# Part IB, 2003

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$3 . \mathrm{II} . 11 \mathrm{~F} \quad$

commentState and prove the Contraction Mapping Theorem.

Let $(X, d)$ be a bounded metric space, and let $F$ denote the set of all continuous maps $X \rightarrow X$. Let $\rho: F \times F \rightarrow \mathbb{R}$ be the function

$\rho(f, g)=\sup \{d(f(x), g(x)): x \in X\}$

Show that $\rho$ is a metric on $F$, and that $(F, \rho)$ is complete if $(X, d)$ is complete. [You may assume that a uniform limit of continuous functions is continuous.]

Now suppose that $(X, d)$ is complete. Let $C \subseteq F$ be the set of contraction mappings, and let $\theta: C \rightarrow X$ be the function which sends a contraction mapping to its unique fixed point. Show that $\theta$ is continuous. [Hint: fix $f \in C$ and consider $d(\theta(g), f(\theta(g)))$, where $g \in C$ is close to $f$.]

1.I.1F

commentLet $E$ be a subset of $\mathbb{R}^{n}$. Prove that the following conditions on $E$ are equivalent:

(i) $E$ is closed and bounded.

(ii) $E$ has the Bolzano-Weierstrass property (i.e., every sequence in $E$ has a subsequence convergent to a point of $E$ ).

(iii) Every continuous real-valued function on $E$ is bounded.

[The Bolzano-Weierstrass property for bounded closed intervals in $\mathbb{R}^{1}$ may be assumed.]

1.II.10F

commentExplain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.

A function $d: X \times X \rightarrow \mathbb{R}$ is called a pseudometric on $X$ if it satisfies all the conditions for a metric except the requirement that $d(x, y)=0$ implies $x=y$. If $d$ is a pseudometric on $X$, show that the binary relation $R$ on $X$ defined by $x R y \Leftrightarrow d(x, y)=0$ is an equivalence relation, and that the function $d$ induces a metric on the set $X / R$ of equivalence classes.

Now let $(X, d)$ be a metric space. If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are Cauchy sequences in $X$, show that the sequence whose $n$th term is $d\left(x_{n}, y_{n}\right)$ is a Cauchy sequence of real numbers. Deduce that the function $\bar{d}$ defined by

$\bar{d}\left(\left(x_{n}\right),\left(y_{n}\right)\right)=\lim _{n \rightarrow \infty} d\left(x_{n}, y_{n}\right)$

is a pseudometric on the set $C$ of all Cauchy sequences in $X$. Show also that there is an isometric embedding (that is, a distance-preserving mapping) $X \rightarrow C / R$, where $R$ is the equivalence relation on $C$ induced by the pseudometric $\bar{d}$ as in the previous paragraph. Under what conditions on $X$ is $X \rightarrow C / R$ bijective? Justify your answer.

2.I.1F

commentExplain what it means for a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{1}$ to be differentiable at a point $(a, b)$. Show that if the partial derivatives $\partial f / \partial x$ and $\partial f / \partial y$ exist in a neighbourhood of $(a, b)$ and are continuous at $(a, b)$ then $f$ is differentiable at $(a, b)$.

Let

$f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad((x, y) \neq(0,0))$

and $f(0,0)=0$. Do the partial derivatives of $f$ exist at $(0,0) ?$ Is $f$ differentiable at $(0,0) ?$ Justify your answers.

2.II.10F

commentLet $V$ be the space of $n \times n$ real matrices. Show that the function

$N(A)=\sup \left\{\|A \mathbf{x}\|: \mathbf{x} \in \mathbb{R}^{n},\|\mathbf{x}\|=1\right\}$

(where $\|-\|$ denotes the usual Euclidean norm on $\mathbb{R}^{n}$ ) defines a norm on $V$. Show also that this norm satisfies $N(A B) \leqslant N(A) N(B)$ for all $A$ and $B$, and that if $N(A)<\epsilon$ then all entries of $A$ have absolute value less than $\epsilon$. Deduce that any function $f: V \rightarrow \mathbb{R}$ such that $f(A)$ is a polynomial in the entries of $A$ is continuously differentiable.

Now let $d: V \rightarrow \mathbb{R}$ be the mapping sending a matrix to its determinant. By considering $d(I+H)$ as a polynomial in the entries of $H$, show that the derivative $d^{\prime}(I)$ is the function $H \mapsto \operatorname{tr} H$. Deduce that, for any $A, d^{\prime}(A)$ is the mapping $H \mapsto \operatorname{tr}((\operatorname{adj} A) H)$, where $\operatorname{adj} A$ is the adjugate of $A$, i.e. the matrix of its cofactors.

[Hint: consider first the case when $A$ is invertible. You may assume the results that the set $U$ of invertible matrices is open in $V$ and that its closure is the whole of $V$, and the identity $(\operatorname{adj} A) A=\operatorname{det} A . I$.]

3.I.1F

commentLet $V$ be the vector space of continuous real-valued functions on $[-1,1]$. Show that the function

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

defines a norm on $V$.

Let $f_{n}(x)=x^{n}$. Show that $\left(f_{n}\right)$ is a Cauchy sequence in $V$. Is $\left(f_{n}\right)$ convergent? Justify your answer.

