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Analysis of Several Concrete Surfaces

In this Chapter, we will calculate the fundamental variants introduced in the previous Chapter -- the shift tensor \(Z_{\alpha}^{i}\), the metric tensors \(S_{\alpha\beta}\) and \(S^{\alpha\beta}\), the area element \(\sqrt{S}\), the components \(N^{i}\) of the unit normal, and the Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) -- for concrete surfaces. Examples always bring theory to life but, in the case of surfaces, they are particularly important: while all Euclidean spaces are essentially the same, each surface represents its own unique space.
We will begin with a sphere of radius \(R\) referred to coordinates illustrated in the following figure.
(3.7)
This example will offer a great deal of insight as we will analyze it with the help of two different ambient coordinate systems: Cartesian and spherical. Since a spherical coordinate system is perfectly suited to the geometry of a sphere, its use will simplify our calculations despite the greater complexity of the ambient metric tensor and Christoffel symbol.
In this Chapter, we will denote matrices corresponding to a system by putting brackets around the associated indicial symbol. For example, \(\left[ S_{\alpha\beta}\right] \) will denote the matrix associated with the covariant metric tensor and \(\left[ Z_{\alpha}^{i}\right] \) will denote the matrix corresponding to the shift tensor.

4.1.1In Cartesian ambient coordinates

Let us operate with spherical coordinates \(\theta,\varphi\) on the surface and Cartesian coordinates \(x,y,z\) in the ambient space. Recall the equations of the surface
\[ \begin{aligned} x\left( \theta,\varphi\right) & =R\sin\theta\cos\varphi\ \ \ \ \ \ \ \ \ \ \left(3.8\right)\\ y\left( \theta,\varphi\right) & =R\sin\theta\sin\varphi\ \ \ \ \ \ \ \ \ \ \left(3.9\right)\\ z\left( \theta,\varphi\right) & =R\cos\theta\ \ \ \ \ \ \ \ \ \ \left(3.10\right) \end{aligned} \]
Recall that all versions of the ambient metric tensor, i.e. \(Z_{ij}\), \(Z^{ij} \), and \(\delta_{j}^{i}\), correspond to the \(3\times3\) identity matrix, i.e.
\[ \left[ Z_{ij}\right] =\left[ Z^{ij}\right] =\left[ \begin{array} {ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right] .\tag{4.1} \]
Thus, the placement of the ambient indices has no effect on the values of any of the variants.
The shift tensors \(Z_{\alpha}^{i}\) and \(Z_{i\alpha}\) are obtained by differentiating the equations of the surface with respect to each of the surface variables, \(\theta\) and \(\varphi\). Therefore, we have
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} \phantom{-} R\cos\theta\cos\varphi & -R\sin\theta\sin\varphi\\ \phantom{-} R\cos\theta\sin\varphi & \phantom{-} R\sin\theta\cos\varphi\\ -R\sin\theta & 0 \end{array} \right] .\tag{4.2} \]
The covariant metric tensor \(S_{\alpha\beta}\) is given by the equation
\[ S_{\alpha\beta}=Z_{i\alpha}Z_{\beta}^{i} \tag{3.113} \]
and is therefore represented by the matrix
\[ \left[ S_{\alpha\beta}\right] =\left[ Z_{i\alpha}\right] ^{T}\left[ Z_{\beta}^{i}\right]\tag{4.3} \]
which yields
\[ \left[ S_{\alpha\beta}\right] =\left[ \begin{array} {cc} R^{2} & 0\\ 0 & R^{2}\sin^{2}\theta \end{array} \right] .\tag{4.4} \]
Since \(\left[ S_{\alpha\beta}\right] \) is diagonal, the covariant basis is orthogonal. Also, you probably recognize \(\left[ S_{\alpha\beta}\right] \) as the bottom right \(2\times2\) submatrix of the metric tensor associated with spherical coordinates in a three-dimensional Euclidean space. It is left as an exercise to explain why this is so by observing the relationship between \(S_{\alpha}\) and the ambient covariant basis corresponding to spherical coordinates.
The contravariant metric tensor \(S^{\alpha\beta}\) corresponds to \(\left[ S_{\alpha\beta}\right] ^{-1}\), i.e.
\[ \left[ S^{\alpha\beta}\right] =\left[ S_{\alpha\beta}\right] ^{-1}=\left[ \begin{array} {cc} R^{-2} & 0\\ 0 & R^{-2}\sin^{-2}\theta \end{array} \right] .\tag{4.5} \]
The area element \(\sqrt{S}\) is the square root of the determinant of \(\left[ S_{\alpha\beta}\right] \), i.e.
\[ \sqrt{S}=\sqrt{\det\left[ S_{\alpha\beta}\right] }=R^{2}\sin\theta.\tag{4.6} \]
If you recall, \(\sqrt{S}\) is used in converting geometric integrals over the surface into arithmetic integrals. For example, the total area of the sphere of radius \(R\) is given by
\[ \int_{S}dS=\int_{0}^{\pi}\int_{0}^{2\pi}\sqrt{S}d\varphi d\theta=\int_{0} ^{\pi}\int_{0}^{2\pi}R^{2}\sin\theta d\varphi d\theta=4\pi R^{2}.\tag{4.7} \]
Next, let us calculate the shift tensors \(Z^{i\alpha}\) and \(Z_{i}^{\alpha}\) with raised surface indices. Since,
\[ \begin{aligned} Z^{i\alpha} & =S^{\alpha\beta}Z_{\beta}^{i}\text{ \ \ \ and}\ \ \ \ \ \ \ \ \ \ \left(4.8\right)\\ Z_{i}^{\alpha} & =S^{\alpha\beta}Z_{i\beta}\ \ \ \ \ \ \ \ \ \ \left(4.9\right) \end{aligned} \]
we have
\[ \begin{aligned} \left[ Z^{i\alpha}\right] & =\left[ Z_{\beta}^{i}\right] \left[ S^{\alpha\beta}\right] \text{\ \ \ \ and}\ \ \ \ \ \ \ \ \ \ \left(4.10\right)\\ \left[ Z_{i}^{\alpha}\right] & =\left[ Z_{i\beta}\right] \left[ S^{\alpha\beta}\right] ,\ \ \ \ \ \ \ \ \ \ \left(4.11\right) \end{aligned} \]
where justifying the order of the terms in the matrix products is left as an exercise. Note that, technically, the above products should involve \(\left[ S^{\alpha\beta}\right] ^{T}\), however the transpose is not necessary since \(\left[ S^{\alpha\beta}\right] \) is symmetric. Carrying out the matrix products, we find
\[ \left[ Z^{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\left[ \begin{array} {ll} \phantom{-} R^{-1}\cos\theta\cos\varphi & -R^{-1}\sin^{-1}\theta\sin\varphi\\ \phantom{-} R^{-1}\cos\theta\sin\varphi & \phantom{-} R^{-1}\sin^{-1}\theta\cos\varphi\\ -R^{-1}\sin\theta & \phantom{-} 0 \end{array} \right] .\tag{4.12} \]
Let us now turn our attention to calculating the components \(N^{i}\) of the unit normal. This can be accomplished in two ways. First, recall the normalization condition
\[ N^{i}Z_{i\alpha}=0 \tag{3.134} \]
which, in matrix form, reads
\[ \left[ N^{i}\right] \left[ Z_{i\alpha}\right] =0.\tag{4.13} \]
It tells us that \(\left[ N^{i}\right] \) is contained in the null space of the matrix \(\left[ Z_{i\alpha}\right] ^{T}\). Thus, \(\left[ N^{i}\right] \) can be calculated by determining the null space of \(\left[ Z_{i\alpha }\right] ^{T}\) and then normalizing it to unit length. It is left as an exercise to show that the transposes of the matrices corresponding to all four forms of the shift tensor, i.e. \(Z_{\alpha}^{i}\), \(Z_{i\alpha}\), \(Z_{i}^{\alpha}\), and \(Z^{i\alpha}\), have the same null space. Furthermore, calculating \(N^{i}\) by this approach is also left as an exercise.
