# Part IB, 2021

### Jump to course

Paper 1, Section II, F

commentLet $f: X \rightarrow Y$ be a map between metric spaces. Prove that the following two statements are equivalent:

(i) $f^{-1}(A) \subset X$ is open whenever $A \subset Y$ is open.

(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.

For $f: X \rightarrow Y$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.

(a) If $X$ is compact and $f$ is continuous, then $f$ is uniformly continuous.

(b) If $X$ is compact and $f$ is continuous, then $Y$ is compact.

(c) If $X$ is connected, $f$ is continuous and $f(X)$ is dense in $Y$, then $Y$ is connected.

(d) If the set $\{(x, y) \in X \times Y: y=f(x)\}$ is closed in $X \times Y$ and $Y$ is compact, then $f$ is continuous.

Paper 2, Section I, $2 F$

commentLet $K:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be a continuous function and let $C([0,1])$ denote the set of continuous real-valued functions on $[0,1]$. Given $f \in C([0,1])$, define the function $T f$ by the expression

$T f(x)=\int_{0}^{1} K(x, y) f(y) d y$

(a) Prove that $T$ is a continuous map $C([0,1]) \rightarrow C([0,1])$ with the uniform metric on $C([0,1])$.

(b) Let $d_{1}$ be the metric on $C([0,1])$ given by

$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$

Is $T$ continuous with respect to $d_{1} ?$

Paper 2, Section II, F

commentLet $k_{n}: \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions satisfying the following properties:

$k_{n}(x) \geqslant 0$ for all $n$ and $x \in \mathbb{R}$ and there is $R>0$ such that $k_{n}$ vanishes outside $[-R, R]$ for all $n$

each $k_{n}$ is continuous and

$\int_{-\infty}^{\infty} k_{n}(t) d t=1$

- given $\varepsilon>0$ and $\delta>0$, there exists a positive integer $N$ such that if $n \geqslant N$, then

$\int_{-\infty}^{-\delta} k_{n}(t) d t+\int_{\delta}^{\infty} k_{n}(t) d t<\varepsilon$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a bounded continuous function and set

$f_{n}(x):=\int_{-\infty}^{\infty} k_{n}(t) f(x-t) d t$

Show that $f_{n}$ converges uniformly to $f$ on any compact subset of $\mathbb{R}$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be a continuous function with $g(0)=g(1)=0$. Show that there is a sequence of polynomials $p_{n}$ such that $p_{n}$ converges uniformly to $g$ on $[0,1]$. $[$ Hint: consider the functions

$k_{n}(t)= \begin{cases}\left(1-t^{2}\right)^{n} / c_{n} & t \in[-1,1] \\ 0 & \text { otherwise }\end{cases}$

where $c_{n}$ is a suitably chosen constant.]

Paper 3, Section II, F

commentDefine the terms connected and path-connected for a topological space. Prove that the interval $[0,1]$ is connected and that if a topological space is path-connected, then it is connected.

Let $X$ be an open subset of Euclidean space $\mathbb{R}^{n}$. Show that $X$ is connected if and only if $X$ is path-connected.

Let $X$ be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in $\mathbb{R}^{n}$. Assume $X$ is connected; must $X$ be also pathconnected? Briefly justify your answer.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} A=\{(x, 0): x \in(0,1]\}, \quad B=\{(0, y): y \in[1 / 2,1]\}, \text { and } \\ C_{n}=\{(1 / n, y): y \in[0,1]\} \text { for } n \geqslant 1 \end{gathered}$

Let

$X=A \cup B \cup \bigcup_{n \geqslant 1} C_{n}$

with the subspace topology. Is $X$ path-connected? Is $X$ connected? Justify your answers.

Paper 4, Section I, $2 F$

commentLet $X$ be a topological space with an equivalence relation, $\tilde{X}$ the set of equivalence classes, $\pi: X \rightarrow \tilde{X}$, the quotient map taking a point in $X$ to its equivalence class.

(a) Define the quotient topology on $\tilde{X}$ and check it is a topology.

(b) Prove that if $Y$ is a topological space, a map $f: \tilde{X} \rightarrow Y$ is continuous if and only if $f \circ \pi$ is continuous.

(c) If $X$ is Hausdorff, is it true that $\tilde{X}$ is also Hausdorff? Justify your answer.

Paper 4, Section II, F

comment(a) Let $g:[0,1] \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a continuous function such that for each $t \in[0,1]$, the partial derivatives $D_{i} g(t, x)(i=1, \ldots, n)$ of $x \mapsto g(t, x)$ exist and are continuous on $[0,1] \times \mathbb{R}^{n}$. Define $G: \mathbb{R}^{n} \rightarrow \mathbb{R}$ by

$G(x)=\int_{0}^{1} g(t, x) d t$

Show that $G$ has continuous partial derivatives $D_{i} G$ given by

$D_{i} G(x)=\int_{0}^{1} D_{i} g(t, x) d t$

for $i=1, \ldots, n$.

(b) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be an infinitely differentiable function, that is, partial derivatives $D_{i_{1}} D_{i_{2}} \cdots D_{i_{k}} f$ exist and are continuous for all $k \in \mathbb{N}$ and $i_{1}, \ldots, i_{k} \in\{1,2\}$. Show that for any $\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$,

$f\left(x_{1}, x_{2}\right)=f\left(x_{1}, 0\right)+x_{2} D_{2} f\left(x_{1}, 0\right)+x_{2}^{2} h\left(x_{1}, x_{2}\right)$

where $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is an infinitely differentiable function.

