Part IA, 2006, Paper 3

# Part IA, 2006, Paper 3

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3.I.1D

commentGive an example of a real $3 \times 3$ matrix $A$ with eigenvalues $-1,(1 \pm i) / \sqrt{2}$. Prove or give a counterexample to the following statements:

(i) any such $A$ is diagonalisable over $\mathbb{C}$;

(ii) any such $A$ is orthogonal;

(iii) any such $A$ is diagonalisable over $\mathbb{R}$.

3.I.2D

commentShow that if $H$ and $K$ are subgroups of a group $G$, then $H \cap K$ is also a subgroup of $G$. Show also that if $H$ and $K$ have orders $p$ and $q$ respectively, where $p$ and $q$ are coprime, then $H \cap K$ contains only the identity element of $G$. [You may use Lagrange's theorem provided it is clearly stated.]

3.II.5D

commentLet $G$ be a group and let $A$ be a non-empty subset of $G$. Show that

$C(A)=\{g \in G: g h=h g \quad \text { for all } h \in A\}$

is a subgroup of $G$.

Show that $\rho: G \times G \rightarrow G$ given by

$\rho(g, h)=g h g^{-1}$

defines an action of $G$ on itself.

Suppose $G$ is finite, let $O_{1}, \ldots, O_{n}$ be the orbits of the action $\rho$ and let $h_{i} \in O_{i}$ for $i=1, \ldots, n$. Using the Orbit-Stabilizer Theorem, or otherwise, show that

$|G|=|C(G)|+\sum_{i}|G| /\left|C\left(\left\{h_{i}\right\}\right)\right|$

where the sum runs over all values of $i$ such that $\left|O_{i}\right|>1$.

Let $G$ be a finite group of order $p^{r}$, where $p$ is a prime and $r$ is a positive integer. Show that $C(G)$ contains more than one element.

3.II.6D

commentLet $\theta: G \rightarrow H$ be a homomorphism between two groups $G$ and $H$. Show that the image of $\theta, \theta(G)$, is a subgroup of $H$; show also that the kernel of $\theta, \operatorname{ker}(\theta)$, is a normal subgroup of $G$.

Show that $G / \operatorname{ker}(\theta)$ is isomorphic to $\theta(G)$.

Let $O(3)$ be the group of $3 \times 3$ real orthogonal matrices and let $S O(3) \subset O(3)$ be the set of orthogonal matrices with determinant 1 . Show that $S O(3)$ is a normal subgroup of $O(3)$ and that $O(3) / S O(3)$ is isomorphic to the cyclic group of order $2 .$

Give an example of a homomorphism from $O(3)$ to $S O(3)$ with kernel of order $2 .$

3.II.7D

commentLet $S L(2, \mathbb{R})$ be the group of $2 \times 2$ real matrices with determinant 1 and let $\sigma: \mathbb{R} \rightarrow S L(2, \mathbb{R})$ be a homomorphism. On $K=\mathbb{R} \times \mathbb{R}^{2}$ consider the product

$(x, \mathbf{v}) *(y, \mathbf{w})=(x+y, \mathbf{v}+\sigma(x) \mathbf{w})$

Show that $K$ with this product is a group.

Find the homomorphism or homomorphisms $\sigma$ for which $K$ is a commutative group.

Show that the homomorphisms $\sigma$ for which the elements of the form $(0, \mathbf{v})$ with $\mathbf{v}=(a, 0), a \in \mathbb{R}$, commute with every element of $K$ are precisely those such that

$\sigma(x)=\left(\begin{array}{cc} 1 & r(x) \\ 0 & 1 \end{array}\right)$

with $r:(\mathbb{R},+) \rightarrow(\mathbb{R},+)$ an arbitrary homomorphism.

3.II.8D

commentShow that every Möbius transformation can be expressed as a composition of maps of the forms: $S_{1}(z)=z+\alpha, S_{2}(z)=\lambda z$ and $S_{3}(z)=1 / z$, where $\alpha, \lambda \in \mathbb{C}$.

Show that if $z_{1}, z_{2}, z_{3}$ and $w_{1}, w_{2}, w_{3}$ are two triples of distinct points in $\mathbb{C} \cup\{\infty\}$, there exists a unique Möbius transformation that takes $z_{j}$ to $w_{j}(j=1,2,3)$.

Let $G$ be the group of those Möbius transformations which map the set $\{0,1, \infty\}$ to itself. Find all the elements of $G$. To which standard group is $G$ isomorphic?

3.I.3A

commentConsider the vector field $\mathbf{F}(\mathbf{x})=\left(\left(3 x^{3}-x^{2}\right) y,\left(y^{3}-2 y^{2}+y\right) x, z^{2}-1\right)$ and let $S$ be the surface of a unit cube with one corner at $(0,0,0)$, another corner at $(1,1,1)$ and aligned with edges along the $x$-, $y$ - and $z$-axes. Use the divergence theorem to evaluate

$I=\int_{S} \mathbf{F} \cdot d \mathbf{S}$

Verify your result by calculating the integral directly.

