Part IB, 2008, Paper 3

Analysis II

• 3.II.13F

  Analysis II | Part IB, 2008

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function, and $\left(x_{0}, y_{0}\right)$ a point of $\mathbb{R}^{2}$. Prove that if the partial derivatives of $f$ exist in some open disc around $\left(x_{0}, y_{0}\right)$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

  Now let $X$ denote the vector space of all $(n \times n)$ real matrices, and let $f: X \rightarrow \mathbb{R}$ be the function assigning to each matrix its determinant. Show that $f$ is differentiable at the identity matrix $I$, and that $\left.D f\right|_{I}$ is the linear map $H \mapsto \operatorname{tr} H$. Deduce that $f$ is differentiable at any invertible matrix $A$, and that $\left.D f\right|_{A}$ is the linear map $H \mapsto \operatorname{det} A \operatorname{tr}\left(A^{-1} H\right) .$

  Show also that if $K$ is a matrix with $\|K\|<1$, then $(I+K)$ is invertible. Deduce that $f$ is twice differentiable at $I$, and find $\left.D^{2} f\right|_{I}$ as a bilinear map $X \times X \rightarrow \mathbb{R}$.

  [You may assume that the norm $\|-\|$ on $X$ is complete, and that it satisfies the inequality $\|A B\| \leqslant\|A\| \cdot\|B\|$ for any two matrices $A$ and $B .]$

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Complex Analysis

• 3.II.14E

  Complex Analysis | Part IB, 2008

  State and prove Rouché's theorem, and use it to count the number of zeros of $3 z^{9}+8 z^{6}+z^{5}+2 z^{3}+1$ inside the annulus $\{z: 1<|z|<2\}$.

  Let $\left(p_{n}\right)_{n=1}^{\infty}$ be a sequence of polynomials of degree at most $d$ with the property that $p_{n}(z)$ converges uniformly on compact subsets of $\mathbb{C}$ as $n \rightarrow \infty$. Prove that there is a polynomial $p$ of degree at most $d$ such that $p_{n} \rightarrow p$ uniformly on compact subsets of $\mathbb{C}$. [If you use any results about uniform convergence of analytic functions, you should prove them.]

  Suppose that $p$ has $d$ distinct roots $z_{1}, \ldots, z_{d}$. Using Rouché's theorem, or otherwise, show that for each $i$ there is a sequence $\left(z_{i, n}\right)_{n=1}^{\infty}$ such that $p_{n}\left(z_{i, n}\right)=0$ and $z_{i, n} \rightarrow z_{i}$ as $n \rightarrow \infty$.

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Complex Methods

• 3.I.5C

  Complex Methods | Part IB, 2008

  Using the contour integration formula for the inversion of Laplace transforms find the inverse Laplace transforms of the following functions: (a) $\frac{s}{s^{2}+a^{2}} \quad(a$ real and non-zero $)$, (b) $\frac{1}{\sqrt{s}}$.

  [You may use the fact that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\pi / b}$.]

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Electromagnetism

• 3.II.17B

  Electromagnetism | Part IB, 2008

  (i) From Maxwell's equations in vacuum,

  $$\begin{array}{ll} \nabla \cdot \mathbf{E}=0 & \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0 & \nabla \times \mathbf{B}=\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t} \end{array}$$

  obtain the wave equation for the electric field E. [You may find the following identity useful: $\left.\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A} .\right]$

  (ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are

  $$\mathbf{E}(z, t)=\mathbf{E}_{0} \mathrm{e}^{i(k z-\omega t)} \text { and } \mathbf{B}(z, t)=\mathbf{B}_{0} \mathrm{e}^{i(k z-\omega t)}$$

  show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).

  (iii) Use Faraday's law for these fields to show that

  $$\mathbf{B}_{0}=\frac{k}{\omega}\left(\hat{\mathbf{e}}_{z} \times \mathbf{E}_{0}\right)$$

  (iv) Explain with symmetry arguments how these results generalise to

  $$\mathbf{E}(\mathbf{r}, t)=E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \hat{\mathbf{n}} \quad \text { and } \quad \mathbf{B}(\mathbf{r}, t)=\frac{1}{c} E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}(\hat{\mathbf{k}} \times \hat{\mathbf{n}})$$

  where $\hat{\mathbf{n}}$ is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and $\hat{\mathbf{k}}$ is the unit vector in the direction of propagation of the wave.

  (v) Using Maxwell's equations in vacuum prove that:

  $$\oint_{\mathcal{A}}\left(1 / \mu_{0}\right)(\mathbf{E} \times \mathbf{B}) \cdot d \mathcal{A}=-\frac{\partial}{\partial t} \int_{\mathcal{V}}\left(\frac{\epsilon_{0} E^{2}}{2}+\frac{B^{2}}{2 \mu_{0}}\right) d V$$

  where $\mathcal{V}$ is the closed volume and $\mathcal{A}$ is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.

