Part IB, 2006, Paper 2

# Part IB, 2006, Paper 2

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2.I.3F

commentDefine uniform convergence for a sequence $f_{1}, f_{2}, \ldots$ of real-valued functions on an interval in $\mathbf{R}$. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to a (necessarily continuous) function $f$ on a closed interval $[a, b]$, show that

$\int_{a}^{b} f_{n}(x) d x \rightarrow \int_{a}^{b} f(x) d x$

as $n \rightarrow \infty$.

Which of the following sequences of functions $f_{1}, f_{2}, \ldots$ converges uniformly on the open interval $(0,1)$ ? Justify your answers.

(i) $f_{n}(x)=1 /(n x)$;

(ii) $f_{n}(x)=e^{-x / n}$.

2.II.13F

commentFor a smooth mapping $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$, the Jacobian $J(F)$ at a point $(x, y)$ is defined as the determinant of the derivative $D F$, viewed as a linear map $\mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$. Suppose that $F$ maps into a curve in the plane, in the sense that $F$ is a composition of two smooth mappings $\mathbf{R}^{2} \rightarrow \mathbf{R} \rightarrow \mathbf{R}^{2}$. Show that the Jacobian of $F$ is identically zero.

Conversely, let $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be a smooth mapping whose Jacobian is identically zero. Write $F(x, y)=(f(x, y), g(x, y))$. Suppose that $\partial f /\left.\partial y\right|_{(0,0)} \neq 0$. Show that $\partial f / \partial y \neq 0$ on some open neighbourhood $U$ of $(0,0)$ and that on $U$

$(\partial g / \partial x, \partial g / \partial y)=e(x, y)(\partial f / \partial x, \partial f / \partial y)$

for some smooth function $e$ defined on $U$. Now suppose that $c: \mathbf{R} \rightarrow U$ is a smooth curve of the form $t \mapsto(t, \alpha(t))$ such that $F \circ c$ is constant. Write down a differential equation satisfied by $\alpha$. Apply an existence theorem for differential equations to show that there is a neighbourhood $V$ of $(0,0)$ such that every point in $V$ lies on a curve $t \mapsto(t, \alpha(t))$ on which $F$ is constant.

[A function is said to be smooth when it is infinitely differentiable. Detailed justification of the smoothness of the functions in question is not expected.]

2.II.14D

commentLet $\Omega$ be the region enclosed between the two circles $C_{1}$ and $C_{2}$, where

$C_{1}=\{z \in \mathbf{C}:|z-i|=1\}, \quad C_{2}=\{z \in \mathbf{C}:|z-2 i|=2\}$

Find a conformal mapping that maps $\Omega$ onto the unit disc.

[Hint: you may find it helpful first to map $\Omega$ to a strip in the complex plane. ]

2.I.6G

commentGiven that the electric field $\mathbf{E}$ and the current density $\mathbf{j}$ within a conducting medium of uniform conductivity $\sigma$ are related by $j=\sigma \mathbf{E}$, use Maxwell's equations to show that the charge density $\rho$ in the medium obeys the equation

$\frac{\partial \rho}{\partial t}=-\frac{\sigma}{\epsilon_{0}} \rho .$

An infinitely long conducting cylinder of uniform conductivity $\sigma$ is set up with a uniform electric charge density $\rho_{0}$ throughout its interior. The region outside the cylinder is a vacuum. Obtain $\rho$ within the cylinder at subsequent times and hence obtain $\mathbf{E}$ and $\mathbf{j}$ within the cylinder as functions of time and radius. Calculate the value of $\mathbf{E}$ outside the cylinder.

2.II.17G

commentDerive from Maxwell's equations the Biot-Savart law

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

giving the magnetic field $\mathbf{B}(\mathbf{r})$ produced by a steady current density $\mathbf{j}(\mathbf{r})$ that vanishes outside a bounded region $V$.

[You may assume that the divergence of the magnetic vector potential is zero.]

A steady current density $\mathbf{j}(\mathbf{r})$ has the form $\mathbf{j}=\left(0, j_{\phi}(\mathbf{r}), 0\right)$ in cylindrical polar coordinates $(r, \phi, z)$ where

$j_{\phi}(\mathbf{r})= \begin{cases}k r & 0 \leqslant r \leqslant b, \quad-h \leqslant z \leqslant h \\ 0 & \text { otherwise },\end{cases}$

and $k$ is a constant. Find the magnitude and direction of the magnetic field at the origin.