Second, the components \(N^{i}\) can be calculated by evaluating the cross product of the vectors \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\), once again followed by normalization to unity. As we discussed in Chapter 3, the components of \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\) are contained in the columns of the matrix
\[ \left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} \phantom{-} R\cos\theta\cos\varphi & -R\sin\theta\sin\varphi\\ \phantom{-} R\cos\theta\sin\varphi & \phantom{-} R\sin\theta\cos\varphi\\ -R\sin\theta & 0 \end{array} \right] .\tag{4.14} \]
In anticipation of the normalization step, we can drop the factors of \(R\) from \(\left[ Z_{i\alpha}\right] \) in the cross product calculation. Therefore, we have
\[ \begin{aligned} \mathbf{N} & \sim\det\left[ \begin{array} {ccc} \mathbf{i} & \phantom{-} \cos\theta\cos\varphi & -\sin\theta\sin\varphi\\ \mathbf{j} & \phantom{-} \cos\theta\sin\varphi & \phantom{-} \sin\theta\cos\varphi\\ \mathbf{k} & -\sin\theta & 0 \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.15\right)\\ & =\mathbf{i}\sin^{2}\theta\cos\varphi+\mathbf{j}\sin^{2}\theta\sin \varphi+\mathbf{k}\cos\theta\sin\theta.\ \ \ \ \ \ \ \ \ \ \left(4.16\right) \end{aligned} \]
The length of the resulting vector is \(\sin\theta\). Thus, dividing by \(\sin\theta\), we find
\[ \mathbf{N}=\mathbf{i}\sin\theta\cos\varphi+\mathbf{j}\sin\theta\sin \varphi+\mathbf{k}\cos\theta\tag{4.17} \]
or
\[ \left[ N^{i}\right] =\left[ \begin{array} {l} \sin\theta\cos\varphi\\ \sin\theta\sin\varphi\\ \cos\theta \end{array} \right] .\tag{4.18} \]
Finally, note that for a sphere, the components \(N^{i}\) could have also been easily guessed from geometric considerations as we know that \(\mathbf{N}\) is a unit vector pointing in the radial direction and is therefore collinear with the position vector \(\mathbf{R}\) if the latter emanates from the center of the sphere.
The last remaining object to be calculated is the surface Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\). Recall that the relationship between the surface and the ambient Christoffel symbols reads
\[ \Gamma_{\alpha\beta}^{\gamma}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j} Z_{k}^{\gamma}+Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i}}{\partial S^{\beta }}, \tag{3.125} \]
which reduces to
\[ \Gamma_{\alpha\beta}^{\gamma}=Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i} }{\partial S^{\beta}} \tag{3.126} \]
when the ambient space is referred to affine coordinates, which is the case here. The above identity has three free indices \(\alpha\), \(\beta\), and \(\gamma\) and, therefore, cannot be captured by a matrix identity. In order to continue using matrix algebra, we need to fix the value of one of the free indices. We will choose \(\beta\) for this role and consider \(\beta=1\) and \(\beta=2\) separately. Since \(S^{1}=\theta\) and \(S^{2}=\varphi\), we have
\[ \Gamma_{\alpha1}^{\gamma}=Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i} }{\partial\theta}\text{\ \ \ and \ \ \ }\Gamma_{\alpha2}^{\gamma} =Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i}}{\partial\varphi}.\tag{4.19} \]
The elements of \(\partial Z_{\alpha}^{i}/\partial\theta\) are obtained by differentiating \(\left[ Z_{\alpha}^{i}\right] \) with respect to \(\theta\), i.e.
\[ \left[ \frac{\partial Z_{\alpha}^{i}}{\partial\theta}\right] =\left[ \begin{array} {ll} -R\sin\theta\cos\varphi & -R\cos\theta\sin\varphi\\ -R\sin\theta\sin\varphi & \phantom{-} R\cos\theta\cos\varphi\\ -R\cos\theta & \phantom{-} 0 \end{array} \right] .\tag{4.20} \]
Since
\[ \left[ \Gamma_{\alpha1}^{\gamma}\right] =\left[ Z_{i}^{\gamma}\right] ^{T}\left[ \frac{\partial Z_{\alpha}^{i}}{\partial\theta}\right] ,\tag{4.21} \]
we find
\[ \left[ \Gamma_{\alpha1}^{\gamma}\right] =\left[ \begin{array} {cc} 0 & 0\\ 0 & \cot\theta \end{array} \right] .\tag{4.22} \]
Meanwhile, the elements of \(\partial Z_{\alpha}^{i}/\partial\varphi\) are obtained by differentiating \(\left[ Z_{\alpha}^{i}\right] \) with respect to \(\varphi\), i.e.
\[ \left[ \frac{\partial Z_{\alpha}^{i}}{\partial\varphi}\right] =\left[ \begin{array} {ll} -R\cos\theta\sin\varphi & -R\sin\theta\cos\varphi\\ \phantom{-} R\cos\theta\cos\varphi & -R\sin\theta\sin\varphi\\ 0 & 0 \end{array} \right] .\tag{4.23} \]
As before, since
\[ \left[ \Gamma_{\alpha2}^{\gamma}\right] =\left[ Z_{i}^{\gamma}\right] ^{T}\left[ \frac{\partial Z_{\alpha}^{i}}{\partial\varphi}\right] ,\tag{4.24} \]
we find
\[ \left[ \Gamma_{\alpha2}^{\gamma}\right] =\left[ \begin{array} {cc} 0 & -\sin\theta\cos\theta\\ \cot\theta & 0 \end{array} \right] .\tag{4.25} \]
In summary, the three nonzero elements of \(\Gamma_{\alpha\beta}^{\gamma}\) are
\[ \begin{aligned} \Gamma_{22}^{1} & =-\sin\theta\cos\theta\text{ \ \ \ and}\ \ \ \ \ \ \ \ \ \ \left(4.26\right)\\ \Gamma_{12}^{2} & =\Gamma_{21}^{2}=\cot\theta.\ \ \ \ \ \ \ \ \ \ \left(4.27\right) \end{aligned} \]
This completes our analysis of a sphere of radius \(R\) in Cartesian ambient coordinates. We will now repeat the same analysis in spherical ambient coordinates and it will be particularly interesting to observe what changes and what stays the same.