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Fluid Dynamics

• 3.II.18B

  Fluid Dynamics | Part IB, 2008

  An ideal liquid contained within a closed circular cylinder of radius $a$ rotates about the axis of the cylinder (assume this axis to be in the vertical $z$-direction).

  (i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity $\mathbf{v}=\boldsymbol{\Omega} \times \mathbf{r}$, where $\boldsymbol{\Omega}=\Omega \hat{\mathbf{e}}_{z}$ is the angular velocity, with $\hat{\mathbf{e}}_{z}$ the unit vector in the $z$-direction, which depends only on time, and $\mathbf{r}$ is the position vector measured from a point on the axis of rotation.

  (ii) Calculate the angular momentum $\mathbf{M}=\rho \int(\mathbf{r} \times \mathbf{v}) d V$ per unit length of the cylinder.

  (iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass $\mathbf{f}=(\alpha x+\beta y, \gamma x+\delta y, 0)$. By taking the curl of the Euler equation, prove that

  $$\frac{d \Omega}{d t}=\frac{1}{2}(\gamma-\beta)$$

  (iv) Find the pressure.

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Geometry

• 3.I.2G

  Geometry | Part IB, 2008

  A smooth surface in $\mathbb{R}^{3}$ has parametrization

  $$\sigma(u, v)=\left(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+u^{2} v, u^{2}-v^{2}\right) .$$

  Show that a unit normal vector at the point $\sigma(u, v)$ is

  $$\left(\frac{-2 u}{1+u^{2}+v^{2}}, \frac{2 v}{1+u^{2}+v^{2}}, \frac{1-u^{2}-v^{2}}{1+u^{2}+v^{2}}\right)$$

  and that the curvature is $\frac{-4}{\left(1+u^{2}+v^{2}\right)^{4}}$.

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• 3.II.12G

  Geometry | Part IB, 2008

  Let $D$ be the unit disc model of the hyperbolic plane, with metric

  $$\frac{4|d \zeta|^{2}}{\left(1-|\zeta|^{2}\right)^{2}}$$

  (i) Show that the group of Möbius transformations mapping $D$ to itself is the group of transformations

  $$\zeta \mapsto \omega \frac{\zeta-\lambda}{\bar{\lambda} \zeta-1},$$

  where $|\lambda|<1$ and $|\omega|=1$.

  (ii) Assuming that the transformations in (i) are isometries of $D$, show that any hyperbolic circle in $D$ is a Euclidean circle.

  (iii) Let $P$ and $Q$ be points on the unit circle with $\angle P O Q=2 \alpha$. Show that the hyperbolic distance from $O$ to the hyperbolic line $P Q$ is given by

  $$2 \tanh ^{-1}\left(\frac{1-\sin \alpha}{\cos \alpha}\right)$$

  (iv) Deduce that if $a>2 \tanh ^{-1}(2-\sqrt{3})$ then no hyperbolic open disc of radius $a$ is contained in a hyperbolic triangle.

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Groups, Rings and Modules

• 3.I.1G

  Groups, Rings and Modules | Part IB, 2008

  Let $G$ be the abelian group generated by elements $a, b, c, d$ subject to the relations

  $$4 a-2 b+2 c+12 d=0, \quad-2 b+2 c=0, \quad 2 b+2 c=0, \quad 8 a+4 c+24 d=0$$

  Express $G$ as a product of cyclic groups, and find the number of elements of $G$ of order 2 .

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Linear Algebra

• 3.II.10E

  Linear Algebra | Part IB, 2008

  Let $k=\mathbb{R}$ or $\mathbb{C}$. What is meant by a quadratic form $q: k^{n} \rightarrow k$ ? Show that there is a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ for $k^{n}$ such that, writing $x=x_{1} v_{1}+\ldots+x_{n} v_{n}$, we have $q(x)=a_{1} x_{1}^{2}+\ldots+a_{n} x_{n}^{2}$ for some scalars $a_{1}, \ldots, a_{n} \in\{-1,0,1\} .$

  Suppose that $k=\mathbb{R}$. Define the rank and signature of $q$ and compute these quantities for the form $q: \mathbb{R}^{3} \rightarrow \mathbb{R}$ given by $q(x)=-3 x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$.

  Suppose now that $k=\mathbb{C}$ and that $q_{1}, \ldots, q_{d}: \mathbb{C}^{n} \rightarrow \mathbb{C}$ are quadratic forms. If $n \geqslant 2^{d}$, show that there is some nonzero $x \in \mathbb{C}^{n}$ such that $q_{1}(x)=\ldots=q_{d}(x)=0$.

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Markov Chains

• 3.I.9H

  Markov Chains | Part IB, 2008

  What does it mean to say that a Markov chain is recurrent?

  Stating clearly any general results to which you appeal, prove that the symmetric simple random walk on $\mathbb{Z}$ is recurrent.

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Methods

• 3.I.6D

  Methods | Part IB, 2008

  Let $\mathcal{L}$ be the operator

  $$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}-k^{2} y$$

  on functions $y(x)$ satisfying $\lim _{x \rightarrow-\infty} \quad y(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$.

