Part IB, 2008, Paper 1

# Part IB, 2008, Paper 1

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1.II.11F

commentState and prove the Contraction Mapping Theorem.

Let $(X, d)$ be a nonempty complete metric space and $f: X \rightarrow X$ a mapping such that, for some $k>0$, the $k$ th iterate $f^{k}$ of $f$ (that is, $f$ composed with itself $k$ times) is a contraction mapping. Show that $f$ has a unique fixed point.

Now let $X$ be the space of all continuous real-valued functions on $[0,1]$, equipped with the uniform norm $\|h\|_{\infty}=\sup \{|h(t)|: t \in[0,1]\}$, and let $\phi: \mathbb{R} \times[0,1] \rightarrow \mathbb{R}$ be a continuous function satisfying the Lipschitz condition

$|\phi(x, t)-\phi(y, t)| \leqslant M|x-y|$

for all $t \in[0,1]$ and all $x, y \in \mathbb{R}$, where $M$ is a constant. Let $F: X \rightarrow X$ be defined by

$F(h)(t)=g(t)+\int_{0}^{t} \phi(h(s), s) d s$

where $g$ is a fixed continuous function on $[0,1]$. Show by induction on $n$ that

$\left|F^{n}(h)(t)-F^{n}(k)(t)\right| \leqslant \frac{M^{n} t^{n}}{n !}\|h-k\|_{\infty}$

for all $h, k \in X$ and all $t \in[0,1]$. Deduce that the integral equation

$f(t)=g(t)+\int_{0}^{t} \phi(f(s), s) d s$

has a unique continuous solution $f$ on $[0,1]$.

1.I.3C

commentGiven that $f(z)$ is an analytic function, show that the mapping $w=f(z)$

(a) preserves angles between smooth curves intersecting at $z$ if $f^{\prime}(z) \neq 0$;

(b) has Jacobian given by $\left|f^{\prime}(z)\right|^{2}$.

1.II.13C

commentBy a suitable choice of contour show the following:

(a)

$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$

where $n>1$,

(b)

$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x=\frac{\pi^{2}}{2 \sqrt{2}}$

1.II.16B

commentSuppose that the current density $\mathbf{J}(\mathbf{r})$ is constant in time but the charge density $\rho(\mathbf{r}, t)$ is not.

(i) Show that $\rho$ is a linear function of time:

$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0)+\dot{\rho}(\mathbf{r}, 0) t$

where $\dot{\rho}(\mathbf{r}, 0)$ is the time derivative of $\rho$ at time $t=0$.

(ii) The magnetic induction due to a current density $\mathbf{J}(\mathbf{r})$ can be written as

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d V^{\prime}$

Show that this can also be written as

$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \nabla \times \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$

(iii) Assuming that $\mathbf{J}$ vanishes at infinity, show that the curl of the magnetic field in (1) can be written as

$\nabla \times \mathbf{B}(\mathbf{r})=\mu_{0} \mathbf{J}(\mathbf{r})+\frac{\mu_{0}}{4 \pi} \nabla \int \frac{\nabla^{\prime} \cdot \mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$

[You may find useful the identities $\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$ and also $\left.\nabla^{2}\left(1 /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)=-4 \pi \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) .\right]$

(iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that $\mathbf{B}$ itself obeys Ampère's law with Maxwell's displacement current term, i.e. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \epsilon_{0} \partial \mathbf{E} / \partial t$.

1.I.5B

commentVerify that the two-dimensional flow given in Cartesian coordinates by

$\mathbf{u}=\left(\mathrm{e}^{y} \sinh x,-\mathrm{e}^{y} \cosh x\right)$

satisfies $\nabla \cdot \mathbf{u}=0$. Find the stream function $\psi(x, y)$. Sketch the streamlines.

1.II.17B

commentTwo incompressible fluids flow in infinite horizontal streams, the plane of contact being $z=0$, with $z$ positive upwards. The flow is given by

$\mathbf{U}(\mathbf{r})= \begin{cases}U_{2} \hat{\mathbf{e}}_{x}, & z>0 \\ U_{1} \hat{\mathbf{e}}_{x}, & z<0\end{cases}$

where $\hat{\mathbf{e}}_{x}$ is the unit vector in the positive $x$ direction. The upper fluid has density $\rho_{2}$ and pressure $p_{0}-g \rho_{2} z$, the lower has density $\rho_{1}$ and pressure $p_{0}-g \rho_{1} z$, where $p_{0}$ is a constant and $g$ is the acceleration due to gravity.

