# Part II, 2014

### Jump to course

Paper 1, Section II, H

commentLet $k$ be an algebraically closed field and $n \geqslant 1$. We say that $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is singular at $p \in \mathbf{A}^{n}$ if either $p$ is a singularity of the hypersurface $\{f=0\}$ or $f$ has an irreducible factor $h$ of multiplicity strictly greater than one with $h(p)=0$. Given $d \geqslant 1$, let $X=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid \operatorname{deg} f \leqslant d\right\}$ and let

$Y=\left\{(f, p) \in X \times \mathbf{A}^{n} \mid f \text { is singular at } p\right\}$

(i) Show that $X \simeq \mathbf{A}^{N}$ for some $N$ (you need not determine $N$ ) and that $Y$ is a Zariski closed subvariety of $X \times \mathbf{A}^{n}$.

(ii) Show that the fibres of the projection map $Y \rightarrow \mathbf{A}^{n}$ are linear subspaces of $\operatorname{dim} e n s i o n N-(n+1)$. Conclude that $\operatorname{dim} Y<\operatorname{dim} X$.

(iii) Hence show that $\{f \in X \mid \operatorname{deg} f=d, Z(f)$ smooth $\}$ is dense in $X$.

[You may use standard results from lectures if they are accurately quoted.]

Paper 2, Section II, H

comment(i) Let $k$ be an algebraically closed field, $n \geqslant 1$, and $S$ a subset of $k^{n}$.

Let $I(S)=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid f(p)=0\right.$ when $\left.p \in S\right\}$. Show that $I(S)$ is an ideal, and that $k\left[x_{1}, \ldots, x_{n}\right] / I(S)$ does not have any non-zero nilpotent elements.

Let $X \subseteq \mathbf{A}^{n}, Y \subseteq \mathbf{A}^{m}$ be affine varieties, and $\Phi: k[Y] \rightarrow k[X]$ be a $k$-algebra homomorphism. Show that $\Phi$ determines a map of sets from $X$ to $Y$.

(ii) Let $X$ be an irreducible affine variety. Define the dimension of $X, \operatorname{dim} X$ (in terms of the tangent spaces of $X$ ) and the transcendence dimension of $X, \operatorname{tr} \cdot \operatorname{dim} X$.

State the Noether normalization theorem. Using this, or otherwise, prove that the transcendence dimension of $X$ equals the dimension of $X$.

Paper 3, Section II, H

commentLet $f \in k[x]$ be a polynomial with distinct roots, $\operatorname{deg} f=d>2$, char $k=0$, and let $C \subseteq \mathbf{P}^{2}$ be the projective closure of the affine curve

$y^{d-1}=f(x)$

Show that $C$ is smooth, with a single point at $\infty$.

Pick an appropriate $\omega \in \Omega_{k(C) / k}^{1}$ and compute the valuation $v_{q}(\omega)$ for all $q \in C$.

Hence determine $\operatorname{deg} \mathcal{K}_{C}$.

Paper 4, Section II, H

commentLet $X$ be a smooth projective curve of genus $g>0$ over an algebraically closed field of characteristic $\neq 2$, and suppose there is a degree 2 morphism $\pi: X \rightarrow \mathbf{P}^{1}$. How many ramification points of $\pi$ are there?

Suppose $Q$ and $R$ are distinct ramification points of $\pi$. Show that $Q \nsim R$, but $2 Q \sim 2 R$.

Now suppose $g=2$. Show that every divisor of degree 2 on $X$ is linearly equivalent to $P+P^{\prime}$ for some $P, P^{\prime} \in X$, and deduce that every divisor of degree 0 is linearly equivalent to $P_{1}-P_{2}$ for some $P_{1}, P_{2} \in X$.

Show that the subgroup $\left\{[D] \in C l^{0}(X) \mid 2[D]=0\right\}$ of the divisor class group of $X$ has order $16 .$

Paper 1, Section II, F

commentDefine what it means for a map $p: \widetilde{X} \rightarrow X$ to be a covering space. State the homotopy lifting lemma.

Let $p:\left(\tilde{X}, \tilde{x}_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based covering space and let $f:\left(Y, y_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based map from a path-connected and locally path-connected space. Show that there is a based lift $\tilde{f}:\left(Y, y_{0}\right) \rightarrow\left(\tilde{X}, \tilde{x}_{0}\right)$ of $f$ if and only if $f_{*}\left(\pi_{1}\left(Y, y_{0}\right)\right) \subseteq p_{*}\left(\pi_{1}\left(\widetilde{X}, \tilde{x}_{0}\right)\right)$.

