Part IA, 2007, Paper 3

# Part IA, 2007, Paper 3

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Paper 3, Section I, D

commentWhat does it mean to say that groups $G$ and $H$ are isomorphic?

Prove that no two of $C_{8}, C_{4} \times C_{2}$ and $C_{2} \times C_{2} \times C_{2}$ are isomorphic. [Here $C_{n}$ denotes the cyclic group of order $n$.]

Give, with justification, a group of order 8 that is not isomorphic to any of those three groups.

Paper 3, Section I, D

commentProve that every permutation of $\{1, \ldots, n\}$ may be expressed as a product of disjoint cycles.

Let $\sigma=(1234)$ and let $\tau=(345)(678)$. Write $\sigma \tau$ as a product of disjoint cycles. What is the order of $\sigma \tau ?$

Paper 3, Section II, $6 \mathrm{D}$

commentWhat does it mean to say that a subgroup $H$ of a group $G$ is normal? Give, with justification, an example of a subgroup of a group that is normal, and also an example of a subgroup of a group that is not normal.

If $H$ is a normal subgroup of $G$, explain carefully how to make the set of (left) cosets of $H$ into a group.

Let $H$ be a normal subgroup of a finite group $G$. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.

(i) If $G$ is cyclic then $H$ and $G / H$ are cyclic.

(ii) If $H$ and $G / H$ are cyclic then $G$ is cyclic.

(iii) If $G$ is abelian then $H$ and $G / H$ are abelian.

(iv) If $H$ and $G / H$ are abelian then $G$ is abelian.

Paper 3, Section II, D

commentLet $x$ be an element of a finite group $G$. What is meant by the order of $x$ ? Prove that the order of $x$ must divide the order of $G$. [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]

If $G$ is a group of order $n$, and $d$ is a divisor of $n$ with $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

Prove that if $m$ and $n$ are coprime then the group $C_{m} \times C_{n}$ is cyclic.

If $m$ and $n$ are not coprime, can it happen that $C_{m} \times C_{n}$ is cyclic?

[Here $C_{n}$ denotes the cyclic group of order $n$.]

Paper 3, Section II, D

commentIn the group of Möbius maps, what is the order of the Möbius map $z \mapsto \frac{1}{z}$ ? What is the order of the Möbius map $z \mapsto \frac{1}{1-z}$ ?

Prove that every Möbius map is conjugate either to a map of the form $z \mapsto \mu z$ (some $\mu \in \mathbb{C}$ ) or to the $\operatorname{map} z \mapsto z+1$. Is $z \mapsto z+1$ conjugate to a map of the form $z \mapsto \mu z ?$

Let $f$ be a Möbius map of order $n$, for some positive integer $n$. Under the action on $\mathbb{C} \cup\{\infty\}$ of the group generated by $f$, what are the various sizes of the orbits? Justify your answer.

Paper 3, Section II, D

commentLet $A$ be a real symmetric $n \times n$ matrix. Prove that every eigenvalue of $A$ is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that $A$ is real.

What does it mean to say that a real $n \times n$ matrix $P$ is orthogonal ? Show that if $P$ is orthogonal and $A$ is as above then $P^{-1} A P$ is symmetric. If $P$ is any real invertible matrix, must $P^{-1} A P$ be symmetric? Justify your answer.

Give, with justification, real $2 \times 2$ matrices $B, C, D, E$ with the following properties:

(i) $B$ has no real eigenvalues;

(ii) $C$ is not diagonalisable over $\mathbb{C}$;

(iii) $D$ is diagonalisable over $\mathbb{C}$, but not over $\mathbb{R}$;

(iv) $E$ is diagonalisable over $\mathbb{R}$, but does not have an orthonormal basis of eigenvectors.

Paper 3 , Section I, A

comment(i) Give definitions for the unit tangent vector $\hat{\mathbf{T}}$ and the curvature $\kappa$ of a parametrised curve $\mathbf{x}(t)$ in $\mathbb{R}^{3}$. Calculate $\hat{\mathbf{T}}$ and $\kappa$ for the circular helix

$\mathbf{x}(t)=(a \cos t, a \sin t, b t),$

where $a$ and $b$ are constants.

(ii) Find the normal vector and the equation of the tangent plane to the surface $S$ in $\mathbb{R}^{3}$ given by

$z=x^{2} y^{3}-y+1$

at the point $x=1, y=1, z=1$.

Paper 3, Section I, A

commentBy using suffix notation, prove the following identities for the vector fields $\mathbf{A}$ and B in $\mathbb{R}^{3}$ :

$\begin{gathered} \nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{A})-\mathbf{A} \cdot(\nabla \times \mathbf{B}) \\ \nabla \times(\mathbf{A} \times \mathbf{B})=(\mathbf{B} \cdot \nabla) \mathbf{A}-\mathbf{B}(\nabla \cdot \mathbf{A})-(\mathbf{A} \cdot \nabla) \mathbf{B}+\mathbf{A}(\nabla \cdot \mathbf{B}) \end{gathered}$

Paper 3, Section II, A

commentShow that any second rank Cartesian tensor $P_{i j}$ in $\mathbb{R}^{3}$ can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that $P_{i j}$ can be decomposed into the following terms

$\tag{†} P_{i j}=P \delta_{i j}+S_{i j}+\epsilon_{i j k} A_{k},$

where $S_{i j}$ is symmetric and traceless. Give expressions for $P, S_{i j}$ and $A_{k}$ explicitly in terms of $P_{i j}$.