4.I.1F

commentExplain what it means for a sequence of functions $\left(f_{n}\right)$ to converge uniformly to a function $f$ on an interval. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to $f$ on a finite interval $[a, b]$, show that

$\int_{a}^{b} f_{n}(x) d x \longrightarrow \int_{a}^{b} f(x) d x \quad \text { as } n \rightarrow \infty$

Let $f_{n}(x)=x \exp (-x / n) / n^{2}, x \geqslant 0$. Does $f_{n} \rightarrow 0$ uniformly on $[0, \infty) ?$ Does $\int_{0}^{\infty} f_{n}(x) d x \rightarrow 0$ ? Justify your answers.

4.II.10F

commentLet $\left(f_{n}\right)_{n \geqslant 1}$ be a sequence of continuous complex-valued functions defined on a set $E \subseteq \mathbb{C}$, and converging uniformly on $E$ to a function $f$. Prove that $f$ is continuous on $E$.

State the Weierstrass $M$-test for uniform convergence of a series $\sum_{n=1}^{\infty} u_{n}(z)$ of complex-valued functions on a set $E$.

Now let $f(z)=\sum_{n=1}^{\infty} u_{n}(z)$, where

$u_{n}(z)=n^{-2} \sec (\pi z / 2 n) .$

Prove carefully that $f$ is continuous on $\mathbb{C} \backslash \mathbb{Z}$.

[You may assume the inequality $|\cos z| \geqslant|\cos (\operatorname{Re} z)| \cdot]$

1.I.7B

commentLet $u(x, y)$ and $v(x, y)$ be a pair of conjugate harmonic functions in a domain $D$.

Prove that

$U(x, y)=e^{-2 u v} \cos \left(u^{2}-v^{2}\right) \quad \text { and } \quad V(x, y)=e^{-2 u v} \sin \left(u^{2}-v^{2}\right)$

also form a pair of conjugate harmonic functions in $D$.

1.II.16B

commentSketch the region $A$ which is the intersection of the discs

$D_{0}=\{z \in \mathbb{C}:|z|<1\} \quad \text { and } \quad D_{1}=\{z \in \mathbb{C}:|z-(1+i)|<1\} .$

Find a conformal mapping that maps $A$ onto the right half-plane $H=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$. Also find a conformal mapping that maps $A$ onto $D_{0}$.

[Hint: You may find it useful to consider maps of the form $w(z)=\frac{a z+b}{c z+d}$.]

2.I.7B

comment(a) Using the residue theorem, evaluate

$\int_{|z|=1}\left(z-\frac{1}{z}\right)^{2 n} \frac{d z}{z}$

(b) Deduce that

$\int_{0}^{2 \pi} \sin ^{2 n} t d t=\frac{\pi}{2^{2 n-1}} \frac{(2 n) !}{(n !)^{2}}$

2.II.16B

comment(a) Show that if $f$ satisfies the equation

$f^{\prime \prime}(x)-x^{2} f(x)=\mu f(x), \quad x \in \mathbb{R},$

where $\mu$ is a constant, then its Fourier transform $\widehat{f}$ satisfies the same equation, i.e.

$\widehat{f}^{\prime \prime}(\lambda)-\lambda^{2} \widehat{f}(\lambda)=\mu \widehat{f}(\lambda) .$

(b) Prove that, for each $n \geq 0$, there is a polynomial $p_{n}(x)$ of degree $n$, unique up to multiplication by a constant, such that

$f_{n}(x)=p_{n}(x) e^{-x^{2} / 2}$

is a solution of $(*)$ for some $\mu=\mu_{n}$.

(c) Using the fact that $g(x)=e^{-x^{2} / 2}$ satisfies $\widehat{g}=c g$ for some constant $c$, show that the Fourier transform of $f_{n}$ has the form

$\widehat{f_{n}}(\lambda)=q_{n}(\lambda) e^{-\lambda^{2} / 2}$

where $q_{n}$ is also a polynomial of degree $n$.

(d) Deduce that the $f_{n}$ are eigenfunctions of the Fourier transform operator, i.e. $\widehat{f_{n}}(x)=c_{n} f_{n}(x)$ for some constants $c_{n} .$

4.I.8B

commentFind the Laurent series centred on 0 for the function

$f(z)=\frac{1}{(z-1)(z-2)}$

in each of the domains (a) $|z|<1$, (b) $1<|z|<2$, (c) $|z|>2$.

4.II.17B

commentLet

$f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},$

and let $C_{R}$ be the boundary of the domain

$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .$

(a) Using the residue theorem, determine

$\int_{C_{R}} f(z) d z$

(b) Show that the integral of $f(z)$ along the circular part $\gamma_{R}$ of $C_{R}$ tends to 0 as $R \rightarrow \infty$.

(c) Deduce that

$\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}$

1.I.6C

commentAn unsteady fluid flow has velocity field given in Cartesian coordinates $(x, y, z)$ by $\mathbf{u}=(1, x t, 0)$, where $t$ denotes time. Dye is released into the fluid from the origin continuously. Find the position at time $t$ of the dye particle that was released at time $s$ and hence show that the dye streak lies along the curve

$y=\frac{1}{2} t x^{2}-\frac{1}{6} x^{3}$

1.II.15C

commentStarting from the Euler equations for incompressible, inviscid flow

$\rho \frac{D \mathbf{u}}{D t}=-\nabla p, \quad \nabla \cdot \mathbf{u}=0$

derive the vorticity equation governing the evolution of the vorticity $\boldsymbol{\omega}=\nabla \times \mathbf{u}$.