Paper 2, Section II, F

commentLet $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be a matrix with integer entries. Considering $S^{1}$ as the quotient space $\mathbb{R} / \mathbb{Z}$, show that the function

$\begin{aligned} \varphi_{A}: S^{1} \times S^{1} & \longrightarrow S^{1} \times S^{1} \\ ([x],[y]) & \longmapsto([a x+b y],[c x+d y]) \end{aligned}$

is well-defined and continuous. If in addition $\operatorname{det}(A)=\pm 1$, show that $\varphi_{A}$ is a homeomorphism.

State the Seifert-van Kampen theorem. Let $X_{A}$ be the space obtained by gluing together two copies of $S^{1} \times D^{2}$ along their boundaries using the homeomorphism $\varphi_{A}$. Show that the fundamental group of $X_{A}$ is cyclic and determine its order.

Paper 3, Section II, F

commentLet $K$ be a simplicial complex in $\mathbb{R}^{N}$, which we may also consider as lying in $\mathbb{R}^{N+1}$ using the first $N$ coordinates. Write $c=(0,0, \ldots, 0,1) \in \mathbb{R}^{N+1}$. Show that if $\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle$ is a simplex of $K$ then $\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle$ is a simplex in $\mathbb{R}^{N+1}$.

Let $L \leqslant K$ be a subcomplex and let $\bar{K}$ be the collection

$K \cup\left\{\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle \mid\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle \in L\right\} \cup\{\langle c\rangle\}$

of simplices in $\mathbb{R}^{N+1}$. Show that $\bar{K}$ is a simplicial complex.

If $|K|$ is a Möbius band, and $|L|$ is its boundary, show that

$H_{i}(\bar{K}) \cong \begin{cases}\mathbb{Z} & \text { if } i=0 \\ \mathbb{Z} / 2 & \text { if } i=1 \\ 0 & \text { if } i \geqslant 2\end{cases}$

Paper 4, Section II, F

commentState the Lefschetz fixed point theorem.

Let $X$ be an orientable surface of genus $g$ (which you may suppose has a triangulation), and let $f: X \rightarrow X$ be a continuous map such that

$f^{3}=\operatorname{Id}_{X}$,

$f$ has no fixed points.

By considering the eigenvalues of the linear map $f_{*}: H_{1}(X ; \mathbb{Q}) \rightarrow H_{1}(X ; \mathbb{Q})$, and their multiplicities, show that $g$ must be congruent to 1 modulo 3 .

Paper 1, Section II, A

commentA particle of mass $m$ scatters on a localised potential well $V(x)$ in one dimension. With reference to the asymptotic behaviour of the wavefunction as $x \rightarrow \pm \infty$, define the reflection and transmission amplitudes, $r$ and $t$, for a right-moving incident particle of wave number $k$. Define also the corresponding amplitudes, $r^{\prime}$ and $t^{\prime}$, for a left-moving incident particle of wave number $k$. Derive expressions for $r^{\prime}$ and $t^{\prime}$ in terms of $r$ and $t$.

(a) Define the $S$-matrix, giving its elements in terms of $r$ and $t$. Using the relation

$|r|^{2}+|t|^{2}=1$

(which you need not derive), show that the S-matrix is unitary. How does the S-matrix simplify if the potential well satisfies $V(-x)=V(x)$ ?

(b) Consider the potential well

$V(x)=-\frac{3 \hbar^{2}}{m} \frac{1}{\cosh ^{2}(x)}$

The corresponding Schrödinger equation has an exact solution

$\psi_{k}(x)=\exp (i k x)\left[3 \tanh ^{2}(x)-3 i k \tanh (x)-\left(1+k^{2}\right)\right]$

with energy $E=\hbar^{2} k^{2} / 2 m$, for every real value of $k$. [You do not need to verify this.] Find the S-matrix for scattering on this potential. What special feature does the scattering have in this case?

(c) Explain the connection between singularities of the S-matrix and bound states of the potential well. By analytic continuation of the solution $\psi_{k}(x)$ to appropriate complex values of $k$, find the wavefunctions and energies of the bound states of the well. [You do not need to normalise the wavefunctions.]

Paper 2, Section II, A

comment(a) A classical particle of mass $m$ scatters on a central potential $V(r)$ with energy $E$, impact parameter $b$, and scattering angle $\theta$. Define the corresponding differential cross-section.