[Hint: You may use the fact that if $u: \mathbb{R} \rightarrow \mathbb{R}$ is infinitely differentiable, then

$\left.u(1)=u(0)+u^{\prime}(0)+\int_{0}^{1}(1-t) u^{\prime \prime}(t) d t .\right]$

Paper 3, Section II, G

commentLet $\gamma$ be a curve (not necessarily closed) in $\mathbb{C}$ and let $[\gamma]$ denote the image of $\gamma$. Let $\phi:[\gamma] \rightarrow \mathbb{C}$ be a continuous function and define

$f(z)=\int_{\gamma} \frac{\phi(\lambda)}{\lambda-z} d \lambda$

for $z \in \mathbb{C} \backslash[\gamma]$. Show that $f$ has a power series expansion about every $a \notin[\gamma]$.

Using Cauchy's Integral Formula, show that a holomorphic function has complex derivatives of all orders. [Properties of power series may be assumed without proof.] Let $f$ be a holomorphic function on an open set $U$ that contains the closed disc $\bar{D}(a, r)$. Obtain an integral formula for the derivative of $f$ on the open disc $D(a, r)$ in terms of the values of $f$ on the boundary of the disc.

Show that if holomorphic functions $f_{n}$ on an open set $U$ converge locally uniformly to a holomorphic function $f$ on $U$, then $f_{n}^{\prime}$ converges locally uniformly to $f^{\prime}$.

Let $D_{1}$ and $D_{2}$ be two overlapping closed discs. Let $f$ be a holomorphic function on some open neighbourhood of $D=D_{1} \cap D_{2}$. Show that there exist open neighbourhoods $U_{j}$ of $D_{j}$ and holomorphic functions $f_{j}$ on $U_{j}, j=1,2$, such that $f(z)=f_{1}(z)+f_{2}(z)$ on $U_{1} \cap U_{2}$.

Paper 4, Section I, $3 G$

commentLet $f$ be a holomorphic function on a neighbourhood of $a \in \mathbb{C}$. Assume that $f$ has a zero of order $k$ at $a$ with $k \geqslant 1$. Show that there exist $\varepsilon>0$ and $\delta>0$ such that for any $b$ with $0<|b|<\varepsilon$ there are exactly $k$ distinct values of $z \in D(a, \delta)$ with $f(z)=b$.

Paper 1, Section I, B

commentLet $x>0, x \neq 2$, and let $C_{x}$ denote the positively oriented circle of radius $x$ centred at the origin. Define

$g(x)=\oint_{C_{x}} \frac{z^{2}+e^{z}}{z^{2}(z-2)} d z$

Evaluate $g(x)$ for $x \in(0, \infty) \backslash\{2\}$.

Paper 1, Section II, G

comment(a) State a theorem establishing Laurent series of analytic functions on suitable domains. Give a formula for the $n^{\text {th }}$Laurent coefficient.

Define the notion of isolated singularity. State the classification of an isolated singularity in terms of Laurent coefficients.

Compute the Laurent series of

$f(z)=\frac{1}{z(z-1)}$

on the annuli $A_{1}=\{z: 0<|z|<1\}$ and $A_{2}=\{z: 1<|z|\}$. Using this example, comment on the statement that Laurent coefficients are unique. Classify the singularity of $f$ at 0 .

(b) Let $U$ be an open subset of the complex plane, let $a \in U$ and let $U^{\prime}=U \backslash\{a\}$. Assume that $f$ is an analytic function on $U^{\prime}$ with $|f(z)| \rightarrow \infty$ as $z \rightarrow a$. By considering the Laurent series of $g(z)=\frac{1}{f(z)}$ at $a$, classify the singularity of $f$ at $a$ in terms of the Laurent coefficients. [You may assume that a continuous function on $U$ that is analytic on $U^{\prime}$ is analytic on $U$.]

Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function with $|f(z)| \rightarrow \infty$ as $z \rightarrow \infty$. By considering Laurent series at 0 of $f(z)$ and of $h(z)=f\left(\frac{1}{z}\right)$, show that $f$ is a polynomial.

(c) Classify, giving reasons, the singularity at the origin of each of the following functions and in each case compute the residue:

$g(z)=\frac{\exp (z)-1}{z \log (z+1)} \quad \text { and } \quad h(z)=\sin (z) \sin (1 / z)$

Paper 2, Section II, B

comment(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function and let $a>0, b>0$ be constants. Show that if

$|f(z)| \leqslant a|z|^{n / 2}+b$

for all $z \in \mathbb{C}$, where $n$ is a positive odd integer, then $f$ must be a polynomial with degree not exceeding $\lfloor n / 2\rfloor$ (closest integer part rounding down).

Does there exist a function $f$, analytic in $\mathbb{C} \backslash\{0\}$, such that $|f(z)| \geqslant 1 / \sqrt{|z|}$ for all nonzero $z ?$ Justify your answer.

(b) State Liouville's Theorem and use it to show the following.

(i) If $u$ is a positive harmonic function on $\mathbb{R}^{2}$, then $u$ is a constant function.

(ii) Let $L=\{z \mid z=a x+b, x \in \mathbb{R}\}$ be a line in $\mathbb{C}$ where $a, b \in \mathbb{C}, a \neq 0$. If $f: \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that $f(\mathbb{C}) \cap L=\emptyset$, then $f$ is a constant function.

Paper 3, Section I, B

commentFind the value of $A$ for which the function

$\phi(x, y)=x \cosh y \sin x+A y \sinh y \cos x$

satisfies Laplace's equation. For this value of $A$, find a complex analytic function of which $\phi$ is the real part.

