Part IA, 2010, Paper 3

# Part IA, 2010, Paper 3

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Paper 3, Section I, D

commentExpress the element $(123)(234)$ in $S_{5}$ as a product of disjoint cycles. Show that it is in $A_{5}$. Write down the elements of its conjugacy class in $A_{5}$.

Paper 3, Section I, D

commentWrite down the matrix representing the following transformations of $\mathbb{R}^{3}$ :

(i) clockwise rotation of $45^{\circ}$ around the $x$ axis,

(ii) reflection in the plane $x=y$,

(iii) the result of first doing (i) and then (ii).

Paper 3, Section II, D

commentLet $G$ be a finite group, $X$ the set of proper subgroups of $G$. Show that conjugation defines an action of $G$ on $X$.

Let $B$ be a proper subgroup of $G$. Show that the orbit of $G$ on $X$ containing $B$ has size at most the index $|G: B|$. Show that there exists a $g \in G$ which is not conjugate to an element of $B$.

Paper 3, Section II, D

commentLet $G$ be a group, $X$ a set on which $G$ acts transitively, $B$ the stabilizer of a point $x \in X$.

Show that if $g \in G$ stabilizes the point $y \in X$, then there exists an $h \in G$ with $h g h^{-1} \in B$.

Let $G=S L_{2}(\mathbb{C})$, acting on $\mathbb{C} \cup\{\infty\}$ by MÃ¶bius transformations. Compute $B=G_{\infty}$, the stabilizer of $\infty$. Given

$g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G$

compute the set of fixed points $\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .$

Show that every element of $G$ is conjugate to an element of $B$.

Paper 3, Section II, D

commentState Lagrange's theorem. Let $p$ be a prime number. Prove that every group of order $p$ is cyclic. Prove that every abelian group of order $p^{2}$ is isomorphic to either $C_{p} \times C_{p}$ or $C_{p^{2} \text {. }}$

Show that $D_{12}$, the dihedral group of order 12 , is not isomorphic to the alternating $\operatorname{group} A_{4}$.

Paper 3, Section II, D

comment(i) State the orbit-stabilizer theorem.

Let $G$ be the group of rotations of the cube, $X$ the set of faces. Identify the stabilizer of a face, and hence compute the order of $G$.

Describe the orbits of $G$ on the set $X \times X$ of pairs of faces.

(ii) Define what it means for a subgroup $N$ of $G$ to be normal. Show that $G$ has a normal subgroup of order 4 .

Paper 3 , Section II, C

commentState the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

The surface $S_{1}$ is defined by the equation $z=3-2 x^{2}-2 y^{2}$ for $1 \leqslant z \leqslant 3$; the surface $S_{2}$ is defined by the equation $x^{2}+y^{2}=1$ for $0 \leqslant z \leqslant 1$; the surface $S_{3}$ is defined by the equation $z=0$ for $x, y$ satisfying $x^{2}+y^{2} \leqslant 1$. The surface $S$ is defined to be the union of the surfaces $S_{1}, S_{2}$ and $S_{3}$. Sketch the surfaces $S_{1}, S_{2}, S_{3}$ and (hence) $S$.

The vector field $\mathbf{F}$ is defined by

$\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)$

Evaluate the integral

$\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}$

where the surface element $\mathrm{d} \mathbf{S}$ points in the direction of the outward normal to $S$.

Paper 3, Section I, C

commentA curve in two dimensions is defined by the parameterised Cartesian coordinates

$x(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u$

where the constants $a, b>0$. Sketch the curve segment corresponding to the range $0 \leqslant u \leqslant 3 \pi$. What is the length of the curve segment between the points $(x(0), y(0))$ and $(x(U), y(U))$, as a function of $U$ ?

A geometrically sensitive ant walks along the curve with varying speed $\kappa(u)^{-1}$, where $\kappa(u)$ is the curvature at the point corresponding to parameter $u$. Find the time taken by the ant to walk from $(x(2 n \pi), y(2 n \pi))$ to $(x(2(n+1) \pi), y(2(n+1) \pi))$, where $n$ is a positive integer, and hence verify that this time is independent of $n$.

[You may quote without proof the formula $\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} .$ ]

Paper 3, Section I, C

commentConsider the vector field

$\mathbf{F}=\left(-y /\left(x^{2}+y^{2}\right), x /\left(x^{2}+y^{2}\right), 0\right)$

defined on all of $\mathbb{R}^{3}$ except the $z$ axis. Compute $\boldsymbol{\nabla} \times \mathbf{F}$ on the region where it is defined.

Let $\gamma_{1}$ be the closed curve defined by the circle in the $x y$-plane with centre $(2,2,0)$ and radius 1 , and $\gamma_{2}$ be the closed curve defined by the circle in the $x y$-plane with centre $(0,0,0)$ and radius 1 .

By using your earlier result, or otherwise, evaluate the line integral $\oint_{\gamma_{1}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$.

By explicit computation, evaluate the line integral $\oint_{\gamma_{2}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$. Is your result consistent with Stokes' theorem? Explain your answer briefly.

Paper 3, Section II, C

commentGiven a spherically symmetric mass distribution with density $\rho$, explain how to obtain the gravitational field $\mathbf{g}=-\nabla \phi$, where the potential $\phi$ satisfies Poisson's equation

$\nabla^{2} \phi=4 \pi G \rho$

The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells $S_{n}$ labelled by integers $n \geqslant 1$. The shell $S_{n}$ has uniform density $2^{n-1} \rho_{0}$, where $\rho_{0}$ is a constant, and occupies the volume between radius $2^{-n+1}$ and $2^{-n}$.

Obtain a closed form expression for the mass of Geometria.

Obtain a closed form expression for the gravitational field $\mathbf{g}$ due to Geometria at a distance $r=2^{-N}$ from its centre of mass, for each positive integer $N \geqslant 1$. What is the potential $\phi(r)$ due to Geometria for $r>1$ ?

Paper 3, Section II, C

commentLet $f(x, y)$ be a function of two variables, and $R$ a region in the $x y$-plane. State the rule for evaluating $\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y$ as an integral with respect to new variables $u(x, y)$ and $v(x, y)$.

Sketch the region $R$ in the $x y$-plane defined by

$R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}$

Sketch the corresponding region in the $u v$-plane, where

$u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}$

Express the integral

$I=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y$

as an integral with respect to $u$ and $v$. Hence, or otherwise, calculate $I$.

Paper 3, Section II, C

comment(a) Define a rank two tensor and show that if two rank two tensors $A_{i j}$ and $B_{i j}$ are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

The quantity $C_{i j}$ has the property that, for every rank two tensor $A_{i j}$, the quantity $C_{i j} A_{i j}$ is a scalar. Is $C_{i j}$ necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

(b) Show that, if a tensor $T_{i j}$ is invariant under rotations about the $x_{3}$-axis, then it has the form

$\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)$

(c) The inertia tensor about the origin of a rigid body occupying volume $V$ and with variable mass density $\rho(\mathbf{x})$ is defined to be

$I_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V$

The rigid body $B$ has uniform density $\rho$ and occupies the cylinder

$\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}$

Show that the inertia tensor of $B$ about the origin is diagonal in the $\left(x_{1}, x_{2}, x_{3}\right)$ coordinate system, and calculate its diagonal elements.