For particle trajectories in the Coulomb potential,

$V_{C}(r)=\frac{e^{2}}{4 \pi \epsilon_{0} r}$

the impact parameter is given by

$b=\frac{e^{2}}{8 \pi \epsilon_{0} E} \cot \left(\frac{\theta}{2}\right)$

Find the differential cross-section as a function of $E$ and $\theta$.

(b) A quantum particle of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m$ scatters in a localised potential $V(\mathbf{r})$. With reference to the asymptotic form of the wavefunction at large $|\mathbf{r}|$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ as a function of the incident and outgoing wavevectors $\mathbf{k}$ and $\mathbf{k}^{\prime}$ (where $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|=k$ ). Define the differential cross-section for this process and express it in terms of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$.

Now consider a potential of the form $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless coupling and $U$ does not depend on $\lambda$. You may assume that the Schrödinger equation for the wavefunction $\psi(\mathbf{k} ; \mathbf{r})$ of a scattering state with incident wavevector $\mathbf{k}$ may be written as the integral equation

$\psi(\mathbf{k} ; \mathbf{r})=\exp (i \mathbf{k} \cdot \mathbf{r})+\frac{2 m \lambda}{\hbar^{2}} \int d^{3} r^{\prime} \mathcal{G}_{0}^{(+)}\left(k ; \mathbf{r}-\mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$

where

$\mathcal{G}_{0}^{(+)}(k ; \mathbf{r})=-\frac{1}{4 \pi} \frac{\exp (i k|\mathbf{r}|)}{|\mathbf{r}|}$

Show that the corresponding scattering amplitude is given by

$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m \lambda}{2 \pi \hbar^{2}} \int d^{3} r^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) U\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k} ; \mathbf{r}^{\prime}\right)$

By expanding the wavefunction in powers of $\lambda$ and keeping only the leading term, calculate the leading-order contribution to the differential cross-section, and evaluate it for the case of the Yukawa potential

$V(\mathbf{r})=\lambda \frac{\exp (-\mu r)}{r}$

By taking a suitable limit, obtain the differential cross-section for quantum scattering in the Coulomb potential $V_{C}(r)$ defined in Part (a) above, correct to leading order in an expansion in powers of the constant $\tilde{\alpha}=e^{2} / 4 \pi \epsilon_{0}$. Express your answer as a function of the particle energy $E$ and scattering angle $\theta$, and compare it to the corresponding classical cross-section calculated in Part (a).

Part II, $2014 \quad$ List of Questions

[TURN OVER

Paper 3, Section II, A

commentIn the nearly-free electron model a particle of mass $m$ moves in one dimension in a periodic potential of the form $V(x)=\lambda U(x)$, where $\lambda \ll 1$ is a dimensionless coupling and $U(x)$ has a Fourier series

$U(x)=\sum_{l=-\infty}^{+\infty} U_{l} \exp \left(\frac{2 \pi i}{a} l x\right)$

with coefficients obeying $U_{-l}=U_{l}^{*}$ for all $l$.

Ignoring any degeneracies in the spectrum, the exact energy $E(k)$ of a Bloch state with wavenumber $k$ can be expanded in powers of $\lambda$ as

$E(k)=E_{0}(k)+\lambda\langle k|U| k\rangle+\lambda^{2} \sum_{k^{\prime} \neq k} \frac{\left\langle k|U| k^{\prime}\right\rangle\left\langle k^{\prime}|U| k\right\rangle}{E_{0}(k)-E_{0}\left(k^{\prime}\right)}+O\left(\lambda^{3}\right)$

where $|k\rangle$ is a normalised eigenstate of the free Hamiltonian $\hat{H}_{0}=\hat{p}^{2} / 2 m$ with momentum $p=\hbar k$ and energy $E_{0}(k)=\hbar^{2} k^{2} / 2 m$.

Working on a finite interval of length $L=N a$, where $N$ is a positive integer, we impose periodic boundary conditions on the wavefunction:

$\psi(x+N a)=\psi(x)$

What are the allowed values of the wavenumbers $k$ and $k^{\prime}$ which appear in (1)? For these values evaluate the matrix element $\left\langle k|U| k^{\prime}\right\rangle$.

For what values of $k$ and $k^{\prime}$ does (1) cease to be a good approximation? Explain your answer. Quoting any results you need from degenerate perturbation theory, calculate to $O(\lambda)$ the location and width of the gaps between allowed energy bands for the periodic potential $V(x)$, in terms of the Fourier coefficients $U_{l}$.