Paper 4, Section II, B

commentLet $f(t)$ be defined for $t \geqslant 0$. Define the Laplace transform $\widehat{f}(s)$ of $f$. Find an expression for the Laplace transform of $\frac{d f}{d t}$ in terms of $\widehat{f}$.

Three radioactive nuclei decay sequentially, so that the numbers $N_{i}(t)$ of the three types obey the equations

$\begin{aligned} \frac{d N_{1}}{d t} &=-\lambda_{1} N_{1} \\ \frac{d N_{2}}{d t} &=\lambda_{1} N_{1}-\lambda_{2} N_{2} \\ \frac{d N_{3}}{d t} &=\lambda_{2} N_{2}-\lambda_{3} N_{3} \end{aligned}$

where $\lambda_{3}>\lambda_{2}>\lambda_{1}>0$ are constants. Initially, at $t=0, N_{1}=N, N_{2}=0$ and $N_{3}=n$. Using Laplace transforms, find $N_{3}(t)$.

By taking an appropriate limit, find $N_{3}(t)$ when $\lambda_{2}=\lambda_{1}=\lambda>0$ and $\lambda_{3}>\lambda$.

Paper 1, Section II, 15D

comment(a) Show that the magnetic flux passing through a simple, closed curve $C$ can be written as

$\Phi=\oint_{C} \mathbf{A} \cdot \mathbf{d} \mathbf{x},$

where $\mathbf{A}$ is the magnetic vector potential. Explain why this integral is independent of the choice of gauge.

(b) Show that the magnetic vector potential due to a static electric current density $\mathbf{J}$, in the Coulomb gauge, satisfies Poisson's equation

$-\nabla^{2} \mathbf{A}=\mu_{0} \mathbf{J}$

Hence obtain an expression for the magnetic vector potential due to a static, thin wire, in the form of a simple, closed curve $C$, that carries an electric current $I$. [You may assume that the electric current density of the wire can be written as

$\mathbf{J}(\mathbf{x})=I \int_{C} \delta^{(3)}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \mathbf{d} \mathbf{x}^{\prime}$

where $\delta^{(3)}$ is the three-dimensional Dirac delta function.]

(c) Consider two thin wires, in the form of simple, closed curves $C_{1}$ and $C_{2}$, that carry electric currents $I_{1}$ and $I_{2}$, respectively. Let $\Phi_{i j}$ (where $i, j \in\{1,2\}$ ) be the magnetic flux passing through the curve $C_{i}$ due to the current $I_{j}$ flowing around $C_{j}$. The inductances are defined by $L_{i j}=\Phi_{i j} / I_{j}$. By combining the results of parts (a) and (b), or otherwise, derive Neumann's formula for the mutual inductance,

$L_{12}=L_{21}=\frac{\mu_{0}}{4 \pi} \oint_{C_{1}} \oint_{C_{2}} \frac{\mathbf{d} \mathbf{x}_{1} \cdot \mathbf{d} \mathbf{x}_{2}}{\left|\mathbf{x}_{1}-\mathbf{x}_{2}\right|} .$

Suppose that $C_{1}$ is a circular loop of radius $a$, centred at $(0,0,0)$ and lying in the plane $z=0$, and that $C_{2}$ is a different circular loop of radius $b$, centred at $(0,0, c)$ and lying in the plane $z=c$. Show that the mutual inductance of the two loops is

$\frac{\mu_{0}}{4} \sqrt{a^{2}+b^{2}+c^{2}} f(q)$

where

$q=\frac{2 a b}{a^{2}+b^{2}+c^{2}}$

and the function $f(q)$ is defined, for $0<q<1$, by the integral

$f(q)=\int_{0}^{2 \pi} \frac{q \cos \theta d \theta}{\sqrt{1-q \cos \theta}}$

Paper 2, Section I, $4 \mathrm{D}$

commentState Gauss's Law in the context of electrostatics.

A simple coaxial cable consists of an inner conductor in the form of a perfectly conducting, solid cylinder of radius $a$, surrounded by an outer conductor in the form of a perfectly conducting, cylindrical shell of inner radius $b>a$ and outer radius $c>b$. The cylinders are coaxial and the gap between them is filled with a perfectly insulating material. The cable may be assumed to be straight and arbitrarily long.

In a steady state, the inner conductor carries an electric charge $+Q$ per unit length, and the outer conductor carries an electric charge $-Q$ per unit length. The charges are distributed in a cylindrically symmetric way and no current flows through the cable.

Determine the electrostatic potential and the electric field as functions of the cylindrical radius $r$, for $0<r<\infty$. Calculate the capacitance $C$ of the cable per unit length and the electrostatic energy $U$ per unit length, and verify that these are related by

$U=\frac{Q^{2}}{2 C}$

Paper 2, Section II, $16 \mathrm{D}$

comment(a) Show that, for $|\mathbf{x}| \gg|\mathbf{y}|$,

$\frac{1}{|\mathbf{x}-\mathbf{y}|}=\frac{1}{|\mathbf{x}|}\left[1+\frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}|^{2}}+\frac{3(\mathbf{x} \cdot \mathbf{y})^{2}-|\mathbf{x}|^{2}|\mathbf{y}|^{2}}{2|\mathbf{x}|^{4}}+O\left(\frac{|\mathbf{y}|^{3}}{|\mathbf{x}|^{3}}\right)\right]$

(b) A particle with electric charge $q>0$ has position vector $(a, 0,0)$, where $a>0$. An earthed conductor (held at zero potential) occupies the plane $x=0$. Explain why the boundary conditions can be satisfied by introducing a fictitious 'image' particle of appropriate charge and position. Hence determine the electrostatic potential and the electric field in the region $x>0$. Find the leading-order approximation to the potential for $|\mathbf{x}| \gg a$ and compare with that of an electric dipole. Directly calculate the total flux of the electric field through the plane $x=0$ and comment on the result. Find the induced charge distribution on the surface of the conductor, and the total induced surface charge. Sketch the electric field lines in the plane $z=0$.

