Part IA, 2018, Paper 3

# Part IA, 2018, Paper 3

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Paper 3, Section I, D

commentProve that every member of $O(3)$ is a product of at most three reflections.

Is every member of $O(3)$ a product of at most two reflections? Justify your answer.

Paper 3, Section I, D

commentFind the order and the sign of the permutation $(13)(2457)(815) \in S_{8}$.

How many elements of $S_{6}$ have order $6 ?$ And how many have order $3 ?$

What is the greatest order of any element of $A_{9}$ ?

Paper 3, Section II, D

commentState and prove the Direct Product Theorem.

Is the group $O(3)$ isomorphic to $S O(3) \times C_{2} ?$ Is $O(2)$ isomorphic to $S O(2) \times C_{2}$ ?

Let $U(2)$ denote the group of all invertible $2 \times 2$ complex matrices $A$ with $A \bar{A}^{\mathrm{T}}=I$, and let $S U(2)$ be the subgroup of $U(2)$ consisting of those matrices with determinant $1 .$

Determine the centre of $U(2)$.

Write down a surjective homomorphism from $U(2)$ to the group $T$ of all unit-length complex numbers whose kernel is $S U(2)$. Is $U(2)$ isomorphic to $S U(2) \times T$ ?

Paper 3, Section II, D

commentDefine the quotient group $G / H$, where $H$ is a normal subgroup of a group $G$. You should check that your definition is well-defined. Explain why, for $G$ finite, the greatest order of any element of $G / H$ is at most the greatest order of any element of $G$.

Show that a subgroup $H$ of a group $G$ is normal if and only if there is a homomorphism from $G$ to some group whose kernel is $H$.

A group is called metacyclic if it has a cyclic normal subgroup $H$ such that $G / H$ is cyclic. Show that every dihedral group is metacyclic.

Which groups of order 8 are metacyclic? Is $A_{4}$ metacyclic? For which $n \leqslant 5$ is $S_{n}$ metacyclic?

Paper 3, Section II, D

commentLet $g$ be an element of a group $G$. We define a map $g^{*}$ from $G$ to $G$ by sending $x$ to $g x g^{-1}$. Show that $g^{*}$ is an automorphism of $G$ (that is, an isomorphism from $G$ to $G$ ).

Now let $A$ denote the group of automorphisms of $G$ (with the group operation being composition), and define a map $\theta$ from $G$ to $A$ by setting $\theta(g)=g^{*}$. Show that $\theta$ is a homomorphism. What is the kernel of $\theta$ ?

Prove that the image of $\theta$ is a normal subgroup of $A$.

Show that if $G$ is cyclic then $A$ is abelian. If $G$ is abelian, must $A$ be abelian? Justify your answer.

Paper 3, Section II, D

commentDefine the sign of a permutation $\sigma \in S_{n}$. You should show that it is well-defined, and also that it is multiplicative (in other words, that it gives a homomorphism from $S_{n}$ to $\{\pm 1\})$.

Show also that (for $n \geqslant 2$ ) this is the only surjective homomorphism from $S_{n}$ to $\{\pm 1\}$.

Paper 3, Section I, $4 \mathbf{C}$

commentIn plane polar coordinates $(r, \theta)$, the orthonormal basis vectors $\mathbf{e}_{r}$ and $\mathbf{e}_{\theta}$ satisfy

$\frac{\partial \mathbf{e}_{r}}{\partial r}=\frac{\partial \mathbf{e}_{\theta}}{\partial r}=\mathbf{0}, \quad \frac{\partial \mathbf{e}_{r}}{\partial \theta}=\mathbf{e}_{\theta}, \quad \frac{\partial \mathbf{e}_{\theta}}{\partial \theta}=-\mathbf{e}_{r}, \quad \text { and } \quad \boldsymbol{\nabla}=\mathbf{e}_{r} \frac{\partial}{\partial r}+\mathbf{e}_{\theta} \frac{1}{r} \frac{\partial}{\partial \theta}$

Hence derive the expression $\nabla \cdot \nabla \phi=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}$ for the Laplacian operator $\nabla^{2}$.

Calculate the Laplacian of $\phi(r, \theta)=\alpha r^{\beta} \cos (\gamma \theta)$, where $\alpha, \beta$ and $\gamma$ are constants. Hence find all solutions to the equation

$\nabla^{2} \phi=0 \quad \text { in } \quad 0 \leqslant r \leqslant a, \quad \text { with } \quad \partial \phi / \partial r=\cos (2 \theta) \text { on } r=a$

Explain briefly how you know that there are no other solutions.

Paper 3, Section I, C

commentDerive a formula for the curvature of the two-dimensional curve $\mathbf{x}(u)=(u, f(u))$.