4.1.2In spherical ambient coordinates

Let us now perform the parallel analysis using the same coordinates \(\theta,\varphi\) on the surface and spherical coordinates \(r,\theta _{1},\varphi_{1}\) in the ambient space. Recall that for this combination of coordinate systems, a sphere of radius \(R\) is described by the exceedingly simple equations
\[ \begin{aligned} r\left( \theta,\varphi\right) & =R\ \ \ \ \ \ \ \ \ \ \left(3.11\right)\\ \theta_{1}\left( \theta,\varphi\right) & =\theta\ \ \ \ \ \ \ \ \ \ \left(3.12\right)\\ \varphi_{1}\left( \theta,\varphi\right) & =\varphi. \ \ \ \ \ \ \ \ \ \ \left(3.13\right) \end{aligned} \]
The ambient covariant metric tensor \(Z_{ij}\), on the other hand, is more complicated than in the case of Cartesian coordinates. It was calculated in Section TBD of Introduction to Tensor Calculus where we showed that it corresponds to the matrix
\[ \left[ Z_{ij}\right] =\left[ \begin{array} {ccc} 1 & 0 & 0\\ 0 & R^{2} & 0\\ 0 & 0 & R^{2}\sin^{2}\theta \end{array} \right] .\tag{4.28} \]
Thus, the contravariant metric tensor \(Z^{ij}\) corresponds to
\[ \left[ Z^{ij}\right] =\left[ \begin{array} {ccc} 1 & 0 & 0\\ 0 & R^{-2} & 0\\ 0 & 0 & R^{-2}\sin^{-2}\theta \end{array} \right] ,\tag{4.29} \]
The shift tensor \(Z_{\alpha}^{i}\) is obtained by differentiating the equations of the surface with respect to \(\theta\) and \(\varphi\) and has the exceedingly simple form
\[ \left[ Z_{\alpha}^{i}\right] =\left[ \begin{array} {rr} 0 & 0\\ 1 & 0\\ 0 & 1 \end{array} \right] .\tag{4.30} \]
Since the metric tensor \(\left[ Z_{ij}\right] \) does not correspond to the identity matrix, the shift tensor \(Z_{i\alpha}\) corresponds to a different matrix compared to \(Z_{\alpha}^{i}\). Since
\[ Z_{i\alpha}=Z_{ij}Z_{\alpha}^{j},\tag{4.31} \]
we have
\[ \left[ Z_{i\alpha}\right] =\left[ Z_{ij}\right] \left[ Z_{\alpha} ^{j}\right]\tag{4.32} \]
which yields
\[ \left[ Z_{i\alpha}\right] =\left[ \begin{array} {cc} 0 & 0\\ R^{2} & 0\\ 0 & R^{2}\sin^{2}\theta \end{array} \right] .\tag{4.33} \]
The covariant metric tensor \(S_{\alpha\beta}\) is related to the shift tensors \(Z_{i\alpha}\) and \(Z_{\beta}^{i}\) by the equation
\[ S_{\alpha\beta}=Z_{i\alpha}Z_{\beta}^{i}. \tag{3.113} \]
Therefore, we have
\[ \left[ S_{\alpha\beta}\right] =\left[ Z_{i\alpha}\right] ^{T}\left[ Z_{\beta}^{i}\right] =\left[ \begin{array} {cc} R^{2} & 0\\ 0 & R^{2}\sin^{2}\theta \end{array} \right] .\tag{4.34} \]
Crucially, we have arrived at the same values as before. This is entirely to be expected since \(S_{\alpha\beta}\) does not depend on the choice of the ambient coordinates. Thus, the contravariant metric tensor \(S^{\alpha\beta}\) and the area element \(\sqrt{S}\) also have the same value as before and need not be documented here.
Now we can calculate the two remaining forms of the shift tensor. The contravariant form \(Z^{i\alpha}\) is given by
\[ Z^{i\alpha}=S^{\alpha\beta}Z_{\beta}^{i}\tag{4.35} \]
and therefore corresponds to the matrix product \(\left[ Z_{\alpha} ^{i}\right] \left[ S^{\alpha\beta}\right] \) which yields
\[ \left[ Z^{i\alpha}\right] =\left[ \begin{array} {ll} 0 & 0\\ R^{-2} & 0\\ 0 & R^{-2}\sin^{-2}\theta \end{array} \right] .\tag{4.36} \]
Finally, \(Z_{i}^{\alpha}\) is given by
\[ Z_{i}^{\alpha}=Z_{ij}Z^{j\alpha}\tag{4.37} \]
and therefore corresponds to the product \(\left[ Z_{ij}\right] \left[ Z^{i\alpha}\right] \) which yields
\[ \left[ Z_{i}^{\alpha}\right] =\left[ \begin{array} {cc} 0 & 0\\ 1 & 0\\ 0 & 1 \end{array} \right] .\tag{4.38} \]
It is interesting to note that \(Z_{\alpha}^{i}\) and \(Z_{i}^{\alpha}\) have identical values, which is not true in general. It is left as an exercise to explain why it makes sense in this case by considering the relationships between the contravariant surface and ambient space bases.
The components \(N^{i}\) of the normal are, once again, easy to guess, since the (exterior) unit normal \(\mathbf{N}\) coincides with the ambient basis vector \(\mathbf{Z}_{1}\) Therefore, the components \(N^{i}\) are given by
\[ \left[ N^{i}\right] =\left[ \begin{array} {c} 1\\ 0\\ 0 \end{array} \right] .\tag{4.39} \]
Note that this result could have also been obtained either from the null space of the matrix
\[ \left[ Z_{i\alpha}\right] ^{T}=\left[ \begin{array} {cc} 0 & 0\\ R^{2} & 0\\ 0 & R^{2}\sin^{2}\theta \end{array} \right] ^{T}=\left[ \begin{array} {ccc} 0 & R^{2} & 0\\ 0 & 0 & R^{2}\sin^{2}\theta \end{array} \right]\tag{4.40} \]
or by calculating the cross product of \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\).
Finally, although we already know the elements of the Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\), it will be interesting to calculate them again since this time, it is the first term on the right in the identity
\[ \Gamma_{\alpha\beta}^{\gamma}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j} Z_{k}^{\gamma}+Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i}}{\partial S^{\beta }} \tag{3.125} \]
that survives. Indeed, the second term vanishes since the elements of the shift tensor \(Z_{\alpha}^{i}\) are constants. Therefore, we have
\[ \Gamma_{\alpha\beta}^{\gamma}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j} Z_{k}^{\gamma}.\tag{4.41} \]
Once again, matrix algebra cannot capture all aspects of the above identity. We therefore again have to fix the values of some of the indices. Let us choose \(\gamma\) and consider \(\gamma=1\) and \(\gamma=2\) separately. For \(\gamma=1\), we have
\[ \Gamma_{\alpha\beta}^{1}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j}Z_{k}^{1}\tag{4.42} \]
while for \(\gamma=2\), we have
\[ \Gamma_{\alpha\beta}^{2}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j}Z_{k}^{2}.\tag{4.43} \]
Recall that
\[ \left[ Z_{k}^{\alpha}\right] =\left[ \begin{array} {cc} 0 & 0\\ 1 & 0\\ 0 & 1 \end{array} \right] . \tag{4.38} \]
Thus, the only nonzero element in \(Z_{k}^{1}\) is \(Z_{2}^{1}=1\) and the only nonzero element of \(Z_{k}^{2}\) is \(Z_{3}^{2}=1\). Thus expressions for \(\Gamma_{\alpha\beta}^{1}\) and \(\Gamma_{\alpha\beta}^{2}\) are given by the simplified equations
\[ \begin{aligned} \Gamma_{\alpha\beta}^{1} & =\Gamma_{ij}^{2}Z_{\alpha}^{i}Z_{\beta} ^{j}\text{, and}\ \ \ \ \ \ \ \ \ \ \left(4.44\right)\\ \Gamma_{\alpha\beta}^{2} & =\Gamma_{ij}^{3}Z_{\alpha}^{i}Z_{\beta}^{j},\ \ \ \ \ \ \ \ \ \ \left(4.45\right) \end{aligned} \]
which can be handled by matrix algebra. Recall that the nonzero elements of the ambient Christoffel symbol \(\Gamma_{ij}^{k}\) are
\[ \begin{aligned} \Gamma_{22}^{1} & =-r\ \ \ \ \ \ \ \ \ \ \left(6.45\right)\\ \Gamma_{33}^{1} & =-r\sin^{2}\theta\ \ \ \ \ \ \ \ \ \ \left(6.45\right)\\ \Gamma_{12}^{2} & =\Gamma_{21}^{2}=r^{-1}\ \ \ \ \ \ \ \ \ \ \left(6.45\right)\\ \Gamma_{33}^{2} & =-\sin\theta\cos\theta\ \ \ \ \ \ \ \ \ \ \left(6.45\right)\\ \Gamma_{13}^{3} & =\Gamma_{31}^{3}=r^{-1}\ \ \ \ \ \ \ \ \ \ \left(6.45\right)\\ \Gamma_{23}^{3} & =\Gamma_{32}^{3}=\cot\theta. \ \ \ \ \ \ \ \ \ \ \left(6.45\right) \end{aligned} \]
Organize the values of the systems \(\Gamma_{ij}^{2}\) and \(\Gamma_{ij}^{3}\) on the surface of the sphere into matrices, i.e.