(c) Now consider instead a particle with charge $q$ at position $(a, b, 0)$, where $a>0$ and $b>0$, with earthed conductors occupying the planes $x=0$ and $y=0$. Find the leading-order approximation to the potential in the region $x, y>0$ for $|\mathbf{x}| \gg a, b$ and state what type of multipole potential this is.

Paper 3, Section II, 15D

comment(a) The energy density stored in the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ is given by

$w=\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}$

Show that, in regions where no electric current flows,

$\frac{\partial w}{\partial t}+\boldsymbol{\nabla} \cdot \mathbf{S}=0$

for some vector field $\mathbf{S}$ that you should determine.

(b) The coordinates $x^{\prime \mu}=\left(c t^{\prime}, \mathbf{x}^{\prime}\right)$ in an inertial frame $\mathcal{S}^{\prime}$ are related to the coordinates $x^{\mu}=(c t, \mathbf{x})$ in an inertial frame $\mathcal{S}$ by a Lorentz transformation $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$, where

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

with $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$. Here $v$ is the relative velocity of $\mathcal{S}^{\prime}$ with respect to $\mathcal{S}$ in the x-direction.

In frame $\mathcal{S}^{\prime}$, there is a static electric field $\mathbf{E}^{\prime}\left(\mathbf{x}^{\prime}\right)$ with $\partial \mathbf{E}^{\prime} / \partial t^{\prime}=0$, and no magnetic field. Calculate the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in frame $\mathcal{S}$. Show that the energy density in frame $\mathcal{S}$ is given in terms of the components of $\mathbf{E}^{\prime}$ by

$w=\frac{\epsilon_{0}}{2}\left[E_{x}^{\prime 2}+\left(\frac{c^{2}+v^{2}}{c^{2}-v^{2}}\right)\left(E_{y}^{\prime 2}+E_{z}^{\prime 2}\right)\right]$

Use the fact that $\partial w / \partial t^{\prime}=0$ to show that

$\frac{\partial w}{\partial t}+\nabla \cdot\left(w v \mathbf{e}_{x}\right)=0$

where $\mathbf{e}_{x}$ is the unit vector in the $x$-direction.

Paper 4, Section I, $5 \mathrm{D}$

commentWrite down Maxwell's equations in a vacuum. Show that they admit wave solutions with

$\mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left[\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right]$

where $\mathbf{B}_{0}, \mathbf{k}$ and $\omega$ must obey certain conditions that you should determine. Find the corresponding electric field $\mathbf{E}(\mathbf{x}, t)$.

A light wave, travelling in the $x$-direction and linearly polarised so that the magnetic field points in the $z$-direction, is incident upon a conductor that occupies the half-space $x>0$. The electric and magnetic fields obey the boundary conditions $\mathbf{E} \times \mathbf{n}=\mathbf{0}$ and $\mathbf{B} \cdot \mathbf{n}=0$ on the surface of the conductor, where $\mathbf{n}$ is the unit normal vector. Determine the contributions to the magnetic field from the incident and reflected waves in the region $x \leqslant 0$. Compute the magnetic field tangential to the surface of the conductor.

Paper 1, Section II, A

commentA two-dimensional flow is given by a velocity potential

$\phi(x, y, t)=\epsilon y \sin (x-t)$

where $\epsilon$ is a constant.

(a) Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\boldsymbol{\nabla} \cdot \mathbf{u}$.

(b) The time-average $\langle\psi\rangle(x, y)$ of a quantity $\psi(x, y, t)$ is defined as

$\langle\psi\rangle(x, y)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t .$

Show that the time-average of this velocity field is zero everywhere. Write down an expression for the acceleration of fluid particles, and find the time-average of this expression at a fixed point $(x, y)$.

(c) Now assume that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at $t=0$ is marked with dye. Write down equations for its subsequent motion. Verify that its position $(x, y)$ for $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) by

$\begin{aligned} x &=\epsilon^{2}\left(\frac{1}{4} \sin 2 t+\frac{t}{2}-\sin t\right) \\ y &=\epsilon(\cos t-1) \end{aligned}$

Deduce the time-average velocity of the dyed particle correct to this order.

Paper 2, Section I, A

commentConsider an axisymmetric container, initially filled with water to a depth $h_{I}$. A small circular hole of radius $r_{0}$ is opened in the base of the container at $z=0$.

(a) Determine how the radius $r$ of the container should vary with $z<h_{I}$ so that the depth of the water will decrease at a constant rate.

(b) For such a container, determine how the cross-sectional area $A$ of the free surface should decrease with time.

[You may assume that the flow rate through the opening is sufficiently small that Bernoulli's theorem for steady flows can be applied.]

Paper 3, Section I, A

commentA two-dimensional flow $\mathbf{u}=(u, v)$ has a velocity field given by

$u=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \quad \text { and } \quad v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}$

(a) Show explicitly that this flow is incompressible and irrotational away from the origin.

