Part IA, 2005, Paper 3

# Part IA, 2005, Paper 3

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3.I.1D

commentLet $A$ be a real $3 \times 3$ symmetric matrix with eigenvalues $\lambda_{1}>\lambda_{2}>\lambda_{3}>0$. Consider the surface $S$ in $\mathbb{R}^{3}$ given by

$x^{T} A x=1$

Find the minimum distance between the origin and $S$. How many points on $S$ realize this minimum distance? Justify your answer.

3.I.2D

commentDefine what it means for a group to be cyclic. If $p$ is a prime number, show that a finite group $G$ of order $p$ must be cyclic. Find all homomorphisms $\varphi: C_{11} \rightarrow C_{14}$, where $C_{n}$ denotes the cyclic group of order $n$. [You may use Lagrange's theorem.]

3.II.5D

commentDefine the notion of an action of a group $G$ on a set $X$. Assuming that $G$ is finite, state and prove the Orbit-Stabilizer Theorem.

Let $G$ be a finite group and $X$ the set of its subgroups. Show that $g(K)=g K g^{-1}$ $(g \in G, K \in X)$ defines an action of $G$ on $X$. If $H$ is a subgroup of $G$, show that the orbit of $H$ has at most $|G| /|H|$ elements.

Suppose $H$ is a subgroup of $G$ and $H \neq G$. Show that there is an element of $G$ which does not belong to any subgroup of the form $g H g^{-1}$ for $g \in G$.

3.II.6D

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

Show that the map $\theta: S L(2, \mathbb{C}) \rightarrow \mathcal{M}$ 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 every $T \in \mathcal{M}$ not equal to the identity is conjugate to a Möbius map $S$ where either $S z=\mu z$ with $\mu \neq 0,1$, or $S z=z \pm 1$. [You may use results about matrices in $S L(2, \mathbb{C})$, provided they are clearly stated.]

Show that if $T \in \mathcal{M}$, then $T$ is the identity, or $T$ has one, or two, fixed points. Also show that if $T \in \mathcal{M}$ has only one fixed point $z_{0}$ then $T^{n} z \rightarrow z_{0}$ as $n \rightarrow \infty$ for any $z \in \mathbb{C} \cup\{\infty\} .$

3.II.7D

commentLet $G$ be a group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G .$

Let $H$ be the set of all $3 \times 3$ real matrices of the form

$\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)$

with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of invertible real matrices under multiplication.

Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ with vector addition.

3.II.8D

commentLet $A$ be a $3 \times 3$ real matrix such that $\operatorname{det}(A)=-1, A \neq-I$, and $A^{T} A=I$, where $A^{T}$ is the transpose of $A$ and $I$ is the identity.

Show that the set $E$ of vectors $x$ for which $A x=-x$ forms a 1-dimensional subspace.

Consider the plane $\Pi$ through the origin which is orthogonal to $E$. Show that $A$ maps $\Pi$ to itself and induces a rotation of $\Pi$ by angle $\theta$, where $\cos \theta=\frac{1}{2}(\operatorname{trace}(A)+1)$. Show that $A$ is a reflection in $\Pi$ if and only if $A$ has trace 1 . [You may use the fact that $\operatorname{trace}\left(B A B^{-1}\right)=\operatorname{trace}(A)$ for any invertible matrix B.]

Prove that $\operatorname{det}(A-I)=4(\cos \theta-1)$.

3.I.3A

commentLet $\mathbf{A}(t, \mathbf{x})$ and $\mathbf{B}(t, \mathbf{x})$ be time-dependent, continuously differentiable vector fields on $\mathbb{R}^{3}$ satisfying

$\frac{\partial \mathbf{A}}{\partial t}=\nabla \times \mathbf{B} \quad \text { and } \quad \frac{\partial \mathbf{B}}{\partial t}=-\nabla \times \mathbf{A}$

Show that for any bounded region $V$,

$\frac{d}{d t}\left[\frac{1}{2} \int_{V}\left(\mathbf{A}^{2}+\mathbf{B}^{2}\right) d V\right]=-\int_{S}(\mathbf{A} \times \mathbf{B}) \cdot d \mathbf{S}$

where $S$ is the boundary of $V$.

