Part IA, 2019, Paper 3

# Part IA, 2019, Paper 3

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Paper 3, Section $I$, D

commentWhat is the orthogonal group $\mathrm{O}(n)$ ? What is the special orthogonal group $\mathrm{SO}(n) ?$

Show that every element of $\mathrm{SO}(3)$ has an eigenvector with eigenvalue $1 .$

Is it true that every element of $\mathrm{O}(3)$ is either a rotation or a reflection? Justify your answer.

Paper 3, Section I, D

commentProve that two elements of $S_{n}$ are conjugate if and only if they have the same cycle type.

Describe a condition on the centraliser (in $S_{n}$ ) of a permutation $\sigma \in A_{n}$ that ensures the conjugacy class of $\sigma$ in $A_{n}$ is the same as the conjugacy class of $\sigma$ in $S_{n}$. Justify your answer.

How many distinct conjugacy classes are there in $A_{5}$ ?

Paper 3, Section II, D

commentLet $\mathcal{M}$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $\mathrm{SL}_{2}(\mathbb{C})$ be the group of all $2 \times 2$ complex matrices of determinant 1 .

Show that the map $\theta: \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathcal{M}$ given by

$\theta\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)(z)=\frac{a z+b}{c z+d}$

is a surjective homomorphism. Find its kernel.

Show that any $T \in \mathcal{M}$ not equal to the identity is conjugate to a Möbius map $S$ where either $S z=\mu z$ with $\mu \neq 0,1$ or $S z=z+1$. [You may use results about matrices in $\mathrm{SL}_{2}(\mathbb{C})$ as long as they are clearly stated.]

Show that any non-identity Möbius map has one or two fixed points. Also show that if $T$ is a Möbius map with just one fixed point $z_{0}$ then $T^{n} z \rightarrow z_{0}$ as $n \rightarrow \infty$ for any $z \in \mathbb{C} \cup\{\infty\}$. [You may assume that Möbius maps are continuous.]

Paper 3, Section II, D

commentState and prove the first isomorphism theorem. [You may assume that kernels of homomorphisms are normal subgroups and images are subgroups.]

Let $G$ be a group with subgroup $H$ and normal subgroup $N$. Prove that $N H=\{n h: n \in N, h \in H\}$ is a subgroup of $G$ and $N \cap H$ is a normal subgroup of $H$. Further, show that $N$ is a normal subgroup of $N H$.

Prove that $\frac{H}{N \cap H}$ is isomorphic to $\frac{N H}{N}$.

If $K$ and $H$ are both normal subgroups of $G$ must $K H$ be a normal subgroup of $G$ ?

If $K$ and $H$ are subgroups of $G$, but not normal subgroups, must $K H$ be a subgroup of $G$ ?

Justify your answers.

Paper 3, Section II, D

commentState and prove Lagrange's Theorem.

Hence show that if $G$ is a finite group and $g \in G$ then the order of $g$ divides the order of $G$.

How many elements are there of order 3 in the following groups? Justify your answers.

(a) $C_{3} \times C_{9}$, where $C_{n}$ denotes the cyclic group of order $n$.

(b) $D_{2 n}$ the dihedral group of order $2 n$.

(c) $S_{7}$ the symmetric group of degree 7 .

(d) $A_{7}$ the alternating group of degree 7 .

Paper 3, Section II, D

commentLet $H$ and $K$ be subgroups of a group $G$ satisfying the following two properties.

(i) All elements of $G$ can be written in the form $h k$ for some $h \in H$ and some $k \in K$.

(ii) $H \cap K=\{e\}$.

Prove that $H$ and $K$ are normal subgroups of $G$ if and only if all elements of $H$ commute with all elements of $K$.

State and prove Cauchy's Theorem.

Let $p$ and $q$ be distinct primes. Prove that an abelian group of order $p q$ is isomorphic to $C_{p} \times C_{q}$. Is it true that all abelian groups of order $p^{2}$ are isomorphic to $C_{p} \times C_{p}$ ?