(i) Consider a perturbation to the flat surface $z=0$ of the form

$z \equiv \zeta(x, y, t)=\zeta_{0} e^{i(k x+\ell y)+s t} .$

State the kinematic boundary conditions on the velocity potentials $\phi_{i}$ that hold on the interface in the two domains, and show by linearising in $\zeta$ that they reduce to

$\frac{\partial \phi_{i}}{\partial z}=\frac{\partial \zeta}{\partial t}+U_{i} \frac{\partial \zeta}{\partial x} \quad(z=0, i=1,2) .$

(ii) State the dynamic boundary condition on the perturbed interface, and show by linearising in $\zeta$ that it reduces to

$\rho_{1}\left(U_{1} \frac{\partial \phi_{1}}{\partial x}+\frac{\partial \phi_{1}}{\partial t}+g \zeta\right)=\rho_{2}\left(U_{2} \frac{\partial \phi_{2}}{\partial x}+\frac{\partial \phi_{2}}{\partial t}+g \zeta\right) \quad(z=0)$

(iii) Use the velocity potentials

$\phi_{1}=U_{1} x+A_{1} e^{q z} e^{i(k x+\ell y)+s t}, \quad \phi_{2}=U_{2} x+A_{2} e^{-q z} e^{i(k x+\ell y)+s t},$

where $q=\sqrt{k^{2}+\ell^{2}}$, and the conditions in (i) and (ii) to perform a stability analysis. Show that the relation between $s, k$ and $\ell$ is

$s=-i k \frac{\rho_{1} U_{1}+\rho_{2} U_{2}}{\rho_{1}+\rho_{2}} \pm\left[\frac{k^{2} \rho_{1} \rho_{2}\left(U_{1}-U_{2}\right)^{2}}{\left(\rho_{1}+\rho_{2}\right)^{2}}-\frac{q g\left(\rho_{1}-\rho_{2}\right)}{\rho_{1}+\rho_{2}}\right]^{1 / 2} .$

Find the criterion for instability.

1.I.2G

commentShow that any element of $S O(3, \mathbb{R})$ is a rotation, and that it can be written as the product of two reflections.

1.II.10G

comment(i) Show that $A_{4}$ is not simple.

(ii) Show that the group Rot $(D)$ of rotational symmetries of a regular dodecahedron is a simple group of order 60 .

(iii) Show that $\operatorname{Rot}(D)$ is isomorphic to $A_{5}$.

1.I.1E

commentLet $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

(i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

(ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

(iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

(iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.

1.II.9E

commentLet $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

Hence, or otherwise, prove that

$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$

1.II.19H

commentThe village green is ringed by a fence with $N$ fenceposts, labelled $0,1, \ldots, N-1$. The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0 , and then chooses one of the two nearest neighbours, 1 or $N-1$, with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.

1.II.14D

commentWrite down the Euler-Lagrange equation for the variational problem for $y(x)$ that extremizes the integral $I$ defined as

$I=\int_{x_{1}}^{x_{2}} f\left(x, y, y^{\prime}\right) d x$

with boundary conditions $y\left(x_{1}\right)=y_{1}, y\left(x_{2}\right)=y_{2}$, where $y_{1}$ and $y_{2}$ are positive constants such that $y_{2}>y_{1}$, with $x_{2}>x_{1}$. Find a first integral of the equation when $f$ is independent of $y$, i.e. $f=f\left(x, y^{\prime}\right)$.

A light ray moves in the $(x, y)$ plane from $\left(x_{1}, y_{1}\right)$ to $\left(x_{2}, y_{2}\right)$ with speed $c(x)$ taking a time $T$. Show that the equation of the path that makes $T$ an extremum satisfies

$\frac{d y}{d x}=\frac{c(x)}{\sqrt{k^{2}-c^{2}(x)}}$

where $k$ is a constant and write down an integral relating $k, x_{1}, x_{2}, y_{1}$ and $y_{2}$.