\[ \begin{aligned} \left[ \Gamma_{ij}^{2}\right] & =\left[ \begin{array} {ccc} 0 & R^{-1} & 0\\ R^{-1} & 0 & 0\\ 0 & 0 & -\sin\theta\cos\theta \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.46\right)\\ \left[ \Gamma_{ij}^{3}\right] & =\left[ \begin{array} {ccc} 0 & 0 & R^{-1}\\ 0 & 0 & \cot\theta\\ R^{-1} & \cot\theta & 0 \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.47\right) \end{aligned} \]
With the help of these matrices, we have
\[ \left[ \Gamma_{\alpha\beta}^{1}\right] =\left[ Z_{\alpha}^{i}\right] ^{T}\left[ \Gamma_{ij}^{2}\right] \left[ Z_{\alpha}^{i}\right] =\left[ \begin{array} {cc} 0 & 0\\ 0 & -\cos\theta\sin\theta \end{array} \right]\tag{4.48} \]
and
\[ \left[ \Gamma_{\alpha\beta}^{2}\right] =\left[ Z_{\alpha}^{i}\right] ^{T}\left[ \Gamma_{ij}^{3}\right] \left[ Z_{\alpha}^{i}\right] =\left[ \begin{array} {cc} 0 & \cot\theta\\ \cot\theta & 0 \end{array} \right] .\tag{4.49} \]
Collecting the nonzero elements of \(\Gamma_{\alpha\beta}^{\gamma}\), we observe that we have arrived at the same values as before, i.e.
\[ \begin{aligned} \Gamma_{22}^{1} & =-\sin\theta\cos\theta\ \ \ \ \ \ \ \ \ \ \left(4.26\right)\\ \Gamma_{12}^{2} & =\Gamma_{21}^{2}=\cot\theta. \ \ \ \ \ \ \ \ \ \ \left(4.27\right) \end{aligned} \]
This completes our analysis of a sphere of radius \(R\).
Consider a cylinder of radius \(R\) referred to coordinates \(z,\theta\) illustrated in the following figure.
(3.14)
As was the case with a sphere, a cylinder can also be effectively analyzed by referring the ambient space either to Cartesian or cylindrical coordinates. We will choose Cartesian coordinates \(x,y,z_{1}\) and save cylindrical coordinates for the exercises. Recall that the equations of the surface read
\[ \begin{aligned} x\left( \theta,z\right) & =R\cos\theta\ \ \ \ \ \ \ \ \ \ \left(3.15\right)\\ y\left( \theta,z\right) & =R\sin\theta\ \ \ \ \ \ \ \ \ \ \left(3.16\right)\\ z_{1}\left( \theta,z\right) & =z. \ \ \ \ \ \ \ \ \ \ \left(3.17\right) \end{aligned} \]
Once again, the ambient metric tensors \(Z_{ij}\) and \(Z^{ij}\) corresponds to the \(3\times3\) identity matrix
\[ \left[ Z_{ij}\right] =\left[ Z^{ij}\right] =\left[ \begin{array} {ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right] ,\tag{4.50} \]
and thus, once again, the placement of the ambient indices has no effect on the values of any of the variants.
The shift tensors \(Z_{\alpha}^{i}\) and \(Z_{i}^{\alpha}\) are obtained by differentiating the equations of the surface with respect to \(\theta\) and \(z\), which yields
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} -R\sin\theta & 0\\ \phantom{-} R\cos\theta & 0\\ \phantom{-} 0 & 1 \end{array} \right] .\tag{4.51} \]
The surface metric tensor \(S_{\alpha\beta}\) is given by
\[ S_{\alpha\beta}=Z_{i\alpha}Z_{\beta}^{i} \tag{3.114} \]
and therefore
\[ \left[ S_{\alpha\beta}\right] =\left[ Z_{i\alpha}\right] ^{T}\left[ Z_{\beta}^{i}\right] =\left[ \begin{array} {cc} R^{2} & 0\\ 0 & 1 \end{array} \right] ,\tag{4.52} \]
while
\[ \left[ S^{\alpha\beta}\right] =\left[ S_{\alpha\beta}\right] ^{-1}=\left[ \begin{array} {cc} \frac{1}{R^{2}} & 0\\ 0 & 1 \end{array} \right] .\tag{4.53} \]
The area element \(\sqrt{S}\) is the square root of the determinant of \(\left[ S_{\alpha\beta}\right] \), i.e.
\[ \sqrt{S}=R.\tag{4.54} \]
Notice that the elements of the metric tensors have constant values, which is a signature feature of affine coordinates. The fact that a cylinder admits such a coordinate system distinguishes it from all the other two-dimensional surfaces discussed in this Chapter. This fact means that, in a certain sense, a cylinder represents a Euclidean space. On the one hand, this makes sense, since a cylinder can be cut along a line of constant \(\theta\) and uncurled, without any distortion, into a flat strip. On the other hand, a cylinder is a curved surface that cannot accommodate straight lines, except in one special direction. Understanding this dichotomy will be one of the goals of our study of curvature.
Let us now return to the calculation of the remaining differential objects. The shift tensors \(Z^{i\alpha}=S^{\alpha\beta}Z_{\beta}^{i}\) and \(Z_{i}^{\alpha}=S^{\alpha\beta}Z_{i\beta}\) correspond to the matrix product \(\left[ Z_{\alpha}^{i}\right] \left[ S^{\alpha\beta}\right] \) which yields
\[ \left[ Z^{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\left[ \begin{array} {ll} -R^{-1}\sin\theta & 0\\ \phantom{-} R^{-1}\cos\theta & 0\\ \phantom{-} 0 & 1 \end{array} \right]\tag{4.55} \]
As was the case with the sphere, the components \(N^{i}\) of the unit normal can be determined from the null space of \(\left[ Z_{i\alpha}\right] \), by calculating the cross product of \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\), or simply by intuiting from general geometric considerations. In any case, the components \(N^{i}\) correspond to the matrix
\[ \left[ N^{i}\right] =\left[ \begin{array} {r} \cos\theta\\ \sin\theta\\ 0 \end{array} \right] .\tag{4.56} \]
Since the elements of the metric tensors are constant, the Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\) vanishes at all points, i.e.
\[ \Gamma_{\alpha\beta}^{\gamma}=0,\tag{4.57} \]
which can also be described as a signature characteristic of an affine coordinate system.
This completes our analysis of a cylinder and we will now turn our attention to the more general surfaces of revolution.
A sphere and a cylinder are both examples of surfaces of revolution. Thus, the analysis presented in this Section represents a generalization of the preceding two Sections.
  (3.21)
We will refer the ambient space to Cartesian coordinates and will invite the reader to repeat the analysis in cylindrical coordinates as an exercise. In Cartesian coordinates \(x,y,z\), the equations of a surface of revolution read
\[ \begin{aligned} x\left( \theta,\gamma\right) & =G\left( \gamma\right) \cos\theta\ \ \ \ \ \ \ \ \ \ \left(3.22\right)\\ y\left( \theta,\gamma\right) & =G\left( \gamma\right) \sin\theta\ \ \ \ \ \ \ \ \ \ \left(3.23\right)\\ z\left( \theta,\gamma\right) & =H\left( \gamma\right) \ \ \ \ \ \ \ \ \ \ \left(3.24\right) \end{aligned} \]
Since we have already performed several similar analyses, we will begin to omit some of the details in the upcoming calculations. Filling in those details will be left as exercises.