Paper 3, Section I, B

commentLet

$\begin{aligned} u &=\left(2 x+x^{2} z+z^{3}\right) \exp ((x+y) z) \\ v &=\left(x^{2} z+z^{3}\right) \exp ((x+y) z) \\ w &=\left(2 z+x^{3}+x^{2} y+x z^{2}+y z^{2}\right) \exp ((x+y) z) \end{aligned}$

Show that $u d x+v d y+w d z$ is an exact differential, clearly stating any criteria that you use.

Show that for any path between $(-1,0,1)$ and $(1,0,1)$

$\int_{(-1,0,1)}^{(1,0,1)}(u d x+v d y+w d z)=4 \sinh 1$

Paper 3, Section I, B

commentApply the divergence theorem to the vector field $\mathbf{u}(\mathbf{x})=\mathbf{a} \phi(\mathbf{x})$ where $\mathbf{a}$ is an arbitrary constant vector and $\phi$ is a scalar field, to show that

$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$

where $V$ is a volume bounded by the surface $S$ and $d \mathbf{S}$ is the outward pointing surface element.

Verify that this result holds when $\phi=x+y$ and $V$ is the spherical volume $x^{2}+$ $y^{2}+z^{2} \leqslant a^{2}$. [You may use the result that $d \mathbf{S}=a^{2} \sin \theta d \theta d \phi(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$, where $\theta$ and $\phi$ are the usual angular coordinates in spherical polars and the components of $d \mathbf{S}$ are with respect to standard Cartesian axes.]

Paper 3, Section II, B

comment(a) The function $u$ satisfies $\nabla^{2} u=0$ in the volume $V$ and $u=0$ on $S$, the surface bounding $V$.

Show that $u=0$ everywhere in $V$.

The function $v$ satisfies $\nabla^{2} v=0$ in $V$ and $v$ is specified on $S$. Show that for all functions $w$ such that $w=v$ on $S$

$\int_{V} \nabla v \cdot \nabla w d V=\int_{V}|\nabla v|^{2} d V$

Hence show that

$\int_{V}|\boldsymbol{\nabla} w|^{2} d V=\int_{V}\left\{|\boldsymbol{\nabla} v|^{2}+|\boldsymbol{\nabla}(w-v)|^{2}\right\} d V \geqslant \int_{V}|\boldsymbol{\nabla} v|^{2} d V$

(b) The function $\phi$ satisfies $\nabla^{2} \phi=\rho(\mathbf{x})$ in the spherical region $|\mathbf{x}|<a$, with $\phi=0$ on $|\mathbf{x}|=a$. The function $\rho(\mathbf{x})$ is spherically symmetric, i.e. $\rho(\mathbf{x})=\rho(|\mathbf{x}|)=\rho(r)$.

Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function $\Phi(r)$. Show that

$4 \pi r^{2} \Phi^{\prime}(r)=4 \pi \int_{0}^{r} s^{2} \rho(s) d s$

Hence find the function $\Phi(r)$ when $\rho(r)$ is given by $\rho(r)=\left\{\begin{array}{ll}\rho_{0} & \text { if } 0 \leqslant r \leqslant b \\ 0 & \text { if } b<r \leqslant a\end{array}\right.$, with $\rho_{0}$ constant.

Explain how the results obtained in part (a) of the question imply that $\Phi(r)$ is the only solution of $\nabla^{2} \phi=\rho(r)$ which satisfies the specified boundary condition on $|\mathbf{x}|=a$.

Use your solution and the results obtained in part (a) of the question to show that, for any function $w$ such that $w=1$ on $r=b$ and $w=0$ on $r=a$,

$\int_{U(b, a)}|\nabla w|^{2} d V \geqslant \frac{4 \pi a b}{a-b}$

where $U(b, a)$ is the region $b<r<a$.

Paper 3, Section II, B

commentShow that for a vector field $\mathbf{A}$

$\nabla \times(\boldsymbol{\nabla} \times \mathbf{A})=\boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$

Hence find an $\mathbf{A}(\mathbf{x})$, with $\boldsymbol{\nabla} \cdot \mathbf{A}=0$, such that $\mathbf{u}=\left(y^{2}, z^{2}, x^{2}\right)=\nabla \times \mathbf{A}$. [Hint: Note that $\mathbf{A}(\mathbf{x})$ is not defined uniquely. Choose your expression for $\mathbf{A}(\mathbf{x})$ to be as simple as possible.