Consider the flow

$\mathbf{u}=\beta(-x,-y, 2 z)+\Omega(t)(-y, x, 0)$

in Cartesian coordinates $(x, y, z)$, where $t$ is time and $\beta$ is a constant. Compute the vorticity and show that it evolves in time according to

$\boldsymbol{\omega}=\omega_{0} \mathrm{e}^{2 \beta t} \mathbf{k}$

where $\omega_{0}$ is the initial magnitude of the vorticity and $\mathbf{k}$ is a unit vector in the $z$-direction.

Show that the material curve $C(t)$ that takes the form

$x^{2}+y^{2}=1 \quad \text { and } \quad z=1$

at $t=0$ is given later by

$x^{2}+y^{2}=a^{2}(t) \quad \text { and } \quad z=\frac{1}{a^{2}(t)},$

where the function $a(t)$ is to be determined.

Calculate the circulation of $\mathbf{u}$ around $C$ and state how this illustrates Kelvin's circulation theorem.

3.I.8C

commentShow that the velocity field

$\mathbf{u}=\mathbf{U}+\frac{\boldsymbol{\Gamma} \times \mathbf{r}}{2 \pi r^{2}},$

where $\mathbf{U}=(U, 0,0), \mathbf{\Gamma}=(0,0, \Gamma)$ and $\mathbf{r}=(x, y, 0)$ in Cartesian coordinates $(x, y, z)$, represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.

Show that

$\oint_{C_{R}}(\mathbf{u} \cdot \mathbf{n}) \mathbf{u} d l \rightarrow \frac{1}{2} \boldsymbol{\Gamma} \times \mathbf{U} \quad \text { as } \quad R \rightarrow \infty$

where $C_{R}$ is a circle $x^{2}+y^{2}=R^{2}, z=$ const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.

3.II.18C

commentState the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.

Water of density $\rho$ is driven through a tube of length $L$ and internal radius $a$ by the pressure exerted by a spherical, water-filled balloon of radius $R(t)$ attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at $2 \gamma / R$ in excess of atmospheric pressure, where $\gamma$ is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that

$R^{3} \ddot{R}+2 R^{2} \dot{R}^{2}=-\frac{\gamma a^{2}}{2 \rho L} .$

Solve equation ( $\dagger$ ), by multiplying through by $2 R \dot{R}$ or otherwise, to obtain

$t=R_{0}^{2}\left(\frac{2 \rho L}{\gamma a^{2}}\right)^{1 / 2}\left[\frac{\pi}{4}-\frac{\theta}{2}+\frac{1}{4} \sin 2 \theta\right]$

where $\theta=\sin ^{-1}\left(R / R_{0}\right)$ and $R_{0}$ is the initial radius of the balloon. Hence find the time when $R=0$.

4.I.7C

commentInviscid fluid issues vertically downwards at speed $u_{0}$ from a circular tube of radius a. The fluid falls onto a horizontal plate a distance $H$ below the end of the tube, where it spreads out axisymmetrically.

Show that while the fluid is falling freely it has speed

$u=u_{0}\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{1 / 2}$

and occupies a circular jet of radius

$R=a\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{-1 / 4},$

where $z$ is the height above the plate and $g$ is the acceleration due to gravity.

Show further that along the plate, at radial distances $r \gg a$ (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height $h(r)$, where

$\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)$

4.II.16C

commentDefine the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction $\psi(x, y)$ as

$\mathbf{u}=(u, v)=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$

in Cartesian coordinates $(x, y)$. Show that the line integral

$\int_{\mathbf{x}_{1}}^{\mathbf{x}_{\mathbf{2}}} \mathbf{u} \cdot \mathbf{n} d l=\psi\left(\mathbf{x}_{\mathbf{2}}\right)-\psi\left(\mathbf{x}_{\mathbf{1}}\right)$

along any path joining the points $\mathbf{x}_{\mathbf{1}}$ and $\mathbf{x}_{\mathbf{2}}$, where $\mathbf{n}$ is the unit normal to the path. Describe how this result is related to the concept of mass conservation.

Inviscid, incompressible fluid is contained in the semi-infinite channel $x>0$, $0<y<1$, which has rigid walls at $x=0$ and at $y=0,1$, apart from a small opening at the origin through which the fluid is withdrawn with volume flux $m$ per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation

$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0$

and boundary conditions

if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain

$\psi=-m y+\frac{2 m}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin n \pi y e^{-n \pi x}$

$\begin{aligned} & \psi=0 \quad \text { on } y=0, x>0, \\ & \psi=-m \quad \text { on } x=0,0<y<1, \\ & \psi=-m \quad \text { on } y=1, x>0, \\ & \psi \rightarrow-m y \quad \text { as } x \rightarrow \infty, \end{aligned}$

$2 . \mathrm{I} . 4 \mathrm{E} \quad$

commentLet $\tau_{1}$ be the collection of all subsets $A \subset \mathbb{N}$ such that $A=\emptyset$ or $\mathbb{N} \backslash A$ is finite. Let $\tau_{2}$ be the collection of all subsets of $\mathbb{N}$ of the form $I_{n}=\{n, n+1, n+2, \ldots\}$, together with the empty set. Prove that $\tau_{1}$ and $\tau_{2}$ are both topologies on $\mathbb{N}$.