Hence work out the allowed energy bands for the following potentials:

$\begin{aligned} &\text { (i) } \quad V(x)=2 \lambda \cos \left(\frac{2 \pi x}{a}\right), \\ &\text { (ii) } \quad V(x)=\lambda a \sum_{n=-\infty}^{+\infty} \delta(x-n a) . \end{aligned}$

Paper 4, Section II, A

commentLet $\Lambda$ be a Bravais lattice in three dimensions. Define the reciprocal lattice $\Lambda^{*}$.

State and prove Bloch's theorem for a particle moving in a potential $V(\mathbf{x})$ obeying

$V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda, \mathbf{x} \in \mathbb{R}^{3}$

Explain what is meant by a Brillouin zone for this potential and how it is related to the reciprocal lattice.

A simple cubic lattice $\Lambda_{1}$ is given by the set of points

$\Lambda_{1}=\left\{\ell \in \mathbb{R}^{3}: \ell=n_{1} \hat{\mathbf{i}}+n_{2} \hat{\mathbf{j}}+n_{3} \hat{\mathbf{k}}, n_{1}, n_{2}, n_{3} \in \mathbb{Z}\right\}$

where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors parallel to the Cartesian coordinate axes in $\mathbb{R}^{3}$. A bodycentred cubic $\left(\mathrm{BCC}\right.$ ) lattice $\Lambda_{B C C}$ is obtained by adding to $\Lambda_{1}$ the points at the centre of each cube, i.e. all points of the form

$\ell+\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}), \quad \ell \in \Lambda_{1}$

Show that $\Lambda_{B C C}$ is Bravais with primitive vectors

$\begin{aligned} \mathbf{a}_{1} &=\frac{1}{2}(\hat{\mathbf{j}}+\hat{\mathbf{k}}-\hat{\mathbf{i}}) \\ \mathbf{a}_{2} &=\frac{1}{2}(\hat{\mathbf{k}}+\hat{\mathbf{i}}-\hat{\mathbf{j}}) \\ \mathbf{a}_{3} &=\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}) \end{aligned}$

Find the reciprocal lattice $\Lambda_{B C C}^{*}$. Hence find a consistent choice for the first Brillouin zone of a potential $V(\mathbf{x})$ obeying

$\begin{aligned} & V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda_{B C C}, \mathbf{x} \in \mathbb{R}^{3} \\ & \text { [Hint: The matrix } \left.M=\frac{1}{2}\left(\begin{array}{rrr}-1 & 1 & 1 \\1 & -1 & 1 \\1 & 1 & -1\end{array}\right) \text { has inverse } M^{-1}=\left(\begin{array}{lll}0 & 1 & 1 \\1 & 0 & 1 \\1 & 1 & 0\end{array}\right) .\right] \end{aligned}$

Paper 1, Section II, J

comment(i) Explain what a $Q$-matrix is. Let $Q$ be a $Q$-matrix. Define the notion of a Markov chain $\left(X_{t}, t \geqslant 0\right)$ in continuous time with $Q$-matrix given by $Q$, and give a construction of $\left(X_{t}, t \geqslant 0\right)$. [You are not required to justify this construction.]

(ii) A population consists of $N_{t}$ individuals at time $t \geqslant 0$. We assume that each individual gives birth to a new individual at constant rate $\lambda>0$. As the population is competing for resources, we assume that for each $n \geqslant 1$, if $N_{t}=n$, then any individual in the population at time $t$ dies in the time interval $[t, t+h)$ with probability $\delta_{n} h+o(h)$, where $\left(\delta_{n}\right)_{n=1}^{\infty}$ is a given sequence satisfying $\delta_{1}=0, \delta_{n}>0$ for $n \geqslant 2$. Formulate a Markov chain model for $\left(N_{t}, t \geqslant 0\right)$ and write down the $Q$-matrix explicitly. Then find a necessary and sufficient condition on $\left(\delta_{n}\right)_{n=1}^{\infty}$ so that the Markov chain has an invariant distribution. Compute the invariant distribution in the case where $\delta_{n}=\mu(n-1)$ and $\mu>0$.

Paper 2, Section II, J

comment(i) Explain what the Moran model and the infinite alleles model are. State Ewens' sampling formula for the distribution of the allelic frequency spectrum $\left(a_{1}, \ldots, a_{n}\right)$ in terms of $\theta$ where $\theta=N u$ with $u$ denoting the mutation rate per individual and $N$ the population size.

Let $K_{n}$ be the number of allelic types in a sample of size $n$. Give, without justification, an expression for $\mathbb{E}\left(K_{n}\right)$ in terms of $\theta$.