For an isotropic material, the stress $P_{i j}$ can be related to the strain $T_{i j}$ through the stress-strain relation, $P_{i j}=c_{i j k l} T_{k l}$, where the elasticity tensor is given by

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

and $\alpha, \beta$ and $\gamma$ are scalars. As in $(†)$, the strain $T_{i j}$ can be decomposed into its trace $T$, a symmetric traceless tensor $W_{i j}$ and a vector $V_{k}$. Use the stress-strain relation to express each of $T, W_{i j}$ and $V_{k}$ in terms of $P, S_{i j}$ and $A_{k}$.

Hence, or otherwise, show that if $T_{i j}$ is symmetric then so is $P_{i j}$. Show also that the stress-strain relation can be written in the form

$P_{i j}=\lambda \delta_{i j} T_{k k}+\mu T_{i j}$

where $\mu$ and $\lambda$ are scalars.

Paper 3, Section II, A

commentThe function $\phi(x, y, z)$ satisfies $\nabla^{2} \phi=0$ in $V$ and $\phi=0$ on $S$, where $V$ is a region of $\mathbb{R}^{3}$ which is bounded by the surface $S$. Prove that $\phi=0$ everywhere in $V$.

Deduce that there is at most one function $\psi(x, y, z)$ satisfying $\nabla^{2} \psi=\rho$ in $V$ and $\psi=f$ on $S$, where $\rho(x, y, z)$ and $f(x, y, z)$ are given functions.

Given that the function $\psi=\psi(r)$ depends only on the radial coordinate $r=|\mathbf{x}|$, use Cartesian coordinates to show that

$\nabla \psi=\frac{1}{r} \frac{d \psi}{d r} \mathbf{x}, \quad \nabla^{2} \psi=\frac{1}{r} \frac{d^{2}(r \psi)}{d r^{2}}$

Find the general solution in this radial case for $\nabla^{2} \psi=c$ where $c$ is a constant.

Find solutions $\psi(r)$ for a solid sphere of radius $r=2$ with a central cavity of radius $r=1$ in the following three regions:

(i) $0 \leqslant r \leqslant 1$ where $\nabla^{2} \psi=0$ and $\psi(1)=1$ and $\psi$ bounded as $r \rightarrow 0$;

(ii) $1 \leqslant r \leqslant 2$ where $\nabla^{2} \psi=1$ and $\psi(1)=\psi(2)=1$;

(iii) $r \geqslant 2$ where $\nabla^{2} \psi=0$ and $\psi(2)=1$ and $\psi \rightarrow 0$ as $r \rightarrow \infty$.

Paper 3, Section II, A

commentFor a given charge distribution $\rho(x, y, z)$ and divergence-free current distribution $\mathbf{J}(x, y, z)$ (i.e. $\nabla \cdot \mathbf{J}=0)$ in $\mathbb{R}^{3}$, the electric and magnetic fields $\mathbf{E}(x, y, z)$ and $\mathbf{B}(x, y, z)$ satisfy the equations

$\nabla \times \mathbf{E}=0, \quad \nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=\rho, \quad \nabla \times \mathbf{B}=\mathbf{J}$

The radiation flux vector $\mathbf{P}$ is defined by $\mathbf{P}=\mathbf{E} \times \mathbf{B}$. For a closed surface $S$ around a region $V$, show using Gauss' theorem that the flux of the vector $\mathbf{P}$ through $S$ can be expressed as

$\iint_{S} \mathbf{P} \cdot \mathbf{d} \mathbf{S}=-\iiint_{V} \mathbf{E} \cdot \mathbf{J} d V$

For electric and magnetic fields given by

$\mathbf{E}(x, y, z)=(z, 0, x), \quad \mathbf{B}(x, y, z)=(0,-x y, x z)$

find the radiation flux through the quadrant of the unit spherical shell given by

$x^{2}+y^{2}+z^{2}=1, \quad \text { with } \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant y \leqslant 1, \quad-1 \leqslant z \leqslant 1$

[If you use (*), note that an open surface has been specified.]

Paper 3, Section II, A

comment(i) Define what is meant by a conservative vector field. Given a vector field $\mathbf{A}=\left(A_{1}(x, y), A_{2}(x, y)\right)$ and a function $\psi(x, y)$ defined in $\mathbb{R}^{2}$, show that, if $\psi \mathbf{A}$ is a conservative vector field, then

$\psi\left(\frac{\partial A_{1}}{\partial y}-\frac{\partial A_{2}}{\partial x}\right)=A_{2} \frac{\partial \psi}{\partial x}-A_{1} \frac{\partial \psi}{\partial y}$

(ii) Given two functions $P(x, y)$ and $Q(x, y)$ defined in $\mathbb{R}^{2}$, prove Green's theorem,

$\oint_{C}(P d x+Q d y)=\iint_{R}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) d x d y$

where $C$ is a simple closed curve bounding a region $R$ in $\mathbb{R}^{2}$.

Through an appropriate choice for $P$ and $Q$, find an expression for the area of the region $R$, and apply this to evaluate the area of the ellipse bounded by the curve

$x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$