(A\) and \(Z_{\beta}^{i}\) corresponds to the matrix \(B\), then the contraction \(Z_{i}^{\alpha}Z_{\beta}^{i}\) corresponds to the product \(A^{T}B\). Since \(\delta_{\beta}^{\alpha}\) corresponds to the \(2\times2\) identity matrix , the matrix equation \(A^{T}B=I\) has the following "shape": Since each of the matrices is rank \(2\), it is quite feasible that the product results in the \(2\times2\) identity matrix.

On the other hand, if the same two matrices were multiplied in the opposite order, i.e. \(BA^{T}\), the shape of the resulting identity would be The rank of the product still cannot be greater than \(2\) and therefore the resulting matrix cannot be the \(3\times3\) identity matrix. Since \(BA^{T}\) corresponds to the contraction \(Z_{\alpha}^{i}Z_{j}^{\alpha}\), we conclude that the latter cannot equal \(\delta_{j}^{i}\), i.e. However, we will soon discover that \(Z_{\alpha}^{i}Z_{j}^{\alpha}\) satisfies the analogous identity which involves the components \(N^{i}\) of the unit normal \(\mathbf{N}\) studied below.

Having related the ambient and the surface metric tensors, let us now relate the surface Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) to its ambient counterpart \(\Gamma_{jk}^{i}\). Our analysis will yield an attractive identity that offers a practical way of calculating the surface Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) in many situations.

Recall that the surface Christoffel symbol is given by while the ambient Christoffel symbol is given by The key to connecting these identities is of course the relationship which expresses the surface basis \(\mathbf{S}_{\alpha}\) in terms of the ambient basis \(\mathbf{Z}_{i}\).

Substituting \(\mathbf{S}^{\gamma}=\mathbf{Z}^{k}Z_{k}^{\gamma}\) and \(\mathbf{S}_{\alpha}=\mathbf{Z}_{i}Z_{\alpha}^{i}\) in the equation we find By the product rule, It is left as an exercise to show that and therefore Meanwhile, Combining the above equations, we find The first term in this attractive identity can be described as the shift of the ambient Christoffel symbol onto the surface.

Finally, note that when the ambient space is referred to affine coordinates where \(\Gamma_{jk}^{i}=0\), the above identity simplifies to the equation which is often used for practical calculations of the surface Christoffel symbol.

Denote the components of the unit normal \(\mathbf{N}\) by \(N^{i}\), i.e. or Later in this Chapter, we will demonstrate that, up to sign, the components \(N^{i}\) are given by the explicit formula For the time being, however, we would like see how much we can learn about \(N^{i}\) without this formula.

Recall that at each point on the surface, there are two unit normals that point in the opposite directions. Correspondingly, the components \(N^{i}\) can have two opposite sets of values depending on the choice of normal. In any given situation, the choice of normal is made a priori, either arbitrarily or according to some geometric rationale. Once the choice is made, the normal \(\mathbf{N}\) may be considered unique and therefore an invariant. The components \(N^{i}\) are then uniquely determined as well and represent a tensor with respect to changes in the ambient coordinates.

Let us now determine the identities for the components \(N^{i}\) that correspond to the unit-length and orthogonality conditions and Of course, the component form of the unit-length condition reads In order to convert the orthogonality condition to component form, note that since \(\mathbf{N}=N^{i}\mathbf{Z}_{i}\), we have Since we have Thus, in component form, the orthogonality condition reads It is left as an exercise to argue that, collectively, the equations determine \(N^{i}\) up to sign.

Earlier in this Chapter, we considered a tangent vector \(\mathbf{U}\) with surface components \(U^{\alpha}\) and showed that its ambient components \(U^{i}\) are given by the equation This time, consider a vector \(\mathbf{U}\) with components \(U^{i}\), i.e. \(\mathbf{U}=U^{i}\mathbf{Z}_{i}\), that is not tangent to the surface. The combination still makes sense from the tensor notation point of view. As our experience shows, feasible tensor combinations are almost always worthwhile. What, then, is the geometric interpretation of the vector with surface components In this Section, we will demonstrate that it is the orthogonal projection of \(\mathbf{U}\) onto the tangent plane.

In Chapter 2, we showed that the surface components \(V^{\alpha}\) of the orthogonal projection \(\mathbf{V}\) of \(\mathbf{U}\) onto the tangent plane are given by the dot product Substituting \(\mathbf{U}=U^{i}\mathbf{Z}_{i}\), we find Since we find Thus, the combination \(U^{i}Z_{i}^{\alpha}\) indeed represents the components \(V^{\alpha}\) of the orthogonal projection of \(\mathbf{U}\) onto the tangent plane. As a result, the shift tensor \(Z_{i}^{\alpha}\) may be thought of geometrically as the orthogonal projection operator onto the tangent plane, as it relates the ambient components of a vector \(\mathbf{U}\) to the surface components of its orthogonal projection \(\mathbf{V}\).