Show that a function $f$ from the topological space $\left(\mathbb{N}, \tau_{1}\right)$ to the topological space $\left(\mathbb{N}, \tau_{2}\right)$ is continuous if and only if one of the following alternatives holds:

(i) $f(n) \rightarrow \infty$ as $n \rightarrow \infty$;

(ii) there exists $N \in \mathbb{N}$ such that $f(n)=N$ for all but finitely many $n$ and $f(n) \leqslant N$ for all $n$.

$3 . \mathrm{I} . 3 \mathrm{E} \quad$

comment(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $|f(z)| \leqslant 1+|z|^{1 / 2}$ for every $z$. Prove that $f$ is constant.

(b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $\operatorname{Re}(f(z)) \geqslant 0$ for every $z$. Prove that $f$ is constant.

2.II.13E

comment(a) Let $f:[1, \infty) \rightarrow \mathbb{C}$ be defined by $f(t)=t^{-1} e^{2 \pi i t}$ and let $X$ be the image of $f$. Prove that $X \cup\{0\}$ is compact and path-connected. [Hint: you may find it helpful to set $\left.s=t^{-1} .\right]$

(b) Let $g:[1, \infty) \rightarrow \mathbb{C}$ be defined by $g(t)=\left(1+t^{-1}\right) e^{2 \pi i t}$, let $Y$ be the image of $g$ and let $\bar{D}$ be the closed unit $\operatorname{disc}\{z \in \mathbb{C}:|z| \leq 1\}$. Prove that $Y \cup \bar{D}$ is connected. Explain briefly why it is not path-connected.

3.II.13E

comment(a) State Taylor's Theorem.

(b) Let $f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}$ and $g(z)=\sum_{n=0}^{\infty} b_{n}\left(z-z_{0}\right)^{n}$ be defined whenever $\left|z-z_{0}\right|<r$. Suppose that $z_{k} \rightarrow z_{0}$ as $k \rightarrow \infty$, that no $z_{k}$ equals $z_{0}$ and that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $a_{n}=b_{n}$ for every $n \geqslant 0$.

(c) Let $D$ be a domain, let $z_{0} \in D$ and let $\left(z_{k}\right)$ be a sequence of points in $D$ that converges to $z_{0}$, but such that no $z_{k}$ equals $z_{0}$. Let $f: D \rightarrow \mathbb{C}$ and $g: D \rightarrow \mathbb{C}$ be analytic functions such that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $f(z)=g(z)$ for every $z \in D$.

(d) Let $D$ be the domain $\mathbb{C} \backslash\{0\}$. Give an example of an analytic function $f: D \rightarrow \mathbb{C}$ such that $f\left(n^{-1}\right)=0$ for every positive integer $n$ but $f$ is not identically 0 .

(e) Show that any function with the property described in (d) must have an essential singularity at the origin.

4.I.4E

comment(a) State and prove Morera's Theorem.

(b) Let $D$ be a domain and for each $n \in \mathbb{N}$ let $f_{n}: D \rightarrow \mathbb{C}$ be an analytic function. Suppose that $f: D \rightarrow \mathbb{C}$ is another function and that $f_{n} \rightarrow f$ uniformly on $D$. Prove that $f$ is analytic.

4.II.13E

comment(a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.

(b) Let $p(z)=z^{5}+z$. Find all $z$ such that $|z|=1$ and $\operatorname{Im}(p(z))=0$. Calculate $\operatorname{Re}(p(z))$ for each such $z$. [It will be helpful to set $z=e^{i \theta}$. You may use the addition formulae $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$ and $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$.]

(c) Let $\gamma:[0,2 \pi] \rightarrow \mathbb{C}$ be the closed path $\theta \mapsto e^{i \theta}$. Use your answer to (b) to give a rough sketch of the path $p \circ \gamma$, paying particular attention to where it crosses the real axis.

(d) Hence, or otherwise, determine for every real $t$ the number of $z$ (counted with multiplicity) such that $|z|<1$ and $p(z)=t$. (You need not give rigorous justifications for your calculations.)

1.I.4F

commentDescribe the geodesics (that is, hyperbolic straight lines) in either the disc model or the half-plane model of the hyperbolic plane. Explain what is meant by the statements that two hyperbolic lines are parallel, and that they are ultraparallel.

Show that two hyperbolic lines $l$ and $l^{\prime}$ have a unique common perpendicular if and only if they are ultraparallel.

[You may assume standard results about the group of isometries of whichever model of the hyperbolic plane you use.]

1.II.13F

commentWrite down the Riemannian metric in the half-plane model of the hyperbolic plane. Show that MÃ¶bius transformations mapping the upper half-plane to itself are isometries of this model.

Calculate the hyperbolic distance from $i b$ to $i c$, where $b$ and $c$ are positive real numbers. Assuming that the hyperbolic circle with centre $i b$ and radius $r$ is a Euclidean circle, find its Euclidean centre and radius.

Suppose that $a$ and $b$ are positive real numbers for which the points $i b$ and $a+i b$ of the upper half-plane are such that the hyperbolic distance between them coincides with the Euclidean distance. Obtain an expression for $b$ as a function of $a$. Hence show that, for any $b$ with $0<b<1$, there is a unique positive value of $a$ such that the hyperbolic distance between $i b$ and $a+i b$ coincides with the Euclidean distance.