(b) Find the stream function for this flow.

(c) Find the velocity potential for this flow.

Paper 3, Section II, A

commentA two-dimensional layer of viscous fluid lies between two rigid boundaries at $y=\pm L_{0}$. The boundary at $y=L_{0}$ oscillates in its own plane with velocity $\left(U_{0} \cos \omega t, 0\right)$, while the boundary at $y=-L_{0}$ oscillates in its own plane with velocity $\left(-U_{0} \cos \omega t, 0\right)$. Assume that there is no pressure gradient and that the fluid flows parallel to the boundary with velocity $(u(y, t), 0)$, where $u(y, t)$ can be written as $u(y, t)=\operatorname{Re}\left[U_{0} f(y) \exp (i \omega t)\right]$.

(a) By exploiting the symmetry of the system or otherwise, show that

$f(y)=\frac{\sinh [(1+i) \Delta \hat{y}]}{\sinh [(1+i) \Delta]}, \text { where } \hat{y}=\frac{y}{L_{0}} \text { and } \Delta=\left(\frac{\omega L_{0}^{2}}{2 \nu}\right)^{1 / 2}$

(b) Hence or otherwise, show that

where $\Delta_{\pm}=\Delta(1 \pm \hat{y})$.

(c) Show that, for $\Delta \ll 1$,

$u(y, t) \simeq \frac{U_{0} y}{L_{0}} \cos \omega t$

and briefly interpret this result physically.

$\begin{aligned} & \frac{u(y, t)}{U_{0}}=\frac{\cos \omega t\left[\cosh \Delta_{+} \cos \Delta_{-}-\cosh \Delta_{-} \cos \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)} \\ & +\frac{\sin \omega t\left[\sinh \Delta_{+} \sin \Delta_{-}-\sinh \Delta_{-} \sin \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)}, \end{aligned}$

Paper 4, Section II, A

commentConsider the spherically symmetric motion induced by the collapse of a spherical cavity of radius $a(t)$, centred on the origin. For $r<a$, there is a vacuum, while for $r>a$, there is an inviscid incompressible fluid with constant density $\rho$. At time $t=0, a=a_{0}$, and the fluid is at rest and at constant pressure $p_{0}$.

(a) Consider the radial volume transport in the fluid $Q(R, t)$, defined as

$Q(R, t)=\int_{r=R} u d S$

where $u$ is the radial velocity, and $d S$ is an infinitesimal element of the surface of a sphere of radius $R \geqslant a$. Use the incompressibility condition to establish that $Q$ is a function of time alone.

(b) Using the expression for pressure in potential flow or otherwise, establish that

$\frac{1}{4 \pi a} \frac{d Q}{d t}-\frac{(\dot{a})^{2}}{2}=-\frac{p_{0}}{\rho}$

where $\dot{a}(t)$ is the radial velocity of the cavity boundary.

(c) By expressing $Q(t)$ in terms of $a$ and $\dot{a}$, show that

$\dot{a}=-\sqrt{\frac{2 p_{0}}{3 \rho}\left(\frac{a_{0}^{3}}{a^{3}}-1\right)}$

[Hint: You may find it useful to assume $\dot{a}(t)$ is an explicit function of a from the outset.]

(d) Hence write down an integral expression for the implosion time $\tau$, i.e. the time for the radius of the cavity $a \rightarrow 0$. [Do not attempt to evaluate the integral.]

Paper 1, Section I, F

commentLet $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be a smooth function and let $\Sigma=f^{-1}(0)$ (assumed not empty). Show that if the differential $D f_{p} \neq 0$ for all $p \in \Sigma$, then $\Sigma$ is a smooth surface in $\mathbb{R}^{3}$.

Is $\left\{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}=\cosh \left(z^{2}\right)\right\}$ a smooth surface? Is every surface $\Sigma \subset \mathbb{R}^{3}$ of the form $f^{-1}(0)$ for some smooth $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ ? Justify your answers.

Paper 1, Section II, F

commentLet $S \subset \mathbb{R}^{3}$ be an oriented surface. Define the Gauss map $N$ and show that the differential $D N_{p}$ of the Gauss map at any point $p \in S$ is a self-adjoint linear map. Define the Gauss curvature $\kappa$ and compute $\kappa$ in a given parametrisation.

A point $p \in S$ is called umbilic if $D N_{p}$ has a repeated eigenvalue. Let $S \subset \mathbb{R}^{3}$ be a surface such that every point is umbilic and there is a parametrisation $\phi: \mathbb{R}^{2} \rightarrow S$ such that $S=\phi\left(\mathbb{R}^{2}\right)$. Prove that $S$ is part of a plane or part of a sphere. $[$ Hint: consider the symmetry of the mixed partial derivatives $n_{u v}=n_{v u}$, where $n(u, v)=N(\phi(u, v))$ for $\left.(u, v) \in \mathbb{R}^{2} .\right]$

Paper 2, Section II, E

commentDefine $\mathbb{H}$, the upper half plane model for the hyperbolic plane, and show that $\operatorname{PSL}_{2}(\mathbb{R})$ acts on $\mathbb{H}$ by isometries, and that these isometries preserve the orientation of $\mathbb{H}$.

Show that every orientation preserving isometry of $\mathbb{H}$ is in $P S L_{2}(\mathbb{R})$, and hence the full group of isometries of $\mathbb{H}$ is $G=P S L_{2}(\mathbb{R}) \cup P S L_{2}(\mathbb{R}) \tau$, where $\tau z=-\bar{z}$.