The shift tensors \(Z_{\alpha}^{i}\) and \(Z_{i\alpha}\) are obtained by differentiating the equations of the surface with respect to \(\theta\) and \(\gamma\), which yields
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} -G\sin\theta & G_{\gamma}\cos\theta\\ \phantom{-} G\cos\theta & G_{\gamma}\sin\theta\\ \phantom{-} 0 & H_{\gamma} \end{array} \right] ,\tag{4.58} \]
where \(G_{\gamma}\) and \(H_{\gamma}\) denote the derivatives of the functions \(G\) and \(H\).
For the covariant and the contravariant metric tensors, we have
\[ \begin{aligned} S_{\alpha\beta} & =\left[ Z_{\alpha}^{i}\right] ^{T}\left[ Z_{\beta} ^{j}\right] =\left[ \begin{array} {cc} G^{2} & 0\\ 0 & G_{\gamma}^{2}+H_{\gamma}^{2} \end{array} \right] \text{ \ \ \ and}\ \ \ \ \ \ \ \ \ \ \left(4.59\right)\\ S^{\alpha\beta} & =\left[ S_{\alpha\beta}\right] ^{-1}=\left[ \begin{array} {cc} \frac{1}{G^{2}} & 0\\ 0 & \frac{1}{G_{\gamma}^{2}+H_{\gamma}^{2}} \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.60\right) \end{aligned} \]
The area element \(\sqrt{S}\) is given by
\[ \sqrt{S}=\sqrt{\det\left[ S_{\alpha\beta}\right] }=G\sqrt{G_{\gamma} ^{2}+H_{\gamma}^{2}}.\tag{4.61} \]
For the remaining forms \(Z^{i\alpha}\) and \(Z_{i}^{\alpha}\) of the shift tensor, we have
\[ \left[ Z^{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\left[ Z_{\alpha}^{i}\right] \left[ S^{\alpha\beta}\right] =\left[ \begin{array} {ll} -\frac{1}{G}\sin\theta & \frac{G_{\gamma}\cos\theta}{G_{\gamma}^{2}+H_{\gamma }^{2}}\\ \phantom{-} \frac{1}{G}\cos\theta & \frac{G_{\gamma}\sin\theta}{G_{\gamma}^{2}+H_{\gamma }^{2}}\\ \phantom{-} 0 & \frac{H_{\gamma}}{G_{\gamma}^{2}+H_{\gamma}^{2}} \end{array} \right] .\tag{4.62} \]
For the components \(N^{i}\) of the normal, which can be determined by any of the approaches described above, we have
\[ \left[ N^{i}\right] =\frac{1}{\sqrt{G_{\gamma}^{2}+H_{\gamma}^{2}}}\left[ \begin{array} {l} \phantom{-} H_{\gamma}\cos\theta\\ \phantom{-} H_{\gamma}\sin\theta\\ -G_{\gamma} \end{array} \right] .\tag{4.63} \]
The nonzero elements of the Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) are
\[ \begin{aligned} \Gamma_{12}^{1} & =\Gamma_{21}^{1}=\frac{G_{\gamma}}{G}\ \ \ \ \ \ \ \ \ \ \left(4.64\right)\\ \Gamma_{11}^{2} & =-\frac{GG_{\gamma}}{G_{\gamma}^{2}+H_{\gamma}^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.65\right)\\ \Gamma_{22}^{2} & =\frac{G_{\gamma}G_{\gamma\gamma}+H_{\gamma} H_{\gamma\gamma}}{G_{\gamma}^{2}+H_{\gamma}^{2}}.\ \ \ \ \ \ \ \ \ \ \left(4.66\right) \end{aligned} \]
The findings of this Section will pay instant dividends as we turn our attention to another shape of revolution: a torus.
A torus with radii \(R\) and \(r\) is a shape of revolution obtained by rotating a circle of radius \(r\) around an axis that is a distance \(R\) from the center of the circle. Note that, unlike other Sections where the symbol \(r\) denotes one of the ambient coordinates, in this Section, \(r\) is a constant parameter.
(4.67)
Within its plane, the circle is described by the equations
\[ \begin{aligned} G\left( \varphi\right) & =R+r\cos\varphi\ \ \ \ \ \ \ \ \ \ \left(4.68\right)\\ H\left( \varphi\right) & =r\sin\varphi.\ \ \ \ \ \ \ \ \ \ \left(4.69\right) \end{aligned} \]
The following figure illustrates the resulting surface coordinates \(\theta\) and \(\varphi\). Note that both coordinates vary from \(0\) to \(2\pi\).
(4.70)
When the ambient space is referred to Cartesian coordinates \(x,y,z\), the equations of the torus read
\[ \begin{aligned} x\left( \theta,\varphi\right) & =\left( R+r\cos\varphi\right) \cos \theta\ \ \ \ \ \ \ \ \ \ \left(3.32\right)\\ y\left( \theta,\varphi\right) & =\left( R+r\cos\varphi\right) \sin \theta\ \ \ \ \ \ \ \ \ \ \left(3.34\right)\\ z\left( \theta,\varphi\right) & =r\sin\varphi. \ \ \ \ \ \ \ \ \ \ \left(3.35\right) \end{aligned} \]
All of the following identities can be obtained as special cases of the equations from the previous Section on surfaces of revolution. Therefore, we will catalogue the results without further clarifications.
The shift tensors \(Z_{\alpha}^{i}\) and \(Z_{i\alpha}\):
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} -\left( R+r\cos\varphi\right) \sin\theta & -r\cos\theta\sin\varphi\\ \phantom{-} \left( R+r\cos\varphi\right) \cos\theta & -r\sin\theta\sin\varphi\\ \phantom{-} 0 & \phantom{-} r\cos\varphi \end{array} \right] .\tag{4.71} \]
The metric tensors \(S_{\alpha\beta}\) and \(S^{\alpha\beta}\), and the area element \(\sqrt{S}\):
\[ \begin{aligned} \left[ S_{\alpha\beta}\right] & =\left[ \begin{array} {cc} \left( R+r\cos\varphi\right) ^{2} & 0\\ 0 & r^{2} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.72\right)\\ \left[ S^{\alpha\beta}\right] & =\ \left[ \begin{array} {cc} \left( R+r\cos\varphi\right) ^{-2} & 0\\ 0 & r^{-2} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.73\right)\\ \sqrt{S} & =r\left( R+r\cos\varphi\right) .\ \ \ \ \ \ \ \ \ \ \left(4.74\right) \end{aligned} \]
With the help of the area element \(\sqrt{S}\), we can calculate the total area of the torus, i.e.
\[ A=\int_{0}^{2\pi}\int_{0}^{2\pi}r\left( R+r\cos\varphi\right) d\theta d\varphi=4\pi^{2}Rr.\tag{4.75} \]
The shift tensors \(Z^{i\alpha}\) and \(Z_{i}^{\alpha}\):
\[ \left[ Z^{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\left[ \begin{array} {ll} -\left( R+r\cos\varphi\right) ^{-1}\sin\theta & -r^{-1}\cos\theta\sin \varphi\\ \phantom{-} \left( R+r\cos\varphi\right) ^{-1}\cos\theta & -r^{-1}\sin\theta\sin \varphi\\ \phantom{-} 0 & \phantom{-} r^{-1}\cos\varphi \end{array} \right] .\tag{4.76} \]
The components \(N^{i}\) of the normal:
\[ \left[ N^{i}\right] =\left[ \begin{array} {r} \cos\theta\cos\varphi\\ \sin\theta\cos\varphi\\ \sin\varphi \end{array} \right] .\tag{4.77} \]
And, finally, the nonzero elements of the Christoffel symbol \(\Gamma _{\alpha\beta}^{\gamma}\):
\[ \begin{aligned} \Gamma_{12}^{1} & =\Gamma_{21}^{1}=-\frac{r\sin\varphi}{R+r\cos\varphi}.\ \ \ \ \ \ \ \ \ \ \left(4.78\right)\\ \Gamma_{11}^{2} & =\frac{\left( R+r\cos\varphi\right) \sin\varphi}{r}\ \ \ \ \ \ \ \ \ \ \left(4.79\right) \end{aligned} \]
This completes our analysis of two-dimensional surfaces embedded in a three-dimensional Euclidean space and we now turn our attention to curves in a two-dimensional Euclidean plane.