(ii) Let $K_{n}$ and $\theta$ be as above. Show that for $1 \leqslant k \leqslant n$ we have that

$P\left(K_{n}=k\right)=C \frac{\theta^{k}}{\theta(\theta+1) \cdots(\theta+n-1)}$

for some constant $C$ that does not depend on $\theta$.

Show that, given $\left\{K_{n}=k\right\}$, the distribution of the allelic frequency spectrum $\left(a_{1}, \ldots, a_{n}\right)$ does not depend on $\theta$.

Show that the value of $\theta$ which maximises $\mathbb{P}\left(K_{n}=k\right)$ is the one for which $k=\mathbb{E}\left(K_{n}\right)$.

Paper 3, Section II, J

comment(i) Define a Poisson process $\left(N_{t}, t \geqslant 0\right)$ with intensity $\lambda$. Specify without justification the distribution of $N_{t}$. Let $T_{1}, T_{2}, \ldots$ denote the jump times of $\left(N_{t}, t \geqslant 0\right)$. Derive the joint distribution of $\left(T_{1}, \ldots, T_{n}\right)$ given $\left\{N_{t}=n\right\}$.

(ii) Let $\left(N_{t}, t \geqslant 0\right)$ be a Poisson process with intensity $\lambda>0$ and let $X_{1}, X_{2}, \ldots$ be a sequence of i.i.d. random variables, independent of $\left(N_{t}, t \geqslant 0\right)$, all having the same distribution as a random variable $X$. Show that if $g(s, x)$ is a real-valued function of real variables $s, x$, and $T_{j}$ are the jump times of $\left(N_{t}, t \geqslant 0\right)$ then

$\mathbb{E}\left[\exp \left\{\theta \sum_{j=1}^{N_{t}} g\left(T_{j}, X_{j}\right)\right\}\right]=\exp \left\{\lambda \int_{0}^{t}\left(\mathbb{E}\left(e^{\theta g(s, X)}\right)-1\right) d s\right\}$

for all $\theta \in \mathbb{R}$. [Hint: Condition on $\left\{N_{t}=n\right\}$ and $T_{1}, \ldots, T_{n}$, using (i).]

(iii) A university library is open from 9 am to $5 \mathrm{pm}$. Students arrive at times of a Poisson process with intensity $\lambda$. Each student spends a random amount of time in the library, independently of the other students. These times are identically distributed for all students and have the same distribution as a random variable $X$. Show that the number of students in the library at $5 \mathrm{pm}$ is a Poisson random variable with a mean that you should specify.

Paper 4, Section II, J

comment(i) Define the $M / M / 1$ queue with arrival rate $\lambda$ and service rate $\mu$. Find conditions on the parameters $\lambda$ and $\mu$ for the queue to be transient, null recurrent, and positive recurrent, briefly justifying your answers. In the last case give with justification the invariant distribution explicitly. Answer the same questions for an $M / M / \infty$ queue.

(ii) At a taxi station, customers arrive at a rate of 3 per minute, and taxis at a rate of 2 per minute. Suppose that a taxi will wait no matter how many other taxis are present. However, if a person arriving does not find a taxi waiting he or she leaves to find alternative transportation.

Find the long-run proportion of arriving customers who get taxis, and find the average number of taxis waiting in the long run.

An agent helps to assign customers to taxis, and so long as there are taxis waiting he is unable to have his coffee. Once a taxi arrives, how long will it take on average before he can have another sip of his coffee?

Paper 1, Section II, C

comment(a) Consider the integral

$I(k)=\int_{0}^{\infty} f(t) e^{-k t} d t, \quad k>0$

Suppose that $f(t)$ possesses an asymptotic expansion for $t \rightarrow 0^{+}$of the form

$f(t) \sim t^{\alpha} \sum_{n=0}^{\infty} a_{n} t^{\beta n}, \quad \alpha>-1, \quad \beta>0$

where $a_{n}$ are constants. Derive an asymptotic expansion for $I(k)$ as $k \rightarrow \infty$ in the form

$I(k) \sim \sum_{n=0}^{\infty} \frac{A_{n}}{k^{\gamma+\beta n}}$

giving expressions for $A_{n}$ and $\gamma$ in terms of $\alpha, \beta, n$ and the gamma function. Hence establish the asymptotic approximation as $k \rightarrow \infty$

$I_{1}(k)=\int_{0}^{1} e^{k t} t^{-a}\left(1-t^{2}\right)^{-b} d t \sim 2^{-b} \Gamma(1-b) e^{k} k^{b-1}\left(1+\frac{(a+b / 2)(1-b)}{k}\right)$

where $a<1, b<1$.