3.I.4A

commentUse suffix notation in Cartesian coordinates to establish the following two identities for the vector field $\mathbf{v}$ :

$\nabla \cdot(\nabla \times \mathbf{v})=0, \quad(\mathbf{v} \cdot \nabla) \mathbf{v}=\nabla\left(\frac{1}{2}|\mathbf{v}|^{2}\right)-\mathbf{v} \times(\nabla \times \mathbf{v})$

3.II.10A

commentState Stokes' theorem for a vector field $\mathbf{A}$.

By applying Stokes' theorem to the vector field $\mathbf{A}=\phi \mathbf{k}$, where $\mathbf{k}$ is an arbitrary constant vector in $\mathbb{R}^{3}$ and $\phi$ is a scalar field defined on a surface $S$ bounded by a curve $\partial S$, show that

$\int_{S} d \mathbf{S} \times \nabla \phi=\int_{\partial S} \phi d \mathbf{x}$

For the vector field $\mathbf{A}=x^{2} y^{4}(1,1,1)$ in Cartesian coordinates, evaluate the line integral

$I=\int \mathbf{A} \cdot d \mathbf{x}$

around the boundary of the quadrant of the unit circle lying between the $x$ - and $y$ axes, that is, along the straight line from $(0,0,0)$ to $(1,0,0)$, then the circular arc $x^{2}+y^{2}=1, z=0$ from $(1,0,0)$ to $(0,1,0)$ and finally the straight line from $(0,1,0)$ back to $(0,0,0)$.

3.II.11A

commentIn a region $R$ of $\mathbb{R}^{3}$ bounded by a closed surface $S$, suppose that $\phi_{1}$ and $\phi_{2}$ are both solutions of $\nabla^{2} \phi=0$, satisfying boundary conditions on $S$ given by $\phi=f$ on $S$, where $f$ is a given function. Prove that $\phi_{1}=\phi_{2}$.

In $\mathbb{R}^{2}$ show that

$\phi(x, y)=\left(a_{1} \cosh \lambda x+a_{2} \sinh \lambda x\right)\left(b_{1} \cos \lambda y+b_{2} \sin \lambda y\right)$

is a solution of $\nabla^{2} \phi=0$, for any constants $a_{1}, a_{2}, b_{1}, b_{2}$ and $\lambda$. Hence, or otherwise, find a solution $\phi(x, y)$ in the region $x \geqslant 0$ and $0 \leqslant y \leqslant a$ which satisfies:

$\begin{aligned} &\phi(x, 0)=0, \quad \phi(x, a)=0, \quad x \geqslant 0 \\ &\phi(0, y)=\sin \frac{n \pi y}{a}, \quad \phi(x, y) \rightarrow 0 \quad \text { as } \quad x \rightarrow \infty, \quad 0 \leqslant y \leqslant a \end{aligned}$

where $a$ is a real constant and $n$ is an integer.

3.II.12A

commentDefine what is meant by an isotropic tensor. By considering a rotation of a second rank isotropic tensor $B_{i j}$ by $90^{\circ}$ about the $z$-axis, show that its components must satisfy $B_{11}=B_{22}$ and $B_{13}=B_{31}=B_{23}=B_{32}=0$. Now consider a second and different rotation to show that $B_{i j}$ must be a multiple of the Kronecker delta, $\delta_{i j}$.

Suppose that a homogeneous but anisotropic crystal has the conductivity tensor

$\sigma_{i j}=\alpha \delta_{i j}+\gamma n_{i} n_{j}$

where $\alpha, \gamma$ are real constants and the $n_{i}$ are the components of a constant unit vector $\mathbf{n}$ $(\mathbf{n} \cdot \mathbf{n}=1)$. The electric current density $\mathbf{J}$ is then given in components by

$J_{i}=\sigma_{i j} E_{j}$

where $E_{j}$ are the components of the electric field $\mathbf{E}$. Show that

(i) if $\alpha \neq 0$ and $\gamma \neq 0$, then there is a plane such that if $\mathbf{E}$ lies in this plane, then $\mathbf{E}$ and $\mathbf{J}$ must be parallel, and

(ii) if $\gamma \neq-\alpha$ and $\alpha \neq 0$, then $\mathbf{E} \neq 0$ implies $\mathbf{J} \neq 0$.

If $D_{i j}=\epsilon_{i j k} n_{k}$, find the value of $\gamma$ such that

$\sigma_{i j} D_{j k} D_{k m}=-\sigma_{i m}$

3.II.9A

commentEvaluate the line integral

$\int \alpha\left(x^{2}+x y\right) d x+\beta\left(x^{2}+y^{2}\right) d y$

with $\alpha$ and $\beta$ constants, along each of the following paths between the points $A=(1,0)$ and $B=(0,1)$ :

(i) the straight line between $A$ and $B$;

(ii) the $x$-axis from $A$ to the origin $(0,0)$ followed by the $y$-axis to $B$;

(iii) anti-clockwise from $A$ to $B$ around the circular path centred at the origin $(0,0)$.

You should obtain the same answer for the three paths when $\alpha=2 \beta$. Show that when $\alpha=2 \beta$, the integral takes the same value along any path between $A$ and $B$.