Let $\ell$ be a hyperbolic line. Define the reflection $\sigma_{\ell}$ in $\ell$. Now let $\ell, \ell^{\prime}$ be two hyperbolic lines which meet at a point $A \in \mathbb{H}$ at an angle $\theta$. What are the possibilities for the group $G$ generated by $\sigma_{\ell}$ and $\sigma_{\ell^{\prime}}$ ? Carefully justify your answer.

Paper 3, Section I, E

commentState the local Gauss-Bonnet theorem for geodesic triangles on a surface. Deduce the Gauss-Bonnet theorem for closed surfaces. [Existence of a geodesic triangulation can be assumed.]

Let $S_{r} \subset \mathbb{R}^{3}$ denote the sphere with radius $r$ centred at the origin. Show that the Gauss curvature of $S_{r}$ is $1 / r^{2}$. An octant is any of the eight regions in $S_{r}$ bounded by arcs of great circles arising from the planes $x=0, y=0, z=0$. Verify directly that the local Gauss-Bonnet theorem holds for an octant. [You may assume that the great circles on $S_{r}$ are geodesics.]

Paper 3, Section II, E

commentLet $S \subset \mathbb{R}^{3}$ be an embedded smooth surface and $\gamma:[0,1] \rightarrow S$ a parameterised smooth curve on $S$. What is the energy of $\gamma$ ? By applying the Euler-Lagrange equations for stationary curves to the energy function, determine the differential equations for geodesics on $S$ explicitly in terms of a parameterisation of $S$.

If $S$ contains a straight line $\ell$, prove from first principles that each segment $[P, Q] \subset \ell$ (with some parameterisation) is a geodesic on $S$.

Let $H \subset \mathbb{R}^{3}$ be the hyperboloid defined by the equation $x^{2}+y^{2}-z^{2}=1$ and let $P=\left(x_{0}, y_{0}, z_{0}\right) \in H$. By considering appropriate isometries, or otherwise, display explicitly three distinct (as subsets of $H$ ) geodesics $\gamma: \mathbb{R} \rightarrow H$ through $P$ in the case when $z_{0} \neq 0$ and four distinct geodesics through $P$ in the case when $z_{0}=0$. Justify your answer.

Let $\gamma: \mathbb{R} \rightarrow H$ be a geodesic, with coordinates $\gamma(t)=(x(t), y(t), z(t))$. Clairaut's relation asserts $\rho(t) \sin \psi(t)$ is constant, where $\rho(t)=\sqrt{x(t)^{2}+y(t)^{2}}$ and $\psi(t)$ is the angle between $\dot{\gamma}(t)$ and the plane through the point $\gamma(t)$ and the $z$-axis. Deduce from Clairaut's relation that there exist infinitely many geodesics $\gamma(t)$ on $H$ which stay in the half-space $\{z>0\}$ for all $t \in \mathbb{R}$.

[You may assume that if $\gamma(t)$ satisfies the geodesic equations on $H$ then $\gamma$ is defined for all $t \in \mathbb{R}$ and the Euclidean norm $\|\dot{\gamma}(t)\|$ is constant. If you use a version of the geodesic equations for a surface of revolution, then that should be proved.]

Paper 4, Section II, F

commentDefine an abstract smooth surface and explain what it means for the surface to be orientable. Given two smooth surfaces $S_{1}$ and $S_{2}$ and a map $f: S_{1} \rightarrow S_{2}$, explain what it means for $f$ to be smooth

For the cylinder

$C=\left\{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}=1\right\},$

let $a: C \rightarrow C$ be the orientation reversing diffeomorphism $a(x, y, z)=(-x,-y,-z)$. Let $S$ be the quotient of $C$ by the equivalence relation $p \sim a(p)$ and let $\pi: C \rightarrow S$ be the canonical projection map. Show that $S$ can be made into an abstract smooth surface so that $\pi$ is smooth. Is $S$ orientable? Justify your answer.

Paper 1, Section II, G

commentShow that a ring $R$ is Noetherian if and only if every ideal of $R$ is finitely generated. Show that if $\phi: R \rightarrow S$ is a surjective ring homomorphism and $R$ is Noetherian, then $S$ is Noetherian.

State and prove Hilbert's Basis Theorem.

Let $\alpha \in \mathbb{C}$. Is $\mathbb{Z}[\alpha]$ Noetherian? Justify your answer.

Give, with proof, an example of a Unique Factorization Domain that is not Noetherian.

Let $R$ be the ring of continuous functions $\mathbb{R} \rightarrow \mathbb{R}$. Is $R$ Noetherian? Justify your answer.

Paper 2, Section I, $1 G$

commentLet $M$ be a module over a Principal Ideal Domain $R$ and let $N$ be a submodule of $M$. Show that $M$ is finitely generated if and only if $N$ and $M / N$ are finitely generated.

Paper 2, Section II, G

commentLet $M$ be a module over a ring $R$ and let $S \subset M$. Define what it means that $S$ freely generates $M$. Show that this happens if and only if for every $R$-module $N$, every function $f: S \rightarrow N$ extends uniquely to a homomorphism $\phi: M \rightarrow N$.

Let $M$ be a free module over a (non-trivial) ring $R$ that is generated (not necessarily freely) by a subset $T \subset M$ of size $m$. Show that if $S$ is a basis of $M$, then $S$ is finite with $|S| \leqslant m$. Hence, or otherwise, deduce that any two bases of $M$ have the same number of elements. Denoting this number $\operatorname{rk} M$ and by quoting any result you need, show that if $R$ is a Euclidean Domain and $N$ is a submodule of $M$, then $N$ is free with $\operatorname{rk} N \leqslant \operatorname{rk} M$.