Now consider the cone $x^{2}+y^{2} \leqslant z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h$. Let $S_{1}$ be the curved part of the surface of the cone $\left(x^{2}+y^{2}=z^{2} \tan ^{2} \alpha, 0 \leqslant z \leqslant h\right)$ and $S_{2}$ be the flat part of the surface of the cone $\left(x^{2}+y^{2} \leqslant h^{2} \tan ^{2} \alpha, z=h\right)$.

Using the variables $z$ and $\phi$ as used in cylindrical polars $(r, \phi, z)$ to describe points on $S_{1}$, give an expression for the surface element $d \mathbf{S}$ in terms of $d z$ and $d \phi$.

Evaluate $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$.

What does the divergence theorem predict about the two surface integrals $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ where in each case the vector $d \mathbf{S}$ is taken outwards from the cone?

What does Stokes theorem predict about the integrals $\int_{S_{1}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ (defined as in the previous paragraph) and the line integral $\int_{C} \mathbf{A} \cdot d \mathbf{l}$ where $C$ is the circle $x^{2}+y^{2}=h^{2} \tan ^{2} \alpha, z=h$ and the integral is taken in the anticlockwise sense, looking from the positive $z$ direction?

Evaluate $\int_{S_{2}} \mathbf{u} \cdot d \mathbf{S}$ and $\int_{C} \mathbf{A} \cdot d \mathbf{l}$, making your method clear and verify that each of these predictions holds.

Paper 3, Section II, B

commentFor a given set of coordinate axes the components of a 2 nd rank tensor $T$ are given by $T_{i j}$.

(a) Show that if $\lambda$ is an eigenvalue of the matrix with elements $T_{i j}$ then it is also an eigenvalue of the matrix of the components of $T$ in any other coordinate frame.

Show that if $T$ is a symmetric tensor then the multiplicity of the eigenvalues of the matrix of components of $T$ is independent of coordinate frame.

A symmetric tensor $T$ in three dimensions has eigenvalues $\lambda, \lambda, \mu$, with $\mu \neq \lambda$.

Show that the components of $T$ can be written in the form

$T_{i j}=\alpha \delta_{i j}+\beta n_{i} n_{j}$

where $n_{i}$ are the components of a unit vector.

(b) The tensor $T$ is defined by

$T_{i j}(\mathbf{y})=\int_{S} x_{i} x_{j} \exp \left(-c|\mathbf{y}-\mathbf{x}|^{2}\right) d A(\mathbf{x})$

where $S$ is the surface of the unit sphere, $\mathbf{y}$ is the position vector of a point on $S$, and $c$ is a constant.

Deduce, with brief reasoning, that the components of $T$ can be written in the form (1) with $n_{i}=y_{i}$. [You may quote any results derived in part (a).]

Using suitable spherical polar coordinates evaluate $T_{k k}$ and $T_{i j} y_{i} y_{j}$.

Explain how to deduce the values of $\alpha$ and $\beta$ from $T_{k k}$ and $T_{i j} y_{i} y_{j}$. [You do not need to write out the detailed formulae for these quantities.]

Paper 3, Section II, B

commentDefine the Jacobian, $J$, of the one-to-one transformation

$(x, y, z) \rightarrow(u, v, w)$

Give a careful explanation of the result

$\int_{D} f(x, y, z) d x d y d z=\int_{\Delta}|J| \phi(u, v, w) d u d v d w$

where

$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$

and the region $D$ maps under the transformation to the region $\Delta$.

Consider the region $D$ defined by

$x^{2}+\frac{y^{2}}{k^{2}}+z^{2} \leqslant 1$

and

$\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{k^{2} \alpha^{2}}-\frac{z^{2}}{\gamma^{2}} \geqslant 1$

where $\alpha, \gamma$ and $k$ are positive constants.

Let $\tilde{D}$ be the intersection of $D$ with the plane $y=0$. Write down the conditions for $\tilde{D}$ to be non-empty. Sketch the geometry of $\tilde{D}$ in $x>0$, clearly specifying the curves that define its boundaries and points that correspond to minimum and maximum values of $x$ and of $z$ on the boundaries.

Use a suitable change of variables to evaluate the volume of the region $D$, clearly explaining the steps in your calculation.