$\left[\text { Hint }: \quad \int_{-h}^{h} \frac{d z}{\left(r^{2}+z^{2}\right)^{3 / 2}}=\frac{2 h}{r^{2}\left(h^{2}+r^{2}\right)^{1 / 2}}\right]$

2.I.8A

commentExplain what is meant by a material time derivative, $D / D t$. Show that if the material velocity is $\mathbf{u}(\mathbf{x}, t)$ then

$D / D t=\partial / \partial t+\mathbf{u} \cdot \nabla$

When glass is processed in its liquid state, its temperature, $\theta(\mathbf{x}, t)$, satisfies the equation

$D \theta / D t=-\theta \text {. }$

The glass flows in a two-dimensional channel $-1<y<1, \quad x>0$ with steady velocity $\mathbf{u}=\left(1-y^{2}, 0\right)$. At $x=0$ the glass temperature is maintained at the constant value $\theta_{0}$. Find the steady temperature distribution throughout the channel.

2.II.12H

commentLet $\sigma: V \rightarrow U \subset \mathbf{R}^{3}$ denote a parametrized smooth embedded surface, where $V$ is an open ball in $\mathbf{R}^{2}$ with coordinates $(u, v)$. Explain briefly the geometric meaning of the second fundamental form

$L d u^{2}+2 M d u d v+N d v^{2},$

where $L=\sigma_{u u} \cdot \mathbf{N}, M=\sigma_{u v} \cdot \mathbf{N}, N=\sigma_{v v} \cdot \mathbf{N}$, with $\mathbf{N}$ denoting the unit normal vector to the surface $U$.

Prove that if the second fundamental form is identically zero, then $\mathbf{N}_{u}=\mathbf{0}=\mathbf{N}_{v}$ as vector-valued functions on $V$, and hence that $\mathbf{N}$ is a constant vector. Deduce that $U$ is then contained in a plane given by $\mathbf{x} \cdot \mathbf{N}=$ constant.

2.I.2E

comment(i) Give the definition of a Euclidean domain and, with justification, an example of a Euclidean domain that is not a field.

(ii) State the structure theorem for finitely generated modules over a Euclidean domain.

(iii) In terms of your answer to (ii), describe the structure of the $\mathbb{Z}$-module $M$ with generators $\left\{m_{1}, m_{2}, m_{3}\right\}$ and relations $2 m_{3}=2 m_{2}, 4 m_{2}=0$.

2.II.11E

comment(i) Prove the first Sylow theorem, that a finite group of order $p^{n} r$ with $p$ prime and $p$ not dividing the integer $r$ has a subgroup of order $p^{n}$.

(ii) State the remaining Sylow theorems.

(iii) Show that if $p$ and $q$ are distinct primes then no group of order $p q$ is simple.

2.I.1E

commentState Sylvester's law of inertia.

Find the rank and signature of the quadratic form $q$ on $\mathbf{R}^{n}$ given by

$q\left(x_{1}, \ldots, x_{n}\right)=\left(\sum_{i=1}^{n} x_{i}\right)^{2}-\sum_{i=1}^{n} x_{i}^{2}$

2.II.10E

commentSuppose that $V$ is the set of complex polynomials of degree at most $n$ in the variable $x$. Find the dimension of $V$ as a complex vector space.

Define

$e_{k}: V \rightarrow \mathbf{C} \quad \text { by } \quad e_{k}(\phi)=\frac{d^{k} \phi}{d x^{k}}(0)$

Find a subset of $\left\{e_{k} \mid k \in \mathbf{N}\right\}$ that is a basis of the dual vector space $V^{*}$. Find the corresponding dual basis of $V$.

Define

$D: V \rightarrow V \quad \text { by } \quad D(\phi)=\frac{d \phi}{d x} .$

Write down the matrix of $D$ with respect to the basis of $V$ that you have just found, and the matrix of the map dual to $D$ with respect to the dual basis.

2.II.20C

commentConsider the Markov chain $\left(X_{n}\right)_{n \geq 0}$ on the integers $\mathbb{Z}$ whose non-zero transition probabilities are given by $p_{0,1}=p_{0,-1}=1 / 2$ and

$\begin{gathered} p_{n, n-1}=1 / 3, \quad p_{n, n+1}=2 / 3, \quad \text { for } n \geq 1 \\ p_{n, n-1}=3 / 4, \quad p_{n, n+1}=1 / 4, \quad \text { for } n \leqslant-1 \end{gathered}$

(a) Show that, if $X_{0}=1$, then $\left(X_{n}\right)_{n \geq 0}$ hits 0 with probability $1 / 2$.

(b) Suppose now that $X_{0}=0$. Show that, with probability 1 , as $n \rightarrow \infty$ either $X_{n} \rightarrow \infty$ or $X_{n} \rightarrow-\infty$.

(c) In the case $X_{0}=0$ compute $\mathbb{P}\left(X_{n} \rightarrow \infty\right.$ as $\left.n \rightarrow \infty\right)$.

2.I.5A

commentDescribe briefly the method of Lagrange multipliers for finding the stationary values of a function $f(x, y)$ subject to a constraint $g(x, y)=0$.

Use the method to find the smallest possible surface area (including both ends) of a circular cylinder that has volume $V$.