Verify your result for the semicircle with radius $a$ given by $f(u)=\sqrt{a^{2}-u^{2}}$.

Paper 3, Section II, C

comment(a) Suppose that a tensor $T_{i j}$ can be decomposed as

$T_{i j}=S_{i j}+\epsilon_{i j k} V_{k}$

where $S_{i j}$ is symmetric. Obtain expressions for $S_{i j}$ and $V_{k}$ in terms of $T_{i j}$, and check that $(*)$ is satisfied.

(b) State the most general form of an isotropic tensor of rank $k$ for $k=0,1,2,3$, and verify that your answers are isotropic.

(c) The general form of an isotropic tensor of rank 4 is

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

Suppose that $A_{i j}$ and $B_{i j}$ satisfy the linear relationship $A_{i j}=T_{i j k l} B_{k l}$, where $T_{i j k l}$ is isotropic. Express $B_{i j}$ in terms of $A_{i j}$, assuming that $\beta^{2} \neq \gamma^{2}$ and $3 \alpha+\beta+\gamma \neq 0$. If instead $\beta=-\gamma \neq 0$ and $\alpha \neq 0$, find all $B_{i j}$ such that $A_{i j}=0$.

(d) Suppose that $C_{i j}$ and $D_{i j}$ satisfy the quadratic relationship $C_{i j}=T_{i j k l m n} D_{k l} D_{m n}$, where $T_{i j k l m n}$ is an isotropic tensor of rank 6 . If $C_{i j}$ is symmetric and $D_{i j}$ is antisymmetric, find the most general non-zero form of $T_{i j k l m n} D_{k l} D_{m n}$ and prove that there are only two independent terms. [Hint: You do not need to use the general form of an isotropic tensor of rank 6.]

Paper 3, Section II, C

commentUse Maxwell's equations,

$\boldsymbol{\nabla} \cdot \mathbf{E}=\rho, \quad \boldsymbol{\nabla} \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}$

to derive expressions for $\frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}$ and $\frac{\partial^{2} \mathbf{B}}{\partial t^{2}}-\nabla^{2} \mathbf{B}$ in terms of $\rho$ and $\mathbf{J}$.

Now suppose that there exists a scalar potential $\phi$ such that $\mathbf{E}=-\nabla \phi$, and $\phi \rightarrow 0$ as $r \rightarrow \infty$. If $\rho=\rho(r)$ is spherically symmetric, calculate $\mathbf{E}$ using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find $\mathbf{E}$ and $\phi$ in the case when $\rho=1$ for $r<a$ and $\rho=0$ otherwise.

For each integer $n \geqslant 0$, let $S_{n}$ be the sphere of radius $4^{-n}$ centred at the point $\left(1-4^{-n}, 0,0\right)$. Suppose that $\rho$ vanishes outside $S_{0}$, and has the constant value $2^{n}$ in the volume between $S_{n}$ and $S_{n+1}$ for $n \geqslant 0$. Calculate $\mathbf{E}$ and $\phi$ at the point $(1,0,0)$.

Paper 3, Section II, C

commentState the formula of Stokes's theorem, specifying any orientation where needed.

Let $\mathbf{F}=\left(y^{2} z, x z+2 x y z, 0\right)$. Calculate $\boldsymbol{\nabla} \times \mathbf{F}$ and verify that $\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \times \mathbf{F}=0$.

Sketch the surface $S$ defined as the union of the surface $z=-1,1 \leqslant x^{2}+y^{2} \leqslant 4$ and the surface $x^{2}+y^{2}+z=3,1 \leqslant x^{2}+y^{2} \leqslant 4$.

Verify Stokes's theorem for $\mathbf{F}$ on $S$.

Paper 3, Section II, C

commentGiven a one-to-one mapping $u=u(x, y)$ and $v=v(x, y)$ between the region $D$ in the $(x, y)$-plane and the region $D^{\prime}$ in the $(u, v)$-plane, state the formula for transforming the integral $\iint_{D} f(x, y) d x d y$ into an integral over $D^{\prime}$, with the Jacobian expressed explicitly in terms of the partial derivatives of $u$ and $v$.

Let $D$ be the region $x^{2}+y^{2} \leqslant 1, y \geqslant 0$ and consider the change of variables $u=x+y$ and $v=x^{2}+y^{2}$. Sketch $D$, the curves of constant $u$ and the curves of constant $v$ in the $(x, y)$-plane. Find and sketch the image $D^{\prime}$ of $D$ in the $(u, v)$-plane.

Calculate $I=\iint_{D}(x+y) d x d y$ using this change of variables. Check your answer by calculating $I$ directly.