Let us now relate the ambient components of \(\mathbf{U}\) to the ambient components of \(\mathbf{V}\). At the end of the previous Section, we mentioned that the ambient coordinates \(V^{i}\) of a tangent vector \(\mathbf{V}\) are obtained by contracting its surface components \(V^{\alpha}\) with the shift tensor, i.e. Since, as we have just established, \(V^{\alpha}=U^{j}Z_{j}^{\alpha}\), we have

Alternatively, we could have arrived at this relationship by starting with the identity derived in Chapter 2. Then, upon substituting the expansions we find Since we have Equating the components of the vectors on either side, we once again arrive at the relationship

Thus, the combination may be thought of as the operator of orthogonal projection onto the tangent plane. It should be noted that compactness of \(Z_{\alpha}^{i}Z_{j}^{\alpha}\) is quite remarkable. Recall from Linear Algebra that the problem of finding the orthogonal projection onto a linear subspace is solved by the technique of least squares. If \(A\) is the rectangular matrix, then the operator of orthogonal projection onto the column space of \(A\) is given by the combination The combination \(Z_{\alpha}^{i}Z_{j}^{\alpha}\) includes all of the same elements -- however, the complexity of the operations is absorbed by the indicial notation. In particular, the matrix \(\left( A^{T}A\right) ^{-1}\) corresponds to the contravariant metric tensor \(S^{\alpha\beta}\) which makes its presence felt via the placement of the surface index in the shift tensor \(Z_{j}^{\alpha}\) as a superscript.

In Chapter 2, we showed that the orthogonal projection \(\mathbf{W}\) of a vector \(\mathbf{U}\) onto the normal direction \(\mathbf{N}\), i.e. away from the tangent plane, is given by Let us convert this identity to component form.

The dot product \(\mathbf{U}\cdot\mathbf{N}\) is given by \(U^{j}N_{j}\). If \(W^{i}\) are the ambient components of \(\mathbf{W}\), then the above equation reads Equating the components, we find which is the relationship that we set out to find. Whereas the combination represented the operator of orthogonal projection onto the tangent plane, the combination represents the operator of orthogonal projection away from the tangent plane. A good question to consider, then, is what does the sum represent? This question will be answered in the following Section.

Recall the identity from Chapter 2 which states that a vector \(\mathbf{U}\) at a point \(P\ \)on a surface is the sum of its orthogonal projection \(\mathbf{V}=\left( \mathbf{S}^{\alpha}\cdot\mathbf{U}\right) \mathbf{S}_{\alpha}\) onto the tangent plane and its orthogonal projection \(\mathbf{W}=\left( \mathbf{U} \cdot\mathbf{N}\right) \mathbf{N}\) away from the tangent plane. In component form, the above identity reads Substituting the combination \(\delta_{j}^{i}U^{j}\) for \(U^{i}\) on the left, we find Since the above identity holds for any \(U^{j}\), we arrive at the relationship known as the projection formula. It is usually written with the Kronecker delta on the right, i.e. Other useful forms of the projection formula read and Earlier in this Chapter, we remarked that the combination \(Z_{\alpha}^{i} Z_{j}^{\alpha}\) cannot equal \(\delta_{j}^{i}\). The projection formula refines that insight and identifies precisely the relationship of \(Z_{\alpha}^{i} Z_{j}^{\alpha}\) to \(\delta_{j}^{i}\).

As our experience so far has shown, we can work quite effectively with the components \(N^{i}\) of the unit normal even though they have only been specified implicitly by the orthogonality condition and the normalization condition Of course, these conditions actually give us a practical way of calculating \(N^{i}\). It is left as an exercise to show that if \(A\) is the \(3\times2\) matrix that represents the shift tensor \(Z_{i}^{\alpha}\), then \(N^{i}\) corresponds to a unit vector from the one-dimensional null space of \(A^{T}\). Thus, an explicit expression for \(N^{i}\) is not needed. However, it is an intriguing challenge to find such an expression and we will do so in this Section.