2.II.15G

commentVerify that $y=e^{-x}$ is a solution of the differential equation

$(x+2) y^{\prime \prime}+(x+1) y^{\prime}-y=0,$

and find a second solution of the form $a x+b$.

Let $L$ be the operator

$L[y]=y^{\prime \prime}+\frac{(x+1)}{(x+2)} y^{\prime}-\frac{1}{(x+2)} y$

on functions $y(x)$ satisfying

$y^{\prime}(0)=y(0) \quad \text { and } \quad \lim _{x \rightarrow \infty} y(x)=0 .$

The Green's function $G(x, \xi)$ for $L$ satisfies

$L[G]=\delta(x-\xi)$

with $\xi>0$. Show that

$G(x, \xi)=-\frac{(\xi+1)}{(\xi+2)} e^{\xi-x}$

for $x>\xi$, and find $G(x, \xi)$ for $x<\xi$.

Hence or otherwise find the solution of

$L[y]=-(x+2) e^{-x},$

for $x \geqslant 0$, with $y(x)$ satisfying the boundary conditions above.

2.I.4F

commentWhich of the following subspaces of Euclidean space are connected? Justify your answers (i) $\left\{(x, y, z) \in \mathbf{R}^{3}: z^{2}-x^{2}-y^{2}=1\right\}$; (ii) $\left\{(x, y) \in \mathbf{R}^{2}: x^{2}=y^{2}\right\}$; (iii) $\left\{(x, y, z) \in \mathbf{R}^{3}: z=x^{2}+y^{2}\right\}$.

2.II.18D

comment(a) For a positive weight function $w$, let

$\int_{-1}^{1} f(x) w(x) d x \approx \sum_{i=0}^{n} a_{i} f\left(x_{i}\right)$

be the corresponding Gaussian quadrature with $n+1$ nodes. Prove that all the coefficients $a_{i}$ are positive.

(b) The integral

$I(f)=\int_{-1}^{1} f(x) w(x) d x$

is approximated by a quadrature

$I_{n}(f)=\sum_{i=0}^{n} a_{i}^{(n)} f\left(x_{i}^{(n)}\right)$

which is exact on polynomials of degree $\leqslant n$ and has positive coefficients $a_{i}^{(n)}$. Prove that, for any $f$ continuous on $[-1,1]$, the quadrature converges to the integral, i.e.,

$\left|I(f)-I_{n}(f)\right| \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$

[You may use the Weierstrass theorem: for any $f$ continuous on $[-1,1]$, and for any $\epsilon>0$, there exists a polynomial $Q$ of degree $n=n(\epsilon, f)$ such that $\left.\max _{x \in[-1,1]}|f(x)-Q(x)|<\epsilon .\right]$

2.II.16B

commentThe spherically symmetric bound state wavefunctions $\psi(r)$, where $r=|\mathbf{x}|$, for an electron orbiting in the Coulomb potential $V(r)=-e^{2} /\left(4 \pi \epsilon_{0} r\right)$ of a hydrogen atom nucleus, can be modelled as solutions to the equation

$\frac{d^{2} \psi}{d r^{2}}+\frac{2}{r} \frac{d \psi}{d r}+\frac{a}{r} \psi(r)-b^{2} \psi(r)=0$

for $r \geqslant 0$, where $a=e^{2} m /\left(2 \pi \epsilon_{0} \hbar^{2}\right), b=\sqrt{-2 m E} / \hbar$, and $E$ is the energy of the corresponding state. Show that there are normalisable and continuous wavefunctions $\psi(r)$ satisfying this equation with energies

$E=-\frac{m e^{4}}{32 \pi^{2} \epsilon_{0}^{2} \hbar^{2} N^{2}}$

for all integers $N \geqslant 1$.

2.I.7B

comment$A_{1}$ moves at speed $v_{1}$ in the $x$-direction with respect to $A_{0} . A_{2}$ moves at speed $v_{2}$ in the $x$-direction with respect to $A_{1}$. By applying a Lorentz transformation between the rest frames of $A_{0}, A_{1}$, and $A_{2}$, calculate the speed at which $A_{0}$ observes $A_{2}$ to travel.

$A_{3}$ moves at speed $v_{3}$ in the $x$-direction with respect to $A_{2}$. Calculate the speed at which $A_{0}$ observes $A_{3}$ to travel.

2.II.19C

commentSuppose that $X_{1}, \ldots, X_{n}$ are independent normal random variables of unknown mean $\theta$ and variance 1 . It is desired to test the hypothesis $H_{0}: \theta \leq 0$ against the alternative $H_{1}: \theta>0$. Show that there is a uniformly most powerful test of size $\alpha=1 / 20$ and identify a critical region for such a test in the case $n=9$. If you appeal to any theoretical result from the course you should also prove it.

[The 95th percentile of the standard normal distribution is 1.65.]