Part IB, 2002, Paper 3

# Part IB, 2002, Paper 3

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3.I.1E

commentLet $f: \mathbb{R} \rightarrow \mathbb{R}^{2}$ be defined by $f=(u, v)$, where $u$ and $v$ are defined by $u(0)=v(0)=0$ and, for $t \neq 0, u(t)=t^{2} \sin (1 / t)$ and $v(t)=t^{2} \cos (1 / t)$. Show that $f$ is differentiable on $\mathbb{R}$.

Show that for any real non-zero $a,\left\|f^{\prime}(a)-f^{\prime}(0)\right\|>1$, where we regard $f^{\prime}(a)$ as the vector $\left(u^{\prime}(a), v^{\prime}(a)\right)$ in $\mathbb{R}^{2}$.

3.I.8C

commentState and prove Kelvin's circulation theorem.

Consider a planar flow in the unbounded region outside a cylinder for which the vorticity vanishes everywhere at time $t=0$. What may be deduced about the circulation around closed loops in the fluid at time $t$ :

(i) that do not enclose the cylinder;

(ii) that enclose the cylinder?

Give a brief justification for your answer in each case.

3.II.18C

commentUse Euler's equation to derive Bernoulli's theorem for the steady flow of an inviscid fluid of uniform density $\rho$ in the absence of body forces.

Such a fluid flows steadily through a long cylindrical elastic tube having circular cross-section. The variable $z$ measures distance downstream along the axis of the tube. The tube wall has thickness $h(z)$, so that if the external radius of the tube is $r(z)$, its internal radius is $r(z)-h(z)$, where $h(z) \geqslant 0$ is a given slowly-varying function that tends to zero as $z \rightarrow \pm \infty$. The elastic tube wall exerts a pressure $p(z)$ on the fluid given as

$p(z)=p_{0}+k[r(z)-R]$

where $p_{0}, k$ and $R$ are positive constants. Far upstream, $r$ has the constant value $R$, the fluid pressure has the constant value $p_{0}$, and the fluid velocity $u$ has the constant value $V$. Assume that gravity is negligible and that $h(z)$ varies sufficiently slowly that the velocity may be taken as uniform across the tube at each value of $z$. Use mass conservation and Bernoulli's theorem to show that $u(z)$ satisfies

$\frac{h}{R}=1-\left(\frac{V}{u}\right)^{1 / 2}+\frac{1}{4} \lambda\left[1-\left(\frac{u}{V}\right)^{2}\right], \quad \text { where } \quad \lambda=\frac{2 \rho V^{2}}{k R}$

Sketch a graph of $h / R$ against $u / V$. Show that if $h(z)$ exceeds a critical value $h_{c}(\lambda)$, no such flow is possible and find $h_{c}(\lambda)$.

Show that if $h<h_{c}(\lambda)$ everywhere, then for given $h$ the equation has two positive solutions for $u$. Explain how the given value of $\lambda$ determines which solution should be chosen.

3.I.3G

commentLet $f: X \rightarrow Y$ be a continuous map between topological spaces. Let

$G_{f}=\{(x, f(x)): x \in X\} .$

(a) Show that if $Y$ is Hausdorff, then $G_{f}$ is closed in $X \times Y$.

(b) Show that if $X$ is compact, then $G_{f}$ is also compact.

3.II.13G

comment(a) Let $f$ and $g$ be two analytic functions on a domain $D$ and let $\gamma \subset D$ be a simple closed curve homotopic in $D$ to a point. If $|g(z)|<|f(z)|$ for every $z$ in $\gamma$, prove that $\gamma$ encloses the same number of zeros of $f$ as of $f+g$.

(b) Let $g$ be an analytic function on the disk $|z|<1+\epsilon$, for some $\epsilon>0$. Suppose that $g$ maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that $g$ has exactly one fixed point in the open unit disk.

(c) Prove that, if $|a|<1$, then

$z^{m}\left(\frac{z-a}{1-\bar{a} z}\right)^{n}-a$

has $m+n$ zeros in $|z|<1$.

3.I.4E

commentState Euler's formula for a graph $\mathcal{G}$ with $F$ faces, $E$ edges and $V$ vertices on the surface of a sphere.

Suppose that every face in $\mathcal{G}$ has at least three edges, and that at least three edges meet at every vertex of $\mathcal{G}$. Let $F_{n}$ be the number of faces of $\mathcal{G}$ that have exactly $n$ edges $(n \geqslant 3)$, and let $V_{m}$ be the number of vertices at which exactly $m$ edges meet $(m \geqslant 3)$. By expressing $6 F-\sum_{n} n F_{n}$ in terms of the $V_{j}$, or otherwise, show that every convex polyhedron has at least four faces each of which is a triangle, a quadrilateral or a pentagon.

3.II.14E

commentShow that every isometry of Euclidean space $\mathbb{R}^{3}$ is a composition of reflections in planes

What is the smallest integer $N$ such that every isometry $f$ of $\mathbb{R}^{3}$ with $f(0)=0$ can be expressed as the composition of at most $N$ reflections? Give an example of an isometry that needs this number of reflections and justify your answer.