The theory that we have developed for surfaces applies, without any changes, to one-dimensional curves in a two-dimensional Euclidean plane. In fact, with the exception of the normal which requires the surface to be a hypersurface, all of the fundamental differential objects considered in this Chapter can be calculated for a curve embedded in a space of any dimension. We will, however, limit ourselves to curves in the plane here, and save the study of curves in a three-dimensional space for Chapter 8.

4.5.1In Cartesian coordinates

A one-dimensional curve is parameterized by a single coordinate \(S^{1}\) which we will denote by the letter \(\gamma\). First, refer a two-dimensional plane to Cartesian coordinates \(x,y\) and consider a general curve given by the equations
\[ \begin{aligned} x & =x\left( \gamma\right)\ \ \ \ \ \ \ \ \ \ \left(4.80\right)\\ y & =y\left( \gamma\right) .\ \ \ \ \ \ \ \ \ \ \left(4.81\right) \end{aligned} \]
In the equations that follow, the symbols \(x\) and \(y\) will refer to the functions \(x\left( \gamma\right) \) and \(y\left( \gamma\right) \) while \(x_{\gamma}\), \(x_{\gamma\gamma}\), \(y_{\gamma}\), and \(y_{\gamma\gamma}\) will refer to the derivatives \(x^{\prime}\left( \gamma\right) \), \(x^{\prime \prime}\left( \gamma\right) \), \(y^{\prime}\left( \gamma\right) \), and \(y^{\prime\prime}\left( \gamma\right) \).
The ambient metric tensors \(Z_{ij}\) and \(Z^{ij}\) correspond to the \(2\times2\) identity matrix, i.e.
\[ \left[ Z_{ij}\right] =\left[ Z^{ij}\right] =\left[ \begin{array} {cc} 1 & 0\\ 0 & 1 \end{array} \right] .\tag{4.82} \]
The shift tensors \(Z_{\alpha}^{i}\) and \(Z_{i\alpha}\) are obtained by differentiating the equations of the curve with respect to \(\gamma\), i.e.
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {c} x_{\gamma}\\ y_{\gamma} \end{array} \right] .\tag{4.83} \]
The surface metric tensor \(S_{\alpha\beta}=Z_{\alpha}^{i}Z_{i\beta}\) corresponds to the matrix product \(\left[ Z_{\alpha}^{i}\right] ^{T}\left[ Z_{i\beta}\right] \). The result is a \(1\times1\) matrix with a single entry. Nevertheless, we will continue to use brackets to denote the matrices. We have
\[ \left[ S_{\alpha\beta}\right] =\left[ x_{\gamma}^{2}+y_{\gamma}^{2}\right] .\tag{4.84} \]
The contravariant surface metric tensor \(S^{\alpha\beta}\) is the matrix inverse of \(S_{\alpha\beta}\). Since \(\left[ S_{\alpha\beta}\right] \) is a \(1\times1\) matrix, its inverse consists of a single entry that is the reciprocal of the sole entry in \(\left[ S_{\alpha\beta}\right] \), i.e.
\[ \left[ S^{\alpha\beta}\right] =\left[ \frac{1}{x_{\gamma}^{2}+y_{\gamma }^{2}}\right] .\tag{4.85} \]
The determinant of the matrix \(\left[ S_{\alpha\beta}\right] \) equals its only entry. Therefore, the length element \(\sqrt{S}\) is given by the equation
\[ \sqrt{S}=\sqrt{x_{\gamma}^{2}+y_{\gamma}^{2}}.\tag{4.86} \]
This is a familiar expression that we have encountered on numerous occasions when studying curves.
The remaining forms of the shift tensor are
\[ \left[ Z^{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\frac {1}{x_{\gamma}^{2}+y_{\gamma}^{2}}\left[ \begin{array} {c} x_{\gamma}\\ y_{\gamma} \end{array} \right] .\tag{4.87} \]
The single entry of the Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) is given by
\[ \Gamma_{11}^{1}=\frac{x_{\gamma}x_{\gamma\gamma}+y_{\gamma}y_{\gamma\gamma} }{x_{\gamma}^{2}+y_{\gamma}^{2}}.\tag{4.88} \]
Finally, the components \(N^{i}\ \)of the normal are given by
\[ \left[ N^{i}\right] =\frac{1}{\sqrt{x_{\gamma}^{2}+y_{\gamma}^{2}}}\left[ \begin{array} {r} -y_{\gamma}\\ x_{\gamma} \end{array} \right] .\tag{4.89} \]
Let us now catalogue the values of the same objects for two special parameterizations. In the first one, use the arc length \(s\) for the coordinate along the curve. Recall that
\[ \mathbf{R}^{\prime}\left( s\right)\tag{4.90} \]
represents a unit tangent vector. Since, \(x_{s}=x^{\prime}\left( s\right) \) and \(y_{s}=y^{\prime}\left( s\right) \) represent its components, we have
\[ x_{s}^{2}+y_{s}^{2}=1.\tag{4.91} \]
This identity leads to the simplifications found in the following table.
\[ \begin{aligned} \left[ Z_{\alpha}^{i}\right] & =\left[ Z_{i\alpha}\right] =\left[ Z_{i}^{\alpha}\right] =\left[ Z^{i\alpha}\right] =\left[ \begin{array} {c} x_{s}\\ y_{s} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.92\right)\\ \left[ S_{\alpha\beta}\right] & =\ \left[ S^{\alpha\beta}\right] =\left[ 1\right]\ \ \ \ \ \ \ \ \ \ \left(4.93\right)\\ \sqrt{S} & =1\ \ \ \ \ \ \ \ \ \ \left(4.94\right)\\ \Gamma_{11}^{1} & =0\ \ \ \ \ \ \ \ \ \ \left(4.95\right)\\ \left[ N^{i}\right] & =\left[ \begin{array} {r} -y_{s}\\ x_{s} \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.96\right) \end{aligned} \]
For the second special case, consider a curve given by the graph of a function \(y=y\left( x\right) \). Recall that such a curve is described by the parametric equations
\[ \begin{aligned} x & =\gamma\ \ \ \ \ \ \ \ \ \ \left(4.97\right)\\ y & =y\left( \gamma\right) .\ \ \ \ \ \ \ \ \ \ \left(4.98\right) \end{aligned} \]
Using \(x\) in place of \(\gamma\), we have the results summarized in the following table.