State the Primary Decomposition Theorem for a finitely generated module $M$ over a Euclidean Domain $R$. Deduce that any finite subgroup of the multiplicative group of a field is cyclic.

Paper 3, Section I, G

commentLet $G$ be a finite group, and let $H$ be a proper subgroup of $G$ of index $n$.

Show that there is a normal subgroup $K$ of $G$ such that $|G / K|$ divides $n$ ! and $|G / K| \geqslant n$.

Show that if $G$ is non-abelian and simple, then $G$ is isomorphic to a subgroup of $A_{n}$.

Paper 3, Section II, 10G

commentLet $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:

(i) $p$ is prime;

(ii) $p$ is irreducible;

(iii) $(p)$ is a maximal ideal of $R$;

(iv) $R /(p)$ is a field;

(v) $R /(p)$ is an Integral Domain.

Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.

Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.

Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.

Paper 4, Section II, G

commentLet $H$ and $P$ be subgroups of a finite group $G$. Show that the sets $H x P, x \in G$, partition $G$. By considering the action of $H$ on the set of left cosets of $P$ in $G$ by left multiplication, or otherwise, show that

$\frac{|H x P|}{|P|}=\frac{|H|}{\left|H \cap x P x^{-1}\right|}$

for any $x \in G$. Deduce that if $G$ has a Sylow $p$-subgroup, then so does $H$.

Let $p, n \in \mathbb{N}$ with $p$ a prime. Write down the order of the group $G L_{n}(\mathbb{Z} / p \mathbb{Z})$. Identify in $G L_{n}(\mathbb{Z} / p \mathbb{Z})$ a Sylow $p$-subgroup and a subgroup isomorphic to the symmetric group $S_{n}$. Deduce that every finite group has a Sylow $p$-subgroup.

State Sylow's theorem on the number of Sylow $p$-subgroups of a finite group.

Let $G$ be a group of order $p q$, where $p>q$ are prime numbers. Show that if $G$ is non-abelian, then $q \mid p-1$.

Paper 1, Section I, $1 \mathrm{E}$

commentLet $V$ be a vector space over $\mathbb{R}, \operatorname{dim} V=n$, and let $\langle,$,$rangle be a non-degenerate anti-$ symmetric bilinear form on $V$.

Let $v \in V, v \neq 0$. Show that $v^{\perp}$ is of dimension $n-1$ and $v \in v^{\perp}$. Show that if $W \subseteq v^{\perp}$ is a subspace with $W \oplus \mathbb{R} v=v^{\perp}$, then the restriction of $\langle,$,$rangle to W$ is nondegenerate.

Conclude that the dimension of $V$ is even.

Paper 1, Section II, E

commentLet $d \geqslant 1$, and let $J_{d}=\left(\begin{array}{ccccc}0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ & & \cdots & \cdots & \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \ldots & 0 & 0\end{array}\right) \in \operatorname{Mat}_{d}(\mathbb{C})$.

(a) (i) Compute $J_{d}^{n}$, for all $n \geqslant 0$.

(ii) Hence, or otherwise, compute $\left(\lambda I+J_{d}\right)^{n}$, for all $n \geqslant 0$.

(b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $\varphi \in \operatorname{End}(V)$. Suppose $\varphi^{n}=0$ for some $n>1$.

(i) Determine the possible eigenvalues of $\varphi$.

(ii) What are the possible Jordan blocks of $\varphi$ ?

(iii) Show that if $\varphi^{2}=0$, there exists a decomposition

$V=U \oplus W_{1} \oplus W_{2}$

where $\varphi(U)=\varphi\left(W_{1}\right)=0, \varphi\left(W_{2}\right)=W_{1}$, and $\operatorname{dim} W_{2}=\operatorname{dim} W_{1}$.

Paper 2, Section II, E

comment(a) Compute the characteristic polynomial and minimal polynomial of

$A=\left(\begin{array}{ccc} -2 & -6 & -9 \\ 3 & 7 & 9 \\ -1 & -2 & -2 \end{array}\right)$

Write down the Jordan normal form for $A$.

(b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}, f: V \rightarrow V$ be a linear map, and for $\alpha \in \mathbb{C}, n \geqslant 1$, write

$W_{\alpha, n}:=\left\{v \in V \mid(f-\alpha I)^{n} v=0\right\}$

(i) Given $v \in W_{\alpha, n}, v \neq 0$, construct a non-zero eigenvector for $f$ in terms of $v$.

(ii) Show that if $w_{1}, \ldots, w_{d}$ are non-zero eigenvectors for $f$ with eigenvalues $\alpha_{1}, \ldots, \alpha_{d}$, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $w_{1}, \ldots, w_{d}$ are linearly independent.

(iii) Show that if $v_{1} \in W_{\alpha_{1}, n}, \ldots, v_{d} \in W_{\alpha_{d}, n}$ are all non-zero, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $v_{1}, \ldots, v_{d}$ are linearly independent.

Paper 3, Section II, 9E

comment(a) (i) State the rank-nullity theorem.

Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.

Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$

Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.

(ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 1 \\ 2 \\ -1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{c} -2 \\ 2 \\ 2 \\ 1 \\ -2 \end{array}\right)$

and $W$ the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 3 \\ 2 \\ -3 \\ 1 \\ 3 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{c} 1 \\ -4 \\ -1 \\ -2 \\ 1 \end{array}\right)$

Determine the dimension of $U \cap W$.