The inspiration for the expression comes from the concept of the cross product. Note that the elements \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\) of the surface covariant basis represent two linearly independent vectors in the tangent plane. Therefore, their cross product is orthogonal to the plane and is therefore collinear with \(\mathbf{N}\). Thus, all that is left to do is to find a vector of unit length that is a scalar multiple of \(\mathbf{S}_{1}\times\mathbf{S}_{2}\).

The ambient components of \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\) are found in the elements of the shift tensor \(Z_{\alpha}^{i}\). Specifically, the components of \(\mathbf{S}_{1}\) are \(Z_{1}^{i}\) and those of \(\mathbf{S}_{2}\) are \(Z_{2}^{j}\). Thus, as was described in Chapter TBD of Introduction to Tensor Calculus, the covariant components \(U_{i}\) of \(\mathbf{U} =\mathbf{S}_{1}\times\mathbf{S}_{2}\) are given by the equation It is left as a straightforward exercise to confirm that \(U_{i}\) satisfies the orthogonality condition However, in order to be geometrically meaningful, \(U^{i}\) must be a tensor. Meanwhile, the expression \(\varepsilon_{ijk}Z_{1}^{j}Z_{2}^{k}\) uses literal subscripts \(1\) and \(2\), which are disallowed by the rules of the tensor notation and thus \(U^{i}\), as defined, is not a tensor. Fortunately, this difficulty is easy to overcome.

It is left as an exercise that the expression \(\varepsilon_{ijk}Z_{1}^{j} Z_{2}^{k}\) is skew-symmetric with respect to exchanging the values of the literal indices, i.e. Therefore, the difference is twice \(U_{i}\) and therefore also represents a vector that is collinear with \(\mathbf{N}\). With the help of the permutation symbol \(e^{\alpha\beta}\), this difference can be written in the form Next, replace the permutation symbol \(e^{\alpha\beta}\) with the Levi-Civita symbol \(\varepsilon^{\alpha\beta}=e^{\alpha\beta}/\sqrt{S}\) in order to arrive at the tensor expression

Now, the only remaining questions are those of length and sign. It is left as an exercise to show that the length of the vector with components above is \(2\). Thus, scaling the expression above by \(1/2\) we arrive at Of course, depending on our a priori choice of \(\mathbf{N}\), the above formula may deliver the components of \(-\mathbf{N}\) rather than \(\mathbf{N}\). Finally, we must note that the variant on the right side of the above formula is a tensor only under orientation-preserving transformations of both ambient and surface coordinates.

Exercise 3.1Show that the equations of the surface \(Z^{i}\left( \theta,\varphi\right) \) for a sphere of radius \(R\) in cylindrical ambient coordinates \(r,\theta_{1},z\) read

Exercise 3.2Show that the equations of the surface \(Z^{i}\left( \theta,z\right) \) for a cylinder of radius \(R\) in spherical ambient coordinates \(r,\theta_{1},\varphi\) read

Exercise 3.3In the text, we showed that the shift tensor \(Z_{\alpha}^{i}\) is a tensor with the help of the identity However, it is important to be able to demonstrate its tensor property on the basis of its definition Do so in three steps. First, consider changes only of ambient coordinates. First justify and then use the identity to show that Second, consider changes of only surface coordinates and use the identity to show that Finally, consider simultaneous changes of ambient and surface coordinates. Use the identity to show that

Exercise 3.4Suppose that the elements of \(Z_{i\alpha}\) are organized into a \(3\times2\) matrix \(A\). Show that the \(3\times1\) matrix consisting of the elements of \(N^{i}\) belongs to the null space of \(A^{T}\). This observation gives a practical way of calculating the components \(N^{i}\).

Exercise 3.5Show that the length of the vector with components is \(2\). In other words, show that if then

Exercise 3.6Let Show that Notice that in the language of matrices, the above identity reads \(P^{2}=P\) which represents the Linear Algebra definition of a projection matrix.

Exercise 3.7Similarly, let and show that Thus, the matrix \(P\) corresponding to \(P_{j}^{i}\) also satisfies the Linear Algebra definition of a projection matrix.

Exercise 3.8Show that the derivatives of the ambient covariant basis with respect to the surface coordinates are given by the formula

Exercise 3.9Derive the identity from the intrinsic definitions of the Christoffel symbols and

Exercise 3.10Notice the striking similarity between the definition of the surface Christoffel symbol and the formula valid for affine ambient coordinates. Derive the latter equation from the former by assuming affine coordinates in the ambient space from the start.

Exercise 3.11Use the explicit expression for \(N^{i}\) to prove the projection formula This is an important, albeit time-consuming, exercise that you should take your time working out.

Exercise 3.12Show that the length of one cycle of the cycloid is \(8r\).