\[ \begin{aligned} \left[ Z_{\alpha}^{i}\right] & =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {c} 1\\ y_{x} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.99\right)\\ \left[ S_{\alpha\beta}\right] & =\left[ 1+y_{x}^{2}\right]\ \ \ \ \ \ \ \ \ \ \left(4.100\right)\\ \left[ S^{\alpha\beta}\right] & =\left[ \frac{1}{1+y_{x}^{2}}\right]\ \ \ \ \ \ \ \ \ \ \left(4.101\right)\\ \left[ Z_{i}^{\alpha}\right] & =\left[ Z^{i\alpha}\right] =\frac {1}{1+y_{x}^{2}}\left[ \begin{array} {c} 1\\ y_{x} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.102\right)\\ \sqrt{S} & =\sqrt{1+y_{x}^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.103\right)\\ \Gamma_{11}^{1} & =\frac{y_{x}y_{xx}}{1+y_{x}^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.104\right)\\ \left[ N^{i}\right] & =\frac{1}{\sqrt{1+y_{x}^{2}}}\left[ \begin{array} {r} -y_{x}\\ 1 \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.105\right) \end{aligned} \]

4.5.2In polar coordinates

Now, suppose that the plane is referred to polar coordinates \(r,\theta\). A general curve is given by the parametric equations
\[ \begin{aligned} r & =r\left( \gamma\right)\ \ \ \ \ \ \ \ \ \ \left(4.106\right)\\ \theta & =\theta\left( \gamma\right) .\ \ \ \ \ \ \ \ \ \ \left(4.107\right) \end{aligned} \]
In the equations that follow, the symbols \(r\) and \(\theta\) will refer to the functions \(r\left( \gamma\right) \) and \(\theta\left( \gamma\right) \) while \(r_{\gamma}\), \(r_{\gamma\gamma}\), \(\theta_{\gamma}\), and \(\theta_{\gamma \gamma}\) will refer to the derivatives \(r^{\prime}\left( \gamma\right) \), \(r^{\prime\prime}\left( \gamma\right) \), \(\theta^{\prime}\left( \gamma\right) \), and \(\theta^{\prime\prime}\left( \gamma\right) \).
The matrices corresponding to the ambient covariant and contravariant metric tensors \(Z_{ij}\) and \(Z^{ij}\) are
\[ \begin{aligned} \left[ Z_{ij}\right] & =\left[ \begin{array} {cc} 1 & 0\\ 0 & r^{2} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.108\right)\\ \left[ Z^{ij}\right] & =\left[ \begin{array} {cc} 1 & 0\\ 0 & \frac{1}{r^{2}} \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.109\right) \end{aligned} \]
For the shift tensor \(Z_{\alpha}^{i}\) and \(Z_{i\alpha}\), we have
\[ \begin{aligned} \left[ Z_{\alpha}^{i}\right] & =\left[ \begin{array} {c} r_{\gamma}\\ \theta_{\gamma} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.110\right)\\ \left[ Z_{i\alpha}\right] & =\left[ \begin{array} {c} r_{\gamma}\\ r^{2}\theta_{\gamma} \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.111\right) \end{aligned} \]
For the surface metric tensors \(S_{\alpha\beta}\) and \(S^{\alpha\beta}\), we have
\[ \begin{aligned} \left[ S_{\alpha\beta}\right] & =\left[ r_{\gamma}^{2}+r^{2} \theta_{\gamma}^{2}\right]\ \ \ \ \ \ \ \ \ \ \left(4.112\right)\\ \left[ S^{\alpha\beta}\right] & =\left[ \frac{1}{r_{\gamma}^{2} +r^{2}\theta_{\gamma}^{2}}\right] .\ \ \ \ \ \ \ \ \ \ \left(4.113\right) \end{aligned} \]
The line element \(\sqrt{S}\) equals the sole entry of \(\left[ S_{\alpha\beta }\right] \), i.e.
\[ \sqrt{S}=\sqrt{r_{\gamma}^{2}+r^{2}\theta_{\gamma}^{2}}.\tag{4.114} \]
For the remaining forms \(Z^{i\alpha}\) and \(Z_{i}^{\alpha}\) of the shift tensor, we have
\[ \begin{aligned} \left[ Z^{i\alpha}\right] & =\frac{1}{r_{\gamma}^{2}+r^{2}\theta_{\gamma }^{2}}\left[ \begin{array} {c} r_{\gamma}\\ \theta_{\gamma} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.115\right)\\ \left[ Z_{i}^{\alpha}\right] & =\frac{1}{r_{\gamma}^{2}+r^{2} \theta_{\gamma}^{2}}\left[ \begin{array} {c} r_{\gamma}\\ r^{2}\theta_{\gamma} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.116\right) \end{aligned} \]
The Christoffel symbol \(\Gamma_{\beta\gamma}^{\alpha}\) has the single entry
\[ \Gamma_{11}^{1}=\frac{rr_{\gamma}\theta_{\gamma}^{2}+r_{\gamma}r_{\gamma \gamma}+r^{2}\theta_{\gamma}\theta_{\gamma\gamma}}{r_{\gamma}^{2}+r^{2} \theta_{\gamma}^{2}}.\tag{4.117} \]
Finally, for the components \(N^{i}\) of the normal, we have
\[ \left[ N^{i}\right] =\frac{1}{\sqrt{r_{\gamma}^{2}+r^{2}\theta_{\gamma}^{2} }}\left[ \begin{array} {r} r\theta_{\gamma}\\ -r^{-1}r_{\gamma} \end{array} \right] .\tag{4.118} \]
Let us again catalogue the same two special cases as in the case of Cartesian ambient coordinates. First, refer the curve to arc length \(s\). In terms of the symbols \(r_{s}=r^{\prime}\left( s\right) \) and \(\theta_{s}=\theta^{\prime }\left( s\right) \), the above expressions simplify to the following:
\[ \begin{aligned} \left[ Z_{\alpha}^{i}\right] & =\left[ Z^{i\alpha}\right] =\left[ \begin{array} {c} r_{s}\\ \theta_{s} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.119\right)\\ \left[ S_{\alpha\beta}\right] & =\left[ S^{\alpha\beta}\right] =\left[ 1\right]\ \ \ \ \ \ \ \ \ \ \left(4.120\right)\\ \left[ Z_{i\alpha}\right] & =\left[ Z_{i}^{\alpha}\right] =\left[ \begin{array} {c} r_{s}\\ r^{2}\theta_{s} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.121\right)\\ \sqrt{S} & =1\ \ \ \ \ \ \ \ \ \ \left(4.122\right)\\ \Gamma_{\alpha\beta}^{\gamma} & =0\ \ \ \ \ \ \ \ \ \ \left(4.123\right)\\ \ \left[ N^{i}\right] & =\left[ \begin{array} {c} r\theta_{s}\\ -r^{-1}r_{s} \end{array} \right] .\ \ \ \ \ \ \ \ \ \ \left(4.124\right) \end{aligned} \]
Finally, for a curve that represents the graph of a function \(r=r\left( \theta\right) \), we have
\[ \begin{aligned} \left[ Z_{\alpha}^{i}\right] & =\left[ \begin{array} {c} r_{\theta}\\ 1 \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.125\right)\\ \left[ Z_{i\alpha}\right] & =\left[ \begin{array} {c} r_{\theta}\\ r^{2} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.126\right)\\ \left[ S_{\alpha\beta}\right] & =\left[ r_{\theta}^{2}+r^{2}\right]\ \ \ \ \ \ \ \ \ \ \left(4.127\right)\\ \left[ S^{\alpha\beta}\right] & =\left[ \frac{1}{r_{\theta}^{2}+r^{2} }\right] .\ \ \ \ \ \ \ \ \ \ \left(4.128\right)\\ \sqrt{S} & =\sqrt{r_{\theta}^{2}+r^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.129\right)\\ \left[ Z^{i\alpha}\right] & =\frac{1}{r_{\theta}^{2}+r^{2}}\left[ \begin{array} {c} r_{\theta}\\ 1 \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.130\right)\\ \left[ Z_{i}^{\alpha}\right] & =\frac{1}{r_{\theta}^{2}+r^{2}}\left[ \begin{array} {c} r_{\theta}\\ r^{2} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.131\right)\\ \Gamma_{11}^{1} & =\frac{r_{\theta}\left( r+r_{\theta\theta}\right) }{r_{\theta}^{2}+r^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.132\right)\\ \left[ N^{i}\right] & =\frac{1}{\sqrt{r_{\theta}^{2}+r^{2}}}\left[ \begin{array} {r} r\\ -r^{-1}r_{\theta} \end{array} \right]\ \ \ \ \ \ \ \ \ \ \left(4.133\right) \end{aligned} \]
Exercise 4.1Calculate all forms of the shift tensor and the covariant metric tensor \(S_{\alpha\beta}\) for a sphere of radius \(R\) described in cylindrical ambient coordinates \(r,\theta_{1},z\) by the equations
\[ \begin{aligned} r\left( \theta,\varphi\right) & =R\sin\theta\ \ \ \ \ \ \ \ \ \ \left(4.134\right)\\ \theta_{1}\left( \theta,\varphi\right) & =\varphi\ \ \ \ \ \ \ \ \ \ \left(4.135\right)\\ z\left( \theta,\varphi\right) & =R\cos\theta.\ \ \ \ \ \ \ \ \ \ \left(4.136\right) \end{aligned} \]
Confirm that \(S_{\alpha\beta}\) agrees with our earlier analyses.