(b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.

Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.

Show that if $k=n$ the polynomial is of degree precisely $n$.

Give an example where $k \geqslant 1$ but this polynomial is zero.

Paper 4, Section I, $1 \mathbf{E}$

commentLet $\operatorname{Mat}_{n}(\mathbb{C})$ be the vector space of $n$ by $n$ complex matrices.

Given $A \in \operatorname{Mat}_{n}(\mathbb{C})$, define the linear $\operatorname{map}_{A}: \operatorname{Mat}_{n}(\mathbb{C}) \rightarrow \operatorname{Mat}_{n}(\mathbb{C})$,

$X \mapsto A X-X A$

(i) Compute a basis of eigenvectors, and their associated eigenvalues, when $A$ is the diagonal matrix

$A=\left(\begin{array}{llll} 1 & & & \\ & 2 & & \\ & & \ddots & \\ & & & n \end{array}\right)$

What is the rank of $\varphi_{A}$ ?

(ii) Now let $A=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$. Write down the matrix of the linear transformation $\varphi_{A}$ with respect to the standard basis of $\operatorname{Mat}_{2}(\mathbb{C})$.

What is its Jordan normal form?

Paper 4, Section II, E

comment(a) Let $V$ be a complex vector space of dimension $n$.

What is a Hermitian form on $V$ ?

Given a Hermitian form, define the matrix $A$ of the form with respect to the basis $v_{1}, \ldots, v_{n}$ of $V$, and describe in terms of $A$ the value of the Hermitian form on two elements of $V$.

Now let $w_{1}, \ldots, w_{n}$ be another basis of $V$. Suppose $w_{i}=\sum_{j} p_{i j} v_{j}$, and let $P=\left(p_{i j}\right)$. Write down the matrix of the form with respect to this new basis in terms of $A$ and $P$.

Let $N=V^{\perp}$. Describe the dimension of $N$ in terms of the matrix $A$.

(b) Write down the matrix of the real quadratic form

$x^{2}+y^{2}+2 z^{2}+2 x y+2 x z-2 y z .$

Using the Gram-Schmidt algorithm, find a basis which diagonalises the form. What are its rank and signature?

(c) Let $V$ be a real vector space, and $\langle,$,$rangle a symmetric bilinear form on it. Let A$ be the matrix of this form in some basis.

Prove that the signature of $\langle,$,$rangle is the number of positive eigenvalues of A$ minus the number of negative eigenvalues.

Explain, using an example, why the eigenvalues themselves depend on the choice of a basis.

Paper 1, Section II, 19H

commentLet $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with transition matrix $P$. What is a stopping time of $\left(X_{n}\right)_{n \geqslant 0}$ ? What is the strong Markov property?

The exciting game of 'Unopoly' is played by a single player on a board of 4 squares. The player starts with $£ m$ (where $m \in \mathbb{N}$ ). During each turn, the player tosses a fair coin and moves one or two places in a clockwise direction $(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1)$ according to whether the coin lands heads or tails respectively. The player collects $£ 2$ each time they pass (or land on) square 1. If the player lands on square 3 however, they immediately lose $£ 1$ and go back to square 2. The game continues indefinitely unless the player is on square 2 with $£ 0$, in which case the player loses the game and the game ends.

(a) By setting up an appropriate Markov chain, show that if the player is at square 2 with $£ m$, where $m \geqslant 1$, the probability that they are ever at square 2 with $£(m-1)$ is $2 / 3 .$

(b) Find the probability of losing the game when the player starts on square 1 with $£ m$, where $m \geqslant 1$.

[Hint: Take the state space of your Markov chain to be $\{1,2,4\} \times\{0,1, \ldots\}$.]

Paper 2, Section II, 18H

commentLet $P$ be a transition matrix on state space $I$. What does it mean for a distribution $\pi$ to be an invariant distribution? What does it mean for $\pi$ and $P$ to be in detailed balance? Show that if $\pi$ and $P$ are in detailed balance, then $\pi$ is an invariant distribution.

(a) Assuming that an invariant distribution exists, state the relationship between this and

(i) the expected return time to a state $i$;

(ii) the expected time spent in a state $i$ between visits to a state $k$.

(b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with transition matrix $P=\left(p_{i j}\right)_{i, j \in I}$ where $I=\{0,1,2, \ldots\}$. The transition probabilities are given for $i \geqslant 1$ by

$p_{i j}= \begin{cases}q^{-(i+2)} & \text { if } j=i+1, \\ q^{-i} & \text { if } j=i-1 \\ 1-q^{-(i+2)}-q^{-i} & \text { if } j=i\end{cases}$

where $q \geqslant 2$. For $p \in(0,1]$ let $p_{01}=p=1-p_{00}$. Compute the following, justifying your answers:

(i) The expected time spent in states $\{2,4,6, \ldots\}$ between visits to state 1 ;

(ii) The expected time taken to return to state 1 , starting from 1 ;

(iii) The expected time taken to hit state 0 starting from $1 .$

Paper 3 , Section I, H

Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ on a state space $I$.

(a) Define the notion of a communicating class. What does it mean for a communicating class to be closed?

(b) Taking $I=\{1, \ldots, 6\}$, find the communicating classes associated with the transition matrix $P$ given by

$P=\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & \frac{1}{4} & \frac{3}{4} \\ \frac{1}{4} & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{4} \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\ \frac{1}{4} & \frac{1}{2} & 0 & 0 & 0 & \frac{1}{4} \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$