3.I.4F

commentShow that any isometry of Euclidean 3 -space which fixes the origin can be written as a composite of at most three reflections in planes through the origin, and give (with justification) an example of an isometry for which three reflections are necessary.

3.II.14F

commentState and prove the Gauss-Bonnet formula for the area of a spherical triangle. Deduce a formula for the area of a spherical $n$-gon with angles $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$. For what range of values of $\alpha$ does there exist a (convex) regular spherical $n$-gon with angle $\alpha$ ?

Let $\Delta$ be a spherical triangle with angles $\pi / p, \pi / q$ and $\pi / r$ where $p, q, r$ are integers, and let $G$ be the group of isometries of the sphere generated by reflections in the three sides of $\Delta$. List the possible values of $(p, q, r)$, and in each case calculate the order of the corresponding group $G$. If $(p, q, r)=(2,3,5)$, show how to construct a regular dodecahedron whose group of symmetries is $G$.

[You may assume that the images of $\Delta$ under the elements of $G$ form a tessellation of the sphere.]

1.I $. 5 \mathrm{E} \quad$

commentLet $V$ be the subset of $\mathbb{R}^{5}$ consisting of all quintuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ such that

$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=0$

and

$a_{1}+2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}=0$

Prove that $V$ is a subspace of $\mathbb{R}^{5}$. Solve the above equations for $a_{1}$ and $a_{2}$ in terms of $a_{3}, a_{4}$ and $a_{5}$. Hence, exhibit a basis for $V$, explaining carefully why the vectors you give form a basis.

1.II.14E

comment(a) Let $U, U^{\prime}$ be subspaces of a finite-dimensional vector space $V$. Prove that $\operatorname{dim}\left(U+U^{\prime}\right)=\operatorname{dim} U+\operatorname{dim} U^{\prime}-\operatorname{dim}\left(U \cap U^{\prime}\right) .$

(b) Let $V$ and $W$ be finite-dimensional vector spaces and let $\alpha$ and $\beta$ be linear maps from $V$ to $W$. Prove that

$\operatorname{rank}(\alpha+\beta) \leqslant \operatorname{rank} \alpha+\operatorname{rank} \beta$

(c) Deduce from this result that

$\operatorname{rank}(\alpha+\beta) \geqslant|\operatorname{rank} \alpha-\operatorname{rank} \beta|$

(d) Let $V=W=\mathbb{R}^{n}$ and suppose that $1 \leqslant r \leqslant s \leqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=s-r$. Suppose that $r+s \geqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=n$.

2.I.6E

commentLet $a_{1}, a_{2}, \ldots, a_{n}$ be distinct real numbers. For each $i$ let $\mathbf{v}_{i}$ be the vector $\left(1, a_{i}, a_{i}^{2}, \ldots, a_{i}^{n-1}\right)$. Let $A$ be the $n \times n$ matrix with rows $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}$ and let $\mathbf{c}$ be a column vector of size $n$. Prove that $A \mathbf{c}=\mathbf{0}$ if and only if $\mathbf{c}=\mathbf{0}$. Deduce that the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} \operatorname{span} \mathbb{R}^{n}$.

[You may use general facts about matrices if you state them clearly.]

2.II.15E

comment(a) Let $A=\left(a_{i j}\right)$ be an $m \times n$ matrix and for each $k \leqslant n$ let $A_{k}$ be the $m \times k$ matrix formed by the first $k$ columns of $A$. Suppose that $n>m$. Explain why the nullity of $A$ is non-zero. Prove that if $k$ is minimal such that $A_{k}$ has non-zero nullity, then the nullity of $A_{k}$ is 1 .

(b) Suppose that no column of $A$ consists entirely of zeros. Deduce from (a) that there exist scalars $b_{1}, \ldots, b_{k}$ (where $k$ is defined as in (a)) such that $\sum_{j=1}^{k} a_{i j} b_{j}=0$ for every $i \leqslant m$, but whenever $\lambda_{1}, \ldots, \lambda_{k}$ are distinct real numbers there is some $i \leqslant m$ such that $\sum_{j=1}^{k} a_{i j} \lambda_{j} b_{j} \neq 0$.

(c) Now let $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m}$ and $\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{m}$ be bases for the same real $m$ dimensional vector space. Let $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ be distinct real numbers such that for every $j$ the vectors $\mathbf{v}_{1}+\lambda_{j} \mathbf{w}_{1}, \ldots, \mathbf{v}_{m}+\lambda_{j} \mathbf{w}_{m}$ are linearly dependent. For each $j$, let $a_{1 j}, \ldots, a_{m j}$ be scalars, not all zero, such that $\sum_{i=1}^{m} a_{i j}\left(\mathbf{v}_{i}+\lambda_{j} \mathbf{w}_{i}\right)=\mathbf{0}$. By applying the result of (b) to the matrix $\left(a_{i j}\right)$, deduce that $n \leqslant m$.

(d) It follows that the vectors $\mathbf{v}_{1}+\lambda \mathbf{w}_{1}, \ldots, \mathbf{v}_{m}+\lambda \mathbf{w}_{m}$ are linearly dependent for at most $m$ values of $\lambda$. Explain briefly how this result can also be proved using determinants.