(b) Using Laplace's method, or otherwise, find the leading-order asymptotic approximation as $k \rightarrow \infty$ for

$I_{2}(k)=\int_{0}^{\infty} e^{-\left(2 k^{2} / t+t^{2} / k\right)} d t$

[You may assume that $\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t$ for $\operatorname{Re} z>0$,

$\text { and that } \left.\int_{-\infty}^{\infty} e^{-q t^{2}} d t=\sqrt{\pi / q} \text { for } q>0 .\right]$

Paper 3, Section II, C

comment(a) Find the Stokes ray for the function $f(z)$ as $z \rightarrow 0$ with $0<\arg z<\pi$, where

$f(z)=\sinh \left(z^{-1}\right)$

(b) Describe how the leading-order asymptotic behaviour as $x \rightarrow \infty$ of

$I(x)=\int_{a}^{b} f(t) e^{i x g(t)} d t$

may be found by the method of stationary phase, where $f$ and $g$ are real functions and the integral is taken along the real line. You should consider the cases for which:

(i) $g^{\prime}(t)$ is non-zero in $[a, b)$ and has a simple zero at $t=b$.

(ii) $g^{\prime}(t)$ is non-zero apart from having one simple zero at $t=t_{0}$, where $a<t_{0}<b$.

(iii) $g^{\prime}(t)$ has more than one simple zero in $(a, b)$ with $g^{\prime}(a) \neq 0$ and $g^{\prime}(b) \neq 0$.

Use the method of stationary phase to find the leading-order asymptotic form as $x \rightarrow \infty$ of

$J(x)=\int_{0}^{1} \cos \left(x\left(t^{4}-t^{2}\right)\right) d t$

[You may assume that $\left.\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4} .\right]$

Paper 4, Section II, C

commentDerive the leading-order Liouville Green (or WKBJ) solution for $\epsilon \ll 1$ to the ordinary differential equation

$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\Phi(y) f=0$

where $\Phi(y)>0$.

The function $f(y ; \epsilon)$ satisfies the ordinary differential equation

$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\left(1+\frac{1}{y}-\frac{2 \epsilon^{2}}{y^{2}}\right) f=0$

subject to the boundary condition $f^{\prime \prime}(0)=2$. Show that the Liouville-Green solution of (1) for $\epsilon \ll 1$ takes the asymptotic forms

where $\alpha_{1}, \alpha_{2}, B$ and $\theta_{2}$ are constants.

$\left[\right.$ Hint: You may assume that $\left.\int_{0}^{y} \sqrt{1+u^{-1}} d u=\sqrt{y(1+y)}+\sinh ^{-1} \sqrt{y} \cdot\right]$

Explain, showing the relevant change of variables, why the leading-order asymptotic behaviour for $0 \leqslant y \ll 1$ can be obtained from the reduced equation

$\frac{d^{2} f}{d x^{2}}+\left(\frac{1}{x}-\frac{2}{x^{2}}\right) f=0$

The unique solution to $(2)$ with $f^{\prime \prime}(0)=2$ is $f=x^{1 / 2} J_{3}\left(2 x^{1 / 2}\right)$, where the Bessel function $J_{3}(z)$ is known to have the asymptotic form

$J_{3}(z) \sim\left(\frac{2}{\pi z}\right)^{1 / 2} \cos \left(z-\frac{7 \pi}{4}\right) \text { as } z \rightarrow \infty .$

Hence find the values of $\alpha_{1}$ and $\alpha_{2}$.

$\begin{aligned} & f \sim \alpha_{1} y^{\frac{1}{4}} \exp (2 i \sqrt{y} / \epsilon)+\alpha_{2} y^{\frac{1}{4}} \exp (-2 i \sqrt{y} / \epsilon) \quad \text { for } \quad \epsilon^{2} \ll y \ll 1 \\ & \text { and } \quad f \sim B \cos \left[\theta_{2}+(y+\log \sqrt{y}) / \epsilon\right] \quad \text { for } \quad y \gg 1, \end{aligned}$

Paper 1, Section I, A

commentConsider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

(a) Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t)$ and $g(q, p, t)$.

(b) Verify the Leibniz rule

$\{f g, h\}=f\{g, h\}+g\{f, h\}$

(c) Explain what is meant by a canonical transformation $(q, p) \rightarrow(Q, P)$.

(d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson bracket $\{Q, P\}$. Use this to determine whether or not the following transformations are canonical:

(i) $Q=\frac{q^{2}}{2}, P=\frac{p}{q}$,

(ii) $Q=\tan q, P=p \cos q$,

(iii) $Q=\sqrt{2 q} e^{t} \cos p, P=\sqrt{2 q} e^{-t} \sin p$.