3.I.4A

commentGiven a curve $\gamma(s)$ in $\mathbb{R}^{3}$, parameterised such that $\left\|\gamma^{\prime}(s)\right\|=1$ and with $\gamma^{\prime \prime}(s) \neq 0$, define the tangent $\mathbf{t}(s)$, the principal normal $\mathbf{p}(s)$, the curvature $\kappa(s)$ and the binormal $\mathbf{b}(s)$.

The torsion $\tau(s)$ is defined by

$\tau=-\mathbf{b}^{\prime} \cdot \mathbf{p}$

Sketch a circular helix showing $\mathbf{t}, \mathbf{p}, \mathbf{b}$ and $\mathbf{b}^{\prime}$ at a chosen point. What is the sign of the torsion for your helix? Sketch a second helix with torsion of the opposite sign.

3.II.11A

commentExplain, with justification, the significance of the eigenvalues of the Hessian in classifying the critical points of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. In what circumstances are the eigenvalues inconclusive in establishing the character of a critical point?

Consider the function on $\mathbb{R}^{2}$,

$f(x, y)=x y e^{-\alpha\left(x^{2}+y^{2}\right)}$

Find and classify all of its critical points, for all real $\alpha$. How do the locations of the critical points change as $\alpha \rightarrow 0$ ?

3.II.12A

commentExpress the integral

$I=\int_{0}^{\infty} d x \int_{0}^{1} d y \int_{0}^{x} d z x e^{-A x / y-B x y-C y z}$

in terms of the new variables $\alpha=x / y, \beta=x y$, and $\gamma=y z$. Hence show that

$I=\frac{1}{2 A(A+B)(A+B+C)}$

You may assume $A, B$ and $C$ are positive. [Hint: Remember to calculate the limits of the integral.]

3.II.9A

commentLet $V$ be a bounded region of $\mathbb{R}^{3}$ and $S$ be its boundary. Let $\phi$ be the unique solution to $\nabla^{2} \phi=0$ in $V$, with $\phi=f(\mathbf{x})$ on $S$, where $f$ is a given function. Consider any smooth function $w$ also equal to $f(\mathbf{x})$ on $S$. Show, by using Green's first theorem or otherwise, that

$\int_{V}|\nabla w|^{2} d V \geqslant \int_{V}|\nabla \phi|^{2} d V$

[Hint: Set $w=\phi+\delta .]$

Consider the partial differential equation

$\frac{\partial}{\partial t} w=\nabla^{2} w$

for $w(t, \mathbf{x})$, with initial condition $w(0, \mathbf{x})=w_{0}(\mathbf{x})$ in $V$, and boundary condition $w(t, \mathbf{x})=$ $f(\mathbf{x})$ on $S$ for all $t \geqslant 0$. Show that

$\frac{\partial}{\partial t} \int_{V}|\nabla w|^{2} d V \leqslant 0$

with equality holding only when $w(t, \mathbf{x})=\phi(\mathbf{x})$.

Show that $(*)$ remains true with the boundary condition

$\frac{\partial w}{\partial t}+\alpha(\mathbf{x}) \frac{\partial w}{\partial n}=0$

on $S$, provided $\alpha(\mathbf{x}) \geqslant 0$.

3/II/10A Vector Calculus

Write down Stokes' theorem for a vector field $\mathbf{B}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Consider the bounded surface $S$ defined by

$z=x^{2}+y^{2}, \quad \frac{1}{4} \leqslant z \leqslant 1$

Sketch the surface and calculate the surface element $d \mathbf{S}$. For the vector field

$\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)$

calculate $I=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}$ directly.

Show using Stokes' theorem that $I$ may be rewritten as a line integral and verify this yields the same result.