3.I.7G

commentLet $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ and let $\beta$ be another endomorphism of $U$ that commutes with $\alpha$. If $\lambda$ is an eigenvalue of $\alpha$, show that $\beta$ maps the kernel of $\alpha-\lambda \iota$ into itself, where $\iota$ is the identity map. Suppose now that $\alpha$ is diagonalizable with $n$ distinct real eigenvalues where $n=\operatorname{dim} U$. Prove that if there exists an endomorphism $\beta$ of $U$ such that $\alpha=\beta^{2}$, then $\lambda \geqslant 0$ for all eigenvalues $\lambda$ of $\alpha$.

3.II.17G

commentDefine the determinant $\operatorname{det}(A)$ of an $n \times n$ complex matrix A. Let $A_{1}, \ldots, A_{n}$ be the columns of $A$, let $\sigma$ be a permutation of $\{1, \ldots, n\}$ and let $A^{\sigma}$ be the matrix whose columns are $A_{\sigma(1)}, \ldots, A_{\sigma(n)}$. Prove from your definition of determinant that $\operatorname{det}\left(A^{\sigma}\right)=\epsilon(\sigma) \operatorname{det}(A)$, where $\epsilon(\sigma)$ is the sign of the permutation $\sigma$. Prove also that $\operatorname{det}(A)=\operatorname{det}\left(A^{t}\right) .$

Define the adjugate matrix $\operatorname{adj}(A)$ and prove from your $\operatorname{definitions}$ that $A \operatorname{adj}(A)=$ $\operatorname{adj}(A) A=\operatorname{det}(A) I$, where $I$ is the identity matrix. Hence or otherwise, prove that if $\operatorname{det}(A) \neq 0$, then $A$ is invertible.

Let $C$ and $D$ be real $n \times n$ matrices such that the complex matrix $C+i D$ is invertible. By considering $\operatorname{det}(C+\lambda D)$ as a function of $\lambda$ or otherwise, prove that there exists a real number $\lambda$ such that $C+\lambda D$ is invertible. [You may assume that if a matrix $A$ is invertible, then $\operatorname{det}(A) \neq 0$.]

Deduce that if two real matrices $A$ and $B$ are such that there exists an invertible complex matrix $P$ with $P^{-1} A P=B$, then there exists an invertible real matrix $Q$ such that $Q^{-1} A Q=B$.

4.I.6G

commentLet $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ such that $\alpha^{2}=\alpha$. Show that $U$ can be written as the direct sum of the kernel of $\alpha$ and the image of $\alpha$. Hence or otherwise, find the characteristic polynomial of $\alpha$ in terms of the dimension of $U$ and the rank of $\alpha$. Is $\alpha$ diagonalizable? Justify your answer.

4.II.15G

commentLet $\alpha \in L(U, V)$ be a linear map between finite-dimensional vector spaces. Let

$\begin{gathered} M^{l}(\alpha)=\{\beta \in L(V, U): \beta \alpha=0\} \quad \text { and } \\ M^{r}(\alpha)=\{\beta \in L(V, U): \alpha \beta=0\} . \end{gathered}$

(a) Prove that $M^{l}(\alpha)$ and $M^{r}(\alpha)$ are subspaces of $L(V, U)$ of dimensions

$\begin{gathered} \operatorname{dim} M^{l}(\alpha)=(\operatorname{dim} V-\operatorname{rank} \alpha) \operatorname{dim} U \quad \text { and } \\ \operatorname{dim} M^{r}(\alpha)=\operatorname{dim} \operatorname{ker}(\alpha) \operatorname{dim} V \end{gathered}$

[You may use the result that there exist bases in $U$ and $V$ so that $\alpha$ is represented by

$\left(\begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array}\right)$

where $I_{r}$ is the $r \times r$ identity matrix and $r$ is the rank of $\left.\alpha .\right]$

(b) Let $\Phi: L(U, V) \rightarrow L\left(V^{*}, U^{*}\right)$ be given by $\Phi(\alpha)=\alpha^{*}$, where $\alpha^{*}$ is the dual map induced by $\alpha$. Prove that $\Phi$ is an isomorphism. [You may assume that $\Phi$ is linear, and you may use the result that a finite-dimensional vector space and its dual have the same dimension.]

(c) Prove that

$\Phi\left(M^{l}(\alpha)\right)=M^{r}\left(\alpha^{*}\right) \quad \text { and } \quad \Phi\left(M^{r}(\alpha)\right)=M^{l}\left(\alpha^{*}\right)$

[You may use the results that $(\beta \alpha)^{*}=\alpha^{*} \beta^{*}$ and that $\beta^{* *}$ can be identified with $\beta$ under the canonical isomorphism between a vector space and its double dual.]

(d) Conclude that $\operatorname{rank}(\alpha)=\operatorname{rank}\left(\alpha^{*}\right)$.

1.I.2D

commentFermat's principle of optics states that the path of a light ray connecting two points will be such that the travel time $t$ is a minimum. If the speed of light varies continuously in a medium and is a function $c(y)$ of the distance from the boundary $y=0$, show that the path of a light ray is given by the solution to

$c(y) y^{\prime \prime}+c^{\prime}(y)\left(1+y^{\prime 2}\right)=0$

where $y^{\prime}=\frac{d y}{d x}$, etc. Show that the path of a light ray in a medium where the speed of light $c$ is a constant is a straight line. Also find the path from $(0,0)$ to $(1,0)$ if $c(y)=y$, and sketch it.