When $c(x)=a x$ where $a$ is a constant and $k=a x_{2}$, show that the path is given by

$\left(y_{2}-y\right)^{2}=x_{2}^{2}-x^{2} .$

1.II.12F

commentWrite down the definition of a topology on a set $X$.

For each of the following families $\mathcal{T}$ of subsets of $\mathbb{Z}$, determine whether $\mathcal{T}$ is a topology on $\mathbb{Z}$. In the cases where the answer is 'yes', determine also whether $(\mathbb{Z}, \mathcal{T})$ is a Hausdorff space and whether it is compact.

(a) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $U$ is finite or $0 \in U\}$.

(b) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : either $\mathbb{Z} \backslash U$ is finite or $0 \notin U\}$.

(c) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : there exists $k>0$ such that, for all $n, n \in U \Leftrightarrow n+k \in U\}$.

(d) $\mathcal{T}=\{U \subseteq \mathbb{Z}$ : for all $n \in U$, there exists $k>0$ such that $\{n+k m: m \in \mathbb{Z}\} \subseteq U\}$.

1.I.6D

commentShow that if $A=L D L^{T}$, where $L \in \mathbb{R}^{m \times m}$ is a lower triangular matrix with all elements on the main diagonal being unity and $D \in \mathbb{R}^{m \times m}$ is a diagonal matrix with positive elements, then $A$ is positive definite. Find $L$ and the corresponding $D$ when

$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 3 \end{array}\right]$

1.I.8H

commentState the Lagrangian Sufficiency Theorem for the maximization over $x$ of $f(x)$ subject to the constraint $g(x)=b$.

For each $p>0$, solve

$\max \sum_{i=1}^{d} x_{i}^{p} \quad \text { subject to } \sum_{i=1}^{d} x_{i}=1, \quad x_{i} \geqslant 0 .$

1.II.15A

commentThe radial wavefunction $g(r)$ for the hydrogen atom satisfies the equation

$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d g(r)}{d r}\right)-\frac{e^{2} g(r)}{4 \pi \epsilon_{0} r}+\hbar^{2} \frac{\ell(\ell+1)}{2 m r^{2}} g(r)=E g(r)$

With reference to the general form for the time-independent Schrödinger equation, explain the origin of each term. What are the allowed values of $\ell$ ?

The lowest-energy bound-state solution of $(*)$, for given $\ell$, has the form $r^{\alpha} e^{-\beta r}$. Find $\alpha$ and $\beta$ and the corresponding energy $E$ in terms of $\ell$.

A hydrogen atom makes a transition between two such states corresponding to $\ell+1$ and $\ell$. What is the frequency of the emitted photon?

1.I.4C

commentIn an inertial frame $S$ a photon of energy $E$ is observed to travel at an angle $\theta$ relative to the $x$-axis. The inertial frame $S^{\prime}$ moves relative to $S$ at velocity $v$ in the $x$ direction and the $x^{\prime}$-axis of $S^{\prime}$ is taken parallel to the $x$-axis of $S$. Observed in $S^{\prime}$, the photon has energy $E^{\prime}$ and travels at an angle $\theta^{\prime}$ relative to the $x^{\prime}$-axis. Show that

$E^{\prime}=\frac{E(1-\beta \cos \theta)}{\sqrt{1-\beta^{2}}}, \quad \cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta},$

where $\beta=v / c$.

1.II.18H

commentSuppose that $X_{1}, \ldots, X_{n}$ is a sample of size $n$ with common $N\left(\mu_{X}, 1\right)$ distribution, and $Y_{1}, \ldots, Y_{n}$ is an independent sample of size $n$ from a $N\left(\mu_{Y}, 1\right)$ distribution.

(i) Find (with careful justification) the form of the size- $\alpha$ likelihood-ratio test of the null hypothesis $H_{0}: \mu_{Y}=0$ against alternative $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted.

(ii) Find the form of the size- $\alpha$ likelihood-ratio test of the hypothesis

$H_{0}: \mu_{X} \geqslant A, \mu_{Y}=0$

against $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted, where $A$ is a given constant.

Compare the critical regions you obtain in (i) and (ii) and comment briefly.