Paper 2, Section I, A

commentThe components of the angular velocity $\omega$ of a rigid body and of the position vector $\mathbf{r}$ are given in a body frame.

(a) The kinetic energy of the rigid body is defined as

$T=\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}$

Given that the centre of mass is at rest, show that $T$ can be written in the form

$T=\frac{1}{2} I_{a b} \omega_{a} \omega_{b},$

where the explicit form of the tensor $I_{a b}$ should be determined.

(b) Explain what is meant by the principal moments of inertia.

(c) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, which are all unequal. Derive Euler's equations of torque-free motion

$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$

(d) The body rotates about the principal axis with moment of inertia $I_{1}$. Derive the condition for stable rotation.

Paper 2, Section II, A

commentA planar pendulum consists of a mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a bead of mass $M$, which slides along a horizontal rod without friction. The bead is connected to the ends of the horizontal rod by two identical springs of force constant $k$. The pivot constrains the pendulum to swing in the vertical plane through the horizontal rod. The horizontal rod is mounted on a bracket, so the system could rotate about the vertical axis which goes through its centre as shown in the figure.

(a) Initially, the system is not allowed to rotate about the vertical axis.

(i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

(ii) Write down expression(s) for any conserved quantities. Justify your answer.

(iii) Derive the equations of motion.

(iv) For $M=m / 2$ and $g m / k l=3$, find the frequencies of small oscillations around the stable equilibrium and the corresponding normal modes. Describe the respective motions of the system.

(b) Assume now that the system is free to rotate about the vertical axis without friction. Write down the Lagrangian of the system. Identify and calculate the additional conserved quantity.

Paper 3, Section I, A

comment(a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$

State the principle of least action. Write the expression for the Hamiltonian in terms of the generalized velocity $\dot{q}$, the generalized momentum $p$ and the Lagrangian $L$. Use it to derive Hamilton's equations from the principle of least action.

(b) The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

$L=\frac{m \dot{\mathbf{r}}^{2}}{2}-q(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$

(i) Write down the Hamiltonian of the particle.

(ii) Consider a particle which moves in three dimensions in a magnetic field with $\mathbf{A}=(0, B x, 0)$, where $B$ is a constant. There is no electric field. Obtain Hamilton's equations for the particle.

Paper 4, Section I, A

commentConsider a heavy symmetric top of mass $M$ with principal moments of inertia $I_{1}$, $I_{2}$ and $I_{3}$, where $I_{1}=I_{2} \neq I_{3}$. The top is pinned at point $P$, which is at a distance $l$ from the centre of mass, $C$, as shown in the figure.

Its angular velocity in a body frame $\left(\mathbf{e}_{\mathbf{1}}, \mathbf{e}_{\mathbf{2}}, \mathbf{e}_{\mathbf{3}}\right)$ is given by

$\boldsymbol{\omega}=[\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi] \mathbf{e}_{1}+[\dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi] \mathbf{e}_{2}+[\dot{\psi}+\dot{\phi} \cos \theta] \mathbf{e}_{3}$

where $\phi, \theta$ and $\psi$ are the Euler angles.

(a) Assuming that $\left\{\mathbf{e}_{a}\right\}, a=1,2,3$, are chosen to be the principal axes, write down the Lagrangian of the top in terms of $\omega_{a}$ and the principal moments of inertia. Hence find the Lagrangian in terms of the Euler angles.

(b) Find all conserved quantities. Show that $\omega_{3}$, the spin of the top, is constant.

(c) By eliminating $\dot{\phi}$ and $\dot{\psi}$, derive a second-order differential equation for $\theta$.

Paper 4, Section II, A

comment(a) Consider a system with one degree of freedom, which undergoes periodic motion in the potential $V(q)$. The system's Hamiltonian is

$H(p, q)=\frac{p^{2}}{2 m}+V(q)$

(i) Explain what is meant by the angle and action variables, $\theta$ and $I$, of the system and write down the integral expression for the action variable $I$. Is $I$ conserved? Is $\theta$ conserved?

(ii) Consider $V(q)=\lambda q^{6}$, where $\lambda$ is a positive constant. Find $I$ in terms of $\lambda$, the total energy $E$, the mass $M$, and a dimensionless constant factor (which you need not compute explicitly).

(iii) Hence describe how $E$ changes with $\lambda$ if $\lambda$ varies slowly with time. Justify your answer.

(b) Consider now a particle which moves in a plane subject to a central force-field $\mathbf{F}=-k r^{-2} \hat{\mathbf{r}}$.

