• # 3.I.1D

State Lagrange's Theorem.

Show that there are precisely two non-isomorphic groups of order 10 . [You may assume that a group whose elements are all of order 1 or 2 has order $2^{k}$.]

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• # 3.I.2D

Define the Möbius group, and describe how it acts on $\mathbb{C} \cup\{\infty\}$.

Show that the subgroup of the Möbius group consisting of transformations which fix 0 and $\infty$ is isomorphic to $\mathbb{C}^{*}=\mathbb{C} \backslash\{0\}$.

Now show that the subgroup of the Möbius group consisting of transformations which fix 0 and 1 is also isomorphic to $\mathbb{C}^{*}$.

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• # 3.II.5D

Let $G=\left\langle g, h \mid h^{2}=1, g^{6}=1, h g h^{-1}=g^{-1}\right\rangle$ be the dihedral group of order 12 .

i) List all the subgroups of $G$ of order 2 . Which of them are normal?

ii) Now list all the remaining proper subgroups of $G$. [There are $6+3$ of them.]

iii) For each proper normal subgroup $N$ of $G$, describe the quotient group $G / N$.

iv) Show that $G$ is not isomorphic to the alternating group $A_{4}$.

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• # 3.II.6D

State the conditions on a matrix $A$ that ensure it represents a rotation of $\mathbb{R}^{3}$ with respect to the standard basis.

Check that the matrix

$A=\frac{1}{3}\left(\begin{array}{ccc} -1 & 2 & -2 \\ 2 & 2 & 1 \\ 2 & -1 & -2 \end{array}\right)$

represents a rotation. Find its axis of rotation $\mathbf{n}$.

Let $\Pi$ be the plane perpendicular to the axis $\mathbf{n}$. The matrix $A$ induces a rotation of $\Pi$ by an angle $\theta$. Find $\cos \theta$.

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• # 3.II.7D

Let $A$ be a real symmetric matrix. Show that all the eigenvalues of $A$ are real, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal to each other.

Find the eigenvalues and eigenvectors of

$A=\left(\begin{array}{ccc} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{array}\right)$

Give an example of a non-zero complex $(2 \times 2)$ symmetric matrix whose only eigenvalues are zero. Is it diagonalisable?

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• # 3.II.8D

Compute the characteristic polynomial of

$A=\left(\begin{array}{ccc} 3 & -1 & 2 \\ 0 & 4-s & 2 s-2 \\ 0 & -2 s+2 & 4 s-1 \end{array}\right)$

Find the eigenvalues and eigenvectors of $A$ for all values of $s$.

For which values of $s$ is $A$ diagonalisable?

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• # 3.I.3C

If $\mathbf{F}$ and $\mathbf{G}$ are differentiable vector fields, show that

(i) $\boldsymbol{\nabla} \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\boldsymbol{\nabla} \cdot \mathbf{G})-\mathbf{G}(\boldsymbol{\nabla} \cdot \mathbf{F})+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}-(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}$,

(ii) $\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})$.

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• # 3.I.4C

Define the curvature, $\kappa$, of a curve in $\mathbb{R}^{3}$.

The curve $C$ is parametrised by

$\mathbf{x}(t)=\left(\frac{1}{2} e^{t} \cos t, \frac{1}{2} e^{t} \sin t, \frac{1}{\sqrt{2}} e^{t}\right) \quad \text { for }-\infty

Obtain a parametrisation of the curve in terms of its arc length, $s$, measured from the origin. Hence obtain its curvature, $\kappa(s)$, as a function of $s$.

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• # 3.II.10C

Explain what is meant by an exact differential. The three-dimensional vector field $\mathbf{F}$ is defined by

$\mathbf{F}=\left(e^{x} z^{3}+3 x^{2}\left(e^{y}-e^{z}\right), e^{y}\left(x^{3}-z^{3}\right), 3 z^{2}\left(e^{x}-e^{y}\right)-e^{z} x^{3}\right)$

Find the most general function that has $\mathbf{F} \cdot \mathbf{d} \mathbf{x}$ as its differential.

