Chapter Questions
Show that the 'unit' basis vector fields for polar coordinates in the Euclidean plane, defined by$$\begin{aligned}& \hat{\mathbf{r}}=\cos \theta \hat{\mathbf{x}}+\sin \theta \hat{\mathbf{y}}, \\& \hat{\boldsymbol{\theta}}=-\sin \theta \hat{\mathbf{x}}+\cos \theta \hat{\mathbf{y}},\end{aligned}$$where $\hat{\mathbf{x}}=\partial / \partial x$ and $\hat{\mathbf{y}}=\partial / \partial y$, are a noncoordinate basis.
(a) Use (2.6) to prove (2.12).(b) Prove that$$\exp [a \mathrm{~d} / \mathrm{d} \lambda+b \mathrm{~d} / \mathrm{d} \mu]=\exp [a \mathrm{~d} / \mathrm{d} \lambda] \exp [b \mathrm{~d} / \mathrm{d} \mu]$$for all $a$ and $b$ if and only if $[\mathrm{d} / \mathrm{d} \lambda, \mathrm{d} / \mathrm{d} \mu]=0$.
Prove that any three twice-differentiable (i.e. $C^2$ ) vector fields $\bar{X}, \bar{Y}$ and $\bar{Z}$ satisfy the Jacobi identity$$[[\bar{X}, \bar{Y}], \bar{Z}]+[[\bar{Y}, \bar{Z}], \bar{X}]+[[\bar{Z}, \bar{X}], \bar{Y}]=0$$
A linear ('active') transformation in matrix algebra (e.g. an orthogonal rotation) transforms one matrix into another. Show that it is therefore a $\left(\frac{2}{2}\right)$ tensor when operating on matrices.
(a) Prove that a general $\binom{2}{0}$ tensor cannot be expressed as a simple outer product of two vectors. (Hint: count the number of components a $\left({ }_0^2\right)$ tensor may have.)(b) Prove that the $\binom{1}{1}$ tensor $\bar{V} \otimes \tilde{\omega}$ has components $V^i \omega_j$.
Prove that the set of all $\binom{2}{0}$ tensors at $P$ is a vector space under addition defined by analogy with equation (2.16b). Show that $e_i \otimes e_j$ is a basis for that space. (Thus, although a general $\binom{2}{0}$ tensor is not a simple outer product, it can be represented as a sum of such tensors.) This vector space is called $T_P \otimes T_P$.
How many different $\binom{2}{1}$ tensors may be made by contraction on pairs of indices of the $\binom{3}{2}$ tensor $Q^{i j h}{ }_{l m}$ ? How many $\binom{1}{0}$ tensors by a second contraction?
Let $\mathbf{A}$ and $B$ be two $\binom{1}{1}$ tensors, and regard them as vector-valued linear functions of vectors: if $\bar{V}$ is a vector then $\mathbf{A}(\bar{V})$ and $\mathbf{B}(\bar{V})$ are vectors. Show that if we define $\mathbf{C}(\bar{V})$ to be$$\mathbf{C}(\bar{V})=\mathbf{B}(\mathbf{A}(\bar{V})) \text {, }$$then $\mathbf{C}$ is a $\left(\frac{1}{1}\right)$ ) tensor as well. Show that its components are$$C^i{ }_j=B^i{ }_k A^k{ }_j \text {. }$$Discuss the relation of this with the linear transformation defined in §1.6.
XShow that a $\left.{ }_0^2\right)$ tensor's components transform as two vectors, i.e. $T^{i^{\prime} i^{\prime}}=\Lambda^{i^{\prime}}{ }_k \Lambda^{j^{\prime}}{ }_l T^{k l}$.Generalize this to type ( $\left.\begin{array}{l}N \\ N\end{array}\right)$.
Show that if a tensor's components are all zero in one basis, they are zero in all bases. (We then say the tensor is zero. It follows that if two tensors have equal components in one basis they are equal in all, and the tensors are said to be equal.)
Associated with a particular basis $\left\{\bar{e}_i\right\}$ of a vector space of dimension $n$, we are given some set of numbers $\left\{A^i{ }_j, i, j=1, \ldots, n\right\}$. We define another set of numbers $A^i{ }_{k^{\prime}}=\Lambda^i{ }_j \Lambda^l{ }_{k^{\prime}} A^j{ }_l$ and call them the components of the 'tensor' $A$ on the new basis $\left\{\bar{e}_j\right\}$. Show that this 'tensor' is indeed a tensor as we have defined it. This shows that one can take the point of view that a tensor is the collection $\left\{A^i{ }_j\right\}$ transforming in the given way. This is an alternative definition to the one we have used.
Find the matrices $\Lambda$ which cast the following matrices into their unit diagonal form:(a) $\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$,(b) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$(c) $\left(\begin{array}{rr}4 & 0 \\ 0 & -1\end{array}\right)$.
(a) Show that $\left\{g^{i j}\right\}$ are the components of a ( $\left.{ }_0^2\right)$ tensor $\left.g\right|^{-1}$, either by showing that they transform properly, or that they define a bilinear function on one-forms.(b) Show that if a vector basis $\left\{\bar{e}_i\right\}$ is orthonormal, so is its dual one-form basis $\left\{\widetilde{\omega}^i\right\}$, in the sense that $\left.\mathrm{g}\right|^{-1}\left(\widetilde{\omega}^i, \widetilde{\omega}^j\right)= \pm \delta^{i j}$.
Show that a $C^{\infty}$ metric tensor field $g$ is locally flat, in the sense that any point $P$ has a neighborhood in which there exists a coordinate system on whose basis the components $g_{i j}$ have the following properties:(i) $g_{i j}(P)= \pm \delta_{i j}$ (orthonormal form at $P$ )(ii) $\left.\frac{\partial g_{i j}}{\partial x^h}\right|_P=0 \quad$ (orthonormal form a good approximation near $P$ )(iii) $\left.\frac{\partial^2 g_{i j}}{\partial x^k \partial x^i}\right|_P$ not necessarily all zero(no truly orthonormal coordinate system)
In polar coordinates in the Euclidean plane, find the components of the metric on(a) the basis $\{\partial / \partial r, \partial / \partial \theta\}$;(b) the basis $\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}$ of exercise 2.1. Express $\hat{\mathbf{r}}, \hat{\theta}$ in terms of $\partial / \partial r$ and $\partial / \partial \theta$.
Find the components of $\tilde{\mathrm{d}} f$ and the vector $\overline{\mathrm{d}} f$ on both bases of exercise 2.15.