1.II.11D

comment(a) Determine the Green's function $G(x, \xi)$ for the operator $\frac{d^{2}}{d x^{2}}+k^{2}$ on $[0, \pi]$ with Dirichlet boundary conditions by solving the boundary value problem

$\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta(x-\xi), \quad G(0)=0, G(\pi)=0$

when $k$ is not an integer.

(b) Use the method of Green's functions to solve the boundary value problem

$\frac{d^{2} y}{d x^{2}}+k^{2} y=f(x), \quad y(0)=a, y(\pi)=b$

when $k$ is not an integer.

2.I.2C

commentExplain briefly why the second-rank tensor

$\int_{S} x_{i} x_{j} d S(\mathbf{x})$

is isotropic, where $S$ is the surface of the unit sphere centred on the origin.

A second-rank tensor is defined by

$T_{i j}(\mathbf{y})=\int_{S}\left(y_{i}-x_{i}\right)\left(y_{j}-x_{j}\right) d S(\mathbf{x})$

where $S$ is the surface of the unit sphere centred on the origin. Calculate $T(\mathbf{y})$ in the form

$T_{i j}=\lambda \delta_{i j}+\mu y_{i} y_{j}$

where $\lambda$ and $\mu$ are to be determined.

By considering the action of $T$ on $\mathbf{y}$ and on vectors perpendicular to $\mathbf{y}$, determine the eigenvalues and associated eigenvectors of $T$.

2.II.11C

commentState the transformation law for an $n$ th-rank tensor $T_{i j \cdots k}$.

Show that the fourth-rank tensor

$c_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$

is isotropic for arbitrary scalars $\alpha, \beta$ and $\gamma$.

The stress $\sigma_{i j}$ and strain $e_{i j}$ in a linear elastic medium are related by

$\sigma_{i j}=c_{i j k l} e_{k l} .$

Given that $e_{i j}$ is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form

$\sigma_{i j}=\lambda e_{k k} \delta_{i j}+2 \mu e_{i j}$

Show that $e_{i j}$ can be written in the form $e_{i j}=p \delta_{i j}+d_{i j}$, where $d_{i j}$ is a traceless tensor and $p$ is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density $E=\frac{1}{2} \sigma_{i j} e_{i j}$ to be non-negative for any deformation of the solid are that

$\mu \geq 0 \quad \text { and } \quad \lambda \geq-\frac{2}{3} \mu .$

3.I.2D

commentConsider the path between two arbitrary points on a cone of interior angle $2 \alpha$. Show that the arc-length of the path $r(\theta)$ is given by

$\int\left(r^{2}+r^{\prime 2} \operatorname{cosec}^{2} \alpha\right)^{1 / 2} d \theta$

where $r^{\prime}=\frac{d r}{d \theta}$. By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.

3.II.12D

commentThe transverse displacement $y(x, t)$ of a stretched string clamped at its ends $x=0, l$ satisfies the equation

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}-2 k \frac{\partial y}{\partial t}, \quad y(x, 0)=0, \frac{\partial y}{\partial t}(x, 0)=\delta(x-a)$

where $c>0$ is the wave velocity, and $k>0$ is the damping coefficient. The initial conditions correspond to a sharp blow at $x=a$ at time $t=0$.

(a) Show that the subsequent motion of the string is given by

$y(x, t)=\frac{1}{\sqrt{\alpha_{n}^{2}-k^{2}}} \sum_{n} 2 e^{-k t} \sin \frac{\alpha_{n} a}{c} \sin \frac{\alpha_{n} x}{c} \sin /\left(\sqrt{\alpha_{n}^{2}-k^{2}} t\right)$

where $\alpha_{n}=\pi c n / l$.

(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?

4.I.2D

commentConsider the wave equation in a spherically symmetric coordinate system

$\frac{\partial^{2} u(r, t)}{\partial t^{2}}=c^{2} \Delta u(r, t)$

where $\Delta u=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$ is the spherically symmetric Laplacian operator.

(a) Show that the general solution to the equation above is

$u(r, t)=\frac{1}{r}[f(r+c t)+g(r-c t)]$

where $f(x), g(x)$ are arbitrary functions.

(b) Using separation of variables, determine the wave field $u(r, t)$ in response to a pulsating source at the origin $u(0, t)=A \sin \omega t$.

4.II.11D

commentThe velocity potential $\phi(r, \theta)$ for inviscid flow in two dimensions satisfies the Laplace equation

$\Delta \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0$

(a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) $0 \leqslant r \leqslant a$; (ii) $a \leqslant r<\infty$.

(b) Assuming $\phi$ is single-valued, solve the Laplace equation subject to the boundary conditions $\frac{\partial \phi}{\partial r}=0$ at $r=a$, and $\frac{\partial \phi}{\partial r} \rightarrow U \cos \theta$ as $r \rightarrow \infty$. Sketch the lines of constant potential.