Describe (geometrically) all twelve orientation-reversing isometries of a regular tetrahedron.

3.I $77 \mathrm{~F} \quad$

commentWhich of the following statements are true, and which false? Give brief justifications for your answers.

(a) If $U$ and $W$ are subspaces of a vector space $V$, then $U \cap W$ is always a subspace of $V$.

(b) If $U$ and $W$ are distinct subspaces of a vector space $V$, then $U \cup W$ is never a subspace of $V$.

(c) If $U, W$ and $X$ are subspaces of a vector space $V$, then $U \cap(W+X)=$ $(U \cap W)+(U \cap X)$.

(d) If $U$ is a subspace of a finite-dimensional space $V$, then there exists a subspace $W$ such that $U \cap W=\{0\}$ and $U+W=V$.

3.II.17F

commentDefine the determinant of an $n \times n$ matrix $A$, and prove from your definition that if $A^{\prime}$ is obtained from $A$ by an elementary row operation (i.e. by adding a scalar multiple of the $i$ th row of $A$ to the $j$ th row, for some $j \neq i$ ), then $\operatorname{det} A^{\prime}=\operatorname{det} A$.

Prove also that if $X$ is a $2 n \times 2 n$ matrix of the form

$\left(\begin{array}{ll} A & B \\ O & C \end{array}\right)$

where $O$ denotes the $n \times n$ zero matrix, then $\operatorname{det} X=\operatorname{det} A$ det $C$. Explain briefly how the $2 n \times 2 n$ matrix

$\left(\begin{array}{ll} B & I \\ O & A \end{array}\right)$

can be transformed into the matrix

$\left(\begin{array}{cc} B & I \\ -A B & O \end{array}\right)$

by a sequence of elementary row operations. Hence or otherwise prove that $\operatorname{det} A B=$ $\operatorname{det} A \operatorname{det} B$.

3.I.2A

commentWrite down the wave equation for the displacement $y(x, t)$ of a stretched string with constant mass density and tension. Obtain the general solution in the form

$y(x, t)=f(x+c t)+g(x-c t)$

where $c$ is the wave velocity. For a solution in the region $0 \leqslant x<\infty$, with $y(0, t)=0$ and $y \rightarrow 0$ as $x \rightarrow \infty$, show that

$E=\int_{0}^{\infty}\left[\frac{1}{2}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{1}{2} c^{2}\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x$

is constant in time. Express $E$ in terms of the general solution in this case.

3.II.12A

commentConsider the real Sturm-Liouville problem

$\mathcal{L} y(x)=-\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=\lambda r(x) y,$

with the boundary conditions $y(a)=y(b)=0$, where $p, q$ and $r$ are continuous and positive on $[a, b]$. Show that, with suitable choices of inner product and normalisation, the eigenfunctions $y_{n}(x), \quad n=1,2,3 \ldots$, form an orthonormal set.

Hence show that the corresponding Green's function $G(x, \xi)$ satisfying

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

where $\mu$ is not an eigenvalue, is

$G(x, \xi)=\sum_{n=1}^{\infty} \frac{y_{n}(x) y_{n}(\xi)}{\lambda_{n}-\mu}$

where $\lambda_{n}$ is the eigenvalue corresponding to $y_{n}$.

Find the Green's function in the case where

$\mathcal{L} y \equiv y^{\prime \prime},$

with boundary conditions $y(0)=y(\pi)=0$, and deduce, by suitable choice of $\mu$, that

$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$

3.I.6B

commentFor numerical integration, a quadrature formula

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

is applied which is exact on $\mathcal{P}_{n}$, i.e., for all polynomials of degree $n$.

Prove that such a formula is exact for all $f \in \mathcal{P}_{2 n+1}$ if and only if $x_{i}, i=0, \ldots, n$, are the zeros of an orthogonal polynomial $p_{n+1} \in \mathcal{P}_{n+1}$ which satisfies $\int_{a}^{b} p_{n+1}(x) r(x) d x=0$ for all $r \in \mathcal{P}_{n}$. [You may assume that $p_{n+1}$ has $(n+1)$ distinct zeros.]

3.II.16B

comment(a) Consider a system of linear equations $A x=b$ with a non-singular square $n \times n$ matrix $A$. To determine its solution $x=x^{*}$ we apply the iterative method

$x^{k+1}=H x^{k}+v .$

Here $v \in \mathbb{R}^{n}$, while the matrix $H \in \mathbb{R}^{n \times n}$ is such that $x^{*}=H x^{*}+v$ implies $A x^{*}=b$. The initial vector $x^{0} \in \mathbb{R}^{n}$ is arbitrary. Prove that, if the matrix $H$ possesses $n$ linearly independent eigenvectors $w_{1}, \ldots, w_{n}$ whose corresponding eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$ satisfy $\max _{i}\left|\lambda_{i}\right|<1$, then the method converges for any choice of $x^{0}$, i.e. $x^{k} \rightarrow x^{*}$ as $k \rightarrow \infty$.