  Given that the Green's function $G(x ; \xi)$ for $\mathcal{L}$ satisfies

  $$\mathcal{L} G=\delta(x-\xi)$$

  show that a solution of

  $$\mathcal{L} y=S(x)$$

  for a given function $S(x)$, is given by

  $$y(x)=\int_{-\infty}^{\infty} G(x ; \xi) S(\xi) d \xi$$

  Indicate why this solution is unique.

  Show further that the Green's function is given by

  $$G(x ; \xi)=-\frac{1}{2|k|} \exp (-|k||x-\xi|)$$

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• 3.II.15D

  Methods | Part IB, 2008

  Let $\lambda_{1}<\lambda_{2}<\ldots \lambda_{n} \ldots$ and $y_{1}(x), y_{2}(x), \ldots y_{n}(x) \ldots$ be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system

  $$\mathcal{L} y_{n}=\lambda_{n} w(x) y_{n},$$

  where

  $$\mathcal{L} y \equiv \frac{d}{d x}\left(-p(x) \frac{d y}{d x}\right)+q(x) y,$$

  with $p(x)>0$ and $w(x)>0$. The boundary conditions on $y$ are that $y(0)=y(1)=0$.

  Show that two distinct eigenfunctions are orthogonal in the sense that

  $$\int_{0}^{1} w y_{n} y_{m} d x=\delta_{n m} \int_{0}^{1} w y_{n}^{2} d x .$$

  Show also that if $y$ has the form

  $$y=\sum_{n=1}^{\infty} a_{n} y_{n},$$

  with $a_{n}$ being independent of $x$, then

  $$\frac{\int_{0}^{1} y \mathcal{L} y d x}{\int_{0}^{1} w y^{2} d x} \geq \lambda_{1}$$

  Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,

  $$\frac{\partial y}{\partial t}=-\frac{1}{w} \mathcal{L} y$$

  that satisfies the boundary conditions given above is such that

  $$\frac{1}{2} \frac{d}{d t}\left(\int_{0}^{1} w y^{2} d x\right) \leq-\lambda_{1} \int_{0}^{1} w y^{2} d x .$$

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Numerical Analysis

• 3.II.19D

  Numerical Analysis | Part IB, 2008

  Starting from the Taylor formula for $f(x) \in C^{k+1}[a, b]$ with an integral remainder term, show that the error of an approximant $L(f)$ can be written in the form (Peano kernel theorem)

  $$L(f)=\frac{1}{k !} \int_{a}^{b} K(\theta) f^{(k+1)}(\theta) d \theta,$$

  when $L(f)$, which is identically zero if $f(x)$ is a polynomial of degree $k$, satisfies conditions that you should specify. Give an expression for $K(\theta)$.

  Hence determine the minimum value of $c$ in the inequality

  $$|L(f)| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty}$$

  when

  $$L(f)=f^{\prime}(1)-\frac{1}{2}(f(2)-f(0)) \text { for } f(x) \in C^{3}[0,2]$$

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Optimization

• 3.II.20H

  Optimization | Part IB, 2008

  Use the simplex algorithm to solve the problem

  $$\max x_{1}+2 x_{2}-6 x_{3}$$

  subject to $x_{1}, x_{2} \geqslant 0,\left|x_{3}\right| \leqslant 5$, and

  $$\begin{array}{r} x_{1}+x_{2}+x_{3} \leqslant 7 \\ 2 x_{2}+x_{3} \geqslant 1 \end{array}$$

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Quantum Mechanics

• 3.I.7A

  Quantum Mechanics | Part IB, 2008

  Write down a formula for the orbital angular momentum operator $\hat{\mathbf{L}}$. Show that its components satisfy

  $$\left[L_{i}, L_{j}\right]=i \hbar \epsilon_{i j k} L_{k} .$$

  If $L_{3} \psi=0$, show that $\left(L_{1} \pm i L_{2}\right) \psi$ are also eigenvectors of $L_{3}$, and find their eigenvalues.

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• 3.II.16A

  Quantum Mechanics | Part IB, 2008

  What is the probability current for a particle of mass $m$, wavefunction $\psi$, moving in one dimension?

  A particle of energy $E$ is incident from $x<0$ on a barrier given by

  $$V(x)=\left\{\begin{array}{cc} 0 & x \leqslant 0 \\ V_{1} & 0<x<a \\ V_{0} & x \geqslant a \end{array}\right.$$

  where $V_{1}>V_{0}>0$. What are the conditions satisfied by $\psi$ at $x=0$ and $x=a$ ? Write down the form taken by the wavefunction in the regions $x \leqslant 0$ and $x \geqslant a$ distinguishing between the cases $E>V_{0}$ and $E<V_{0}$. For both cases, use your expressions for $\psi$ to calculate the probability currents in these two regions.

  Define the reflection and transmission coefficients, $R$ and $T$. Using current conservation, show that the expressions you have derived satisfy $R+T=1$. Show that $T=0$ if $0<E<V_{0}$.

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