Exercise 4.2Calculate all forms of the shift tensor and the covariant metric tensor \(S_{\alpha\beta}\) for a cylinder of radius \(R\) described in cylindrical ambient coordinates \(r,\theta_{1},z_{1}\) by the equations
\[ \begin{aligned} r\left( \theta,z\right) & =R\ \ \ \ \ \ \ \ \ \ \left(4.137\right)\\ \theta_{1}\left( \theta,z\right) & =\theta\ \ \ \ \ \ \ \ \ \ \left(4.138\right)\\ z_{1}\left( \theta,z\right) & =z.\ \ \ \ \ \ \ \ \ \ \left(4.139\right) \end{aligned} \]
Confirm that \(S_{\alpha\beta}\) agrees with our earlier analyses.
Exercise 4.3Calculate all forms of the shift tensor and the covariant metric tensor \(S_{\alpha\beta}\) for a cylinder of radius \(R\) described in spherical ambient coordinates \(r,\theta_{1},\varphi\) by the equations
\[ \begin{aligned} r\left( \theta,z\right) & =\sqrt{R^{2}+z^{2}}\ \ \ \ \ \ \ \ \ \ \left(4.140\right)\\ \theta_{1}\left( \theta,z\right) & =\arctan\frac{R}{z}\ \ \ \ \ \ \ \ \ \ \left(4.141\right)\\ \varphi\left( \theta,z\right) & =\theta.\ \ \ \ \ \ \ \ \ \ \left(4.142\right) \end{aligned} \]
Confirm that \(S_{\alpha\beta}\) agrees with our earlier analyses.
Exercise 4.4Explain why the transposes of the matrices \(\left[ Z_{i\alpha}\right] \) and \(\left[ Z_{i}^{\alpha}\right] \) have the same null space.
Exercise 4.5Calculate the components \(N^{i}\) of the unit normal to the sphere of radius \(R\) by finding a nonzero vector in the null space of the matrix
\[ \left[ Z_{\alpha}^{i}\right] =\left[ Z_{i\alpha}\right] =\left[ \begin{array} {ll} \phantom{-} R\cos\theta\cos\varphi & -R\sin\theta\sin\varphi\\ \phantom{-} R\cos\theta\sin\varphi & \phantom{-} R\sin\theta\cos\varphi\\ -R\sin\theta & 0 \end{array} \right] \tag{4.2} \]
and normalizing it to \(1\).
Exercise 4.6Use matrix algebra to calculate the elements of the surface Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\) on the sphere of radius \(R\) by considering the cases \(\alpha=1\) and \(\alpha=2\) separately.
Exercise 4.7Use matrix algebra to calculate the elements of the surface Christoffel symbol \(\Gamma_{\alpha\beta}^{\gamma}\) on the sphere of radius \(R\) by considering the cases \(\gamma=1\) and \(\gamma=2\) separately.
Exercise 4.8Fill in the details of the calculation in Section 4.3.
Exercise 4.9Repeat the calculation in Section 4.3 in cylindrical ambient components.
Exercise 4.10Derive the equation
\[ \Gamma_{11}^{1}=\frac{x^{\prime}\left( \gamma\right) x^{\prime\prime}\left( \gamma\right) +y^{\prime}\left( \gamma\right) y^{\prime\prime}\left( \gamma\right) }{x^{\prime}\left( \gamma\right) ^{2}+y^{\prime}\left( \gamma\right) ^{2}} \tag{4.88} \]
in two ways. First, by utilizing the formula
\[ \Gamma_{\alpha\beta}^{\gamma}=Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i} }{\partial S^{\beta}}, \tag{3.126} \]
and, second, by utilizing the formula
\[ \Gamma_{\beta\gamma}^{\alpha}=\frac{1}{2}S^{\alpha\omega}\left( \frac{\partial S_{\omega\beta}}{\partial S^{\gamma}}+\frac{\partial S_{\omega\gamma}}{\partial S^{\beta}}-\frac{\partial S_{\beta\gamma}}{\partial S^{\omega}}\right) . \tag{2.57} \]
Exercise 4.11Repeat the previous exercise for the equation
\[ \Gamma_{11}^{1}=\frac{rr_{\gamma}\theta_{\gamma}^{2}+r_{\gamma}r_{\gamma \gamma}+r^{2}\theta_{\gamma}\theta_{\gamma\gamma}}{r_{\gamma}^{2}+r^{2} \theta_{\gamma}^{2}} \tag{4.117} \]
except note that, in this case, you will need to use the equation
\[ \Gamma_{\alpha\beta}^{\gamma}=\Gamma_{ij}^{k}Z_{\alpha}^{i}Z_{\beta}^{j} Z_{k}^{\gamma}+Z_{i}^{\gamma}\frac{\partial Z_{\alpha}^{i}}{\partial S^{\beta }}. \tag{3.125} \]
Exercise 4.12Show that the surface Laplacian on a sphere with radius \(R\) is given by
\[ \nabla_{\alpha}\nabla^{\alpha}F=\frac{1}{R^{2}\sin\theta}\frac{\partial }{\partial\theta}\left( \sin\theta\frac{\partial F}{\partial\theta}\right) +\frac{1}{R^{2}\sin^{2}\theta}\frac{\partial^{2}F}{\partial\varphi^{2}}.\tag{4.143} \]
Exercise 4.13Show that the surface Laplacian on a cylinder with radius \(R\) is given by
\[ \nabla_{\alpha}\nabla^{\alpha}F=\frac{1}{R^{2}}\frac{\partial^{2}F} {\partial\theta^{2}}+\frac{\partial^{2}F}{\partial z^{2}}.\tag{4.144} \]
Exercise 4.14Show that the surface Laplacian on a torus with radii \(R\) and \(r\) is given by
\[ \nabla_{\alpha}\nabla^{\alpha}F=\frac{1}{\left( R+r\cos\varphi\right) ^{2} }\frac{\partial^{2}F}{\partial\theta^{2}}+\frac{1}{r^{2}\left( R+r\cos \varphi\right) }\frac{\partial}{\partial\varphi}\left( \left( R+r\cos\varphi\right) \frac{\partial F}{\partial\varphi}\right) .\tag{4.145} \]
Exercise 4.15Show that the surface Laplacian on the surface of a revolution whose profile is described by a function \(r\left( z\right) \) is given by
\[ \nabla_{\alpha}\nabla^{\alpha}F=\frac{1}{r\left( z\right) \sqrt{1+r^{\prime }\left( z\right) ^{2}}}\frac{\partial}{\partial z}\left( \frac{r\left( z\right) }{\sqrt{1+r^{\prime}\left( z\right) ^{2}}}\frac{\partial F}{\partial z}\right) +\frac{1}{r\left( z\right) ^{2}}\frac{\partial^{2} F}{\partial\theta^{2}}.\tag{4.146} \]
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