Hence show that the line integral

$\int_{P_{1}}^{P_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$

along any path in $\mathbb{R}^{3}$ between points $P_{1}=(0, a, 0)$ and $P_{2}=(b, b, b)$ vanishes for any values of $a$ and $b$.

The two-dimensional vector field $\mathbf{G}$ is defined at all points in $\mathbb{R}^{2}$ except $(0,0)$ by

$\mathbf{G}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right)$

$(\mathbf{G}$ is not defined at $(0,0)$.) Show that

$\oint_{C} \mathbf{G} \cdot \mathbf{d} \mathbf{x}=2 \pi$

for any closed curve $C$ in $\mathbb{R}^{2}$ that goes around $(0,0)$ anticlockwise precisely once without passing through $(0,0)$.

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• # 3.II.11C

Let $S_{1}$ be the 3 -dimensional sphere of radius 1 centred at $(0,0,0), S_{2}$ be the sphere of radius $\frac{1}{2}$ centred at $\left(\frac{1}{2}, 0,0\right)$ and $S_{3}$ be the sphere of radius $\frac{1}{4}$ centred at $\left(\frac{-1}{4}, 0,0\right)$. The eccentrically shaped planet Zog is composed of rock of uniform density $\rho$ occupying the region within $S_{1}$ and outside $S_{2}$ and $S_{3}$. The regions inside $S_{2}$ and $S_{3}$ are empty. Give an expression for Zog's gravitational potential at a general coordinate $\mathbf{x}$ that is outside $S_{1}$. Is there a point in the interior of $S_{3}$ where a test particle would remain stably at rest? Justify your answer.

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• # 3.II.12C

State (without proof) the divergence theorem for a vector field $\mathbf{F}$ with continuous first-order partial derivatives throughout a volume $V$ enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface $S$.

By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field

$\mathbf{F}=\left(x^{3}+2 x y^{2}, y^{3}+2 y z^{2}, z^{3}+2 z x^{2}\right)$

defined within a sphere of radius $R$ centred at the origin.

Suppose that functions $\phi, \psi$ are continuous and that their first and second partial derivatives are all also continuous in a region $V$ bounded by a smooth surface $S$.

Show that

\begin{aligned} \int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S} \\ \int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S}-\int_{S} \psi \boldsymbol{\nabla} \phi \cdot \mathbf{d} \mathbf{S} \end{aligned}

Hence show that if $\rho(\mathbf{x})$ is a continuous function on $V$ and $g(\mathbf{x})$ a continuous function on $S$ and $\phi_{1}$ and $\phi_{2}$ are two continuous functions such that

\begin{aligned} \nabla^{2} \phi_{1}(\mathbf{x}) &=\nabla^{2} \phi_{2}(\mathbf{x})=\rho(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { in } V, \text { and } \\ \phi_{1}(\mathbf{x}) &=\phi_{2}(\mathbf{x})=g(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { on } S \end{aligned}

then $\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x})$ for all $\mathbf{x}$ in $V$.

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• # 3.II.9C

For a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ state if the following implications are true or false. (No justification is required.)

(i) $f$ is differentiable $\Rightarrow f$ is continuous.

(ii) $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist $\Rightarrow f$ is continuous.

(iii) directional derivatives $\frac{\partial f}{\partial \mathbf{n}}$ exist for all unit vectors $\mathbf{n} \in \mathbb{R}^{2} \Rightarrow f$ is differentiable.

(iv) $f$ is differentiable $\Rightarrow \frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous.

(v) all second order partial derivatives of $f$ exist $\Rightarrow \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial y \partial x}$.

Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

$f(x, y)= \begin{cases}\frac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Show that $f$ is continuous at $(0,0)$ and find the partial derivatives $\frac{\partial f}{\partial x}(0, y)$ and $\frac{\partial f}{\partial y}(x, 0)$. Then show that $f$ is differentiable at $(0,0)$ and find its derivative. Investigate whether the second order partial derivatives $\frac{\partial^{2} f}{\partial x \partial y}(0,0)$ and $\frac{\partial^{2} f}{\partial y \partial x}(0,0)$ are the same. Are the second order partial derivatives of $f$ at $(0,0)$ continuous? Justify your answer.

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