(b) Describe the Jacobi iteration method for solving $A x=b$. Show directly from the definition of the method that, if the matrix $A$ is strictly diagonally dominant by rows, i.e.

$\left|a_{i i}\right|^{-1} \sum_{j=1, j \neq i}^{n}\left|a_{i j}\right| \leq \gamma<1, \quad i=1, \ldots, n,$

then the method converges.

3.II.15H

commentConsider the following linear programming problem

$\begin{array}{ll} \text { maximise } & -2 x_{1}+3 x_{2} \\ \text { subject to } & x_{1}-x_{2} \geq 1 \\ & 4 x_{1}-x_{2} \geq 10 \\ & x_{2} \leq 6 \\ & x_{i} \geq 0, i=1,2 \end{array}$

Write down the Phase One problem for (1) and solve it.

By using the solution of the Phase One problem as an initial basic feasible solution for the Phase Two simplex algorithm, solve (1), i.e., find the optimal tableau and read the optimal solution $\left(x_{1}, x_{2}\right)$ and optimal value from it.

3.I $. 9 \mathrm{~F} \quad$

commentExplain what is meant by a quadratic residue modulo an odd prime $p$, and show that $a$ is a quadratic residue modulo $p$ if and only if $a^{\frac{1}{2}(p-1)} \equiv 1(\bmod p)$. Hence characterize the odd primes $p$ for which $-1$ is a quadratic residue.

State the law of quadratic reciprocity, and use it to determine whether 73 is a quadratic residue (mod 127).

3.II.19F

commentExplain what is meant by saying that a positive definite integral quadratic form $f(x, y)=a x^{2}+b x y+c y^{2}$ is reduced, and show that every positive definite form is equivalent to a reduced form

State a criterion for a prime number $p$ to be representable by some form of discriminant $d$, and deduce that $p$ is representable by a form of discriminant $-32$ if and only if $p \equiv 1,2$ or $3(\bmod 8)$. Find the reduced forms of discriminant $-32$, and hence or otherwise show that a prime $p$ is representable by the form $3 x^{2}+2 x y+3 y^{2}$ if and only if $p \equiv 3(\bmod 8)$.

[Standard results on when $-1$ and 2 are squares (mod $p$ ) may be assumed.]

3.II.20D

commentA quantum mechanical system has two states $\chi_{0}$ and $\chi_{1}$, which are normalised energy eigenstates of a Hamiltonian $\mathrm{H}_{3}$, with

$H_{3} \chi_{0}=-\chi_{0}, \quad H_{3} \chi_{1}=+\chi_{1} .$

A general time-dependent state may be written

$\Psi(t)=a_{0}(t) \chi_{0}+a_{1}(t) \chi_{1},$

where $a_{0}(t)$ and $a_{1}(t)$ are complex numbers obeying $\left|a_{0}(t)\right|^{2}+\left|a_{1}(t)\right|^{2}=1$.

(a) Write down the time-dependent Schrödinger equation for $\Psi(t)$, and show that if the Hamiltonian is $\mathrm{H}_{3}$, then

$i \hbar \frac{d a_{0}}{d t}=-a_{0}, \quad i \hbar \frac{d a_{1}}{d t}=+a_{1} .$

For the general state given in equation (1) above, write down the probability to observe the system, at time $t$, in a state $\alpha \chi_{0}+\beta \chi_{1}$, properly normalised so that $|\alpha|^{2}+|\beta|^{2}=1$.

(b) Now consider starting the system in the state $\chi_{0}$ at time $t=0$, and evolving it with a different Hamiltonian $H_{1}$, which acts on the states $\chi_{0}$ and $\chi_{1}$ as follows:

$H_{1} \chi_{0}=\chi_{1}, \quad H_{1} \chi_{1}=\chi_{0}$

By solving the time-dependent Schrödinger equation for the Hamiltonian $H_{1}$, find $a_{0}(t)$ and $a_{1}(t)$ in this case. Hence determine the shortest time $T>0$ such that $\Psi(T)$ is an eigenstate of $H_{3}$ with eigenvalue $+1$.

(c) Now consider taking the state $\Psi(T)$ from part (b), and evolving it for further length of time $T$, with Hamiltonian $H_{2}$, which acts on the states $\chi_{0}$ and $\chi_{1}$ as follows:

$H_{2} \chi_{0}=-i \chi_{1}, \quad H_{2} \chi_{1}=i \chi_{0}$

What is the final state of the system? Is this state observationally distinguishable from the original state $\chi_{0}$ ?

3.I.10D

commentWrite down the formulae for a Lorentz transformation with velocity $v$ taking one set of co-ordinates $(t, x)$ to another $\left(t^{\prime}, x^{\prime}\right)$.

Imagine you observe a train travelling past Cambridge station at a relativistic speed $u$. Someone standing still on the train throws a ball in the direction the train is moving, with speed $v$. How fast do you observe the ball to be moving? Justify your answer.