2.I.5B

commentLet

$A=\left(\begin{array}{cccc} 1 & a & a^{2} & a^{3} \\ a^{3} & 1 & a & a^{2} \\ a^{2} & a^{3} & 1 & a \\ a & a^{2} & a^{3} & 1 \end{array}\right), \quad b=\left(\begin{array}{c} \gamma \\ 0 \\ 0 \\ \gamma a \end{array}\right), \quad \gamma=1-a^{4} \neq 0$

Find the LU factorization of the matrix $A$ and use it to solve the system $A x=b$.

2.II.14B

commentLet

$f^{\prime \prime}(0) \approx a_{0} f(-1)+a_{1} f(0)+a_{2} f(1)=\mu(f)$

be an approximation of the second derivative which is exact for $f \in \mathcal{P}_{2}$, the set of polynomials of degree $\leq 2$, and let

$e(f)=f^{\prime \prime}(0)-\mu(f)$

be its error.

(a) Determine the coefficients $a_{0}, a_{1}, a_{2}$.

(b) Using the Peano kernel theorem prove that, for $f \in C^{3}[-1,1]$, the set of threetimes continuously differentiable functions, the error satisfies the inequality

$|e(f)| \leq \frac{1}{3} \max _{x \in[-1,1]}\left|f^{\prime \prime \prime}(x)\right| .$

3.I.6B

commentGiven $(n+1)$ distinct points $x_{0}, x_{1}, \ldots, x_{n}$, let

$\ell_{i}(x)=\prod_{\substack{k=0 \\ k \neq i}}^{n} \frac{x-x_{k}}{x_{i}-x_{k}}$

be the fundamental Lagrange polynomials of degree $n$, let

$\omega(x)=\prod_{i=0}^{n}\left(x-x_{i}\right)$

and let $p$ be any polynomial of degree $\leq n$.

(a) Prove that $\sum_{i=0}^{n} p\left(x_{i}\right) \ell_{i}(x) \equiv p(x)$.

(b) Hence or otherwise derive the formula

$\frac{p(x)}{\omega(x)}=\sum_{i=0}^{n} \frac{A_{i}}{x-x_{i}}, \quad A_{i}=\frac{p\left(x_{i}\right)}{\omega^{\prime}\left(x_{i}\right)}$

which is the decomposition of $p(x) / \omega(x)$ into partial fractions.

3.II.16B

commentThe functions $H_{0}, H_{1}, \ldots$ are generated by the Rodrigues formula:

$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$

(a) Show that $H_{n}$ is a polynomial of degree $n$, and that the $H_{n}$ are orthogonal with respect to the scalar product

$(f, g)=\int_{-\infty}^{\infty} f(x) g(x) e^{-x^{2}} d x$

(b) By induction or otherwise, prove that the $H_{n}$ satisfy the three-term recurrence relation

$H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) .$

[Hint: you may need to prove the equality $H_{n}^{\prime}(x)=2 n H_{n-1}(x)$ as well.]

$3 . \mathrm{I} . 5 \mathrm{H} \quad$

commentTwo players A and B play a zero-sum game with the pay-off matrix

\begin{tabular}{r|rrr} & $B_{1}$ & $B_{2}$ & $B_{3}$ \ \hline$A_{1}$ & 4 & $-2$ & $-5$ \ $A_{2}$ & $-2$ & 4 & 3 \ $A_{3}$ & $-3$ & 6 & 2 \ $A_{4}$ & 3 & $-8$ & $-6$ \end{tabular}

Here, the $(i, j)$ entry of the matrix indicates the pay-off to player A if he chooses move $A_{i}$ and player $\mathrm{B}$ chooses move $B_{j}$. Show that the game can be reduced to a zero-sum game with $2 \times 2$ pay-off matrix.

Determine the value of the game and the optimal strategy for player A.

3.II.15H

commentExplain what is meant by a transportation problem where the total demand equals the total supply. Write the Lagrangian and describe an algorithm for solving such a problem. Starting from the north-west initial assignment, solve the problem with three sources and three destinations described by the table

\begin{tabular}{|rrr|r|} \hline 5 & 9 & 1 & 36 \ 3 & 10 & 6 & 84 \ 7 & 2 & 5 & 40 \ \hline 14 & 68 & 78 & \ \hline \end{tabular}

where the figures in the $3 \times 3$ box denote the transportation costs (per unit), the right-hand column denotes supplies, and the bottom row demands.

4.I.5H

commentState and prove the Lagrangian sufficiency theorem for a general optimization problem with constraints.

4.II.14H

commentUse the two-phase simplex method to solve the problem

$\begin{array}{llllll} \operatorname{minimize} & 5 x_{1}-12 x_{2}+13 x_{3} & & \\ \text { subject to } & 4 x_{1}+5 x_{2} & & \leq & 9 \\ & 6 x_{1}+4 x_{2}+ & x_{3} & \geq & 12 \\ & 3 x_{1}+2 x_{2}-x_{3} & \leq & 3 \\ & x_{i} \geq 0, & i=1,2,3 \end{array}$

1.I.8G

commentLet $U$ and $V$ be finite-dimensional vector spaces. Suppose that $b$ and $c$ are bilinear forms on $U \times V$ and that $b$ is non-degenerate. Show that there exist linear endomorphisms $S$ of $U$ and $T$ of $V$ such that $c(x, y)=b(S(x), y)=b(x, T(y))$ for all $(x, y) \in U \times V$.

1.II.17G

(a) Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$ and state Euler's criterion.

(b) Compute $\left(\frac{-1}{p}\right)$ and prove that

$\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$