(i) Working in plane polar coordinates $(r, \phi)$, write down the Hamiltonian of the system. Hence deduce two conserved quantities. Prove that the system is integrable and state the number of action variables.

(ii) For a particle which moves on an elliptic orbit find the action variables associated with radial and tangential motions. Can the relationship between the frequencies of the two motions be deduced from this result? Justify your answer.

(iii) Describe how $E$ changes with $m$ and $k$ if one or both of them vary slowly with time.

[You may use

$\int_{r_{1}}^{r_{2}}\left\{\left(1-\frac{r_{1}}{r}\right)\left(\frac{r_{2}}{r}-1\right)\right\}^{\frac{1}{2}} d r=\frac{\pi}{2}\left(r_{1}+r_{2}\right)-\pi \sqrt{r_{1} r_{2}}$

where $0<r_{1}<r_{2}$.]

Paper 1, Section I, $4 \mathrm{I}$

commentState and prove Gibbs' inequality.

Show that, for a pair of discrete random variables $X$ and $Y$, each taking finitely many values, the joint entropy $H(X, Y)$ satisfies

$H(X, Y) \leqslant H(X)+H(Y)$

with equality precisely when $X$ and $Y$ are independent.

Paper 1, Section II, I

commentDescribe, briefly, either the RSA or the Elgamal public key cipher. You should explain, without proof, why it is believed to be difficult to break the cipher you describe.

How can such a cipher be used to sign messages? You should explain how the intended recipient of the message can (a) know from whom it came; (b) know that the message has not been changed; and (c) demonstrate that the sender must have signed it.

Let $I_{0}, I_{1}, \ldots, I_{N}$ be friendly individuals each of whom has a public key cipher. $I_{0}$ wishes to send a message to $I_{N}$ by passing it first to $I_{1}$, then $I_{1}$ passes it to $I_{2}, I_{2}$ to $I_{3}$, until finally it is received by $I_{N}$. At each stage the message can be modified to show from whom it was received and to whom it is sent. Devise a way in which these modifications can be made so that $I_{N}$ can be confident both of the content of the original message and that the message has been passed through the intermediaries $I_{1}, I_{2}, \ldots, I_{N-1}$ in that order and has not been modified by an enemy agent. Assume that it takes a negligible time to transmit a message from $I_{k}$ to $I_{k+1}$ for each $k$, but the time needed to modify a message is not negligible.

Paper 2, Section $I$, I

commentLet $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ be a decodable binary code defined on a finite alphabet $\mathcal{A}$. Let $l(a)$ be the length of the code word $c(a)$. Prove that

$\sum_{a \in \mathcal{A}} 2^{-l(a)} \leqslant 1$

Show that, for the decodable code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ described above, there is a prefixfree code $p: \mathcal{A} \rightarrow\{0,1\}^{*}$ with each code word $p(a)$ having length $l(a)$. [You may use, without proof, any standard results from the course.]

Paper 2, Section II, I

commentWhat is the information capacity of a memoryless, time-independent channel? Compute the information capacity of a binary symmetric channel with probability $p$ of error. Show the steps in your computation.

Binary digits are transmitted through a noisy channel, which is memoryless and time-independent. With probability $\alpha(0<\alpha<1)$ the digit is corrupted and noise is received, otherwise the digit is transmitted unchanged. So, if we denote the input by 0 and 1 and the output as $0, *$ and 1 with $*$ denoting the noise, the transition matrix is

$\left(\begin{array}{cc} 1-\alpha & 0 \\ \alpha & \alpha \\ 0 & 1-\alpha \end{array}\right)$

Compute the information capacity of this channel.

Explain how to code a message for transmission through the channel described above, and how to decode it, so that the probability of error for each bit is arbitrarily small.

Paper 3, Section I, I

commentLet $A$ be a random variable that takes values in the finite alphabet $\mathcal{A}$. Prove that there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ that satisfies

$H(A) \leqslant \mathbb{E}(l(A)) \leqslant H(A)+1$

where $l(a)$ is the length of the code word $c(a)$ and $H(A)$ is the entropy of $A$.

Is it always possible to find such a code with $\mathbb{E}(l(A))=H(A) ?$ Justify your answer.

Paper 4, Section I, I

commentExplain what is meant by a Bose-Ray Chaudhuri-Hocquenghem (BCH) code with design distance $\delta$. Prove that, for such a code, the minimum distance between code words is at least $\delta$. How many errors will the code detect? How many errors will it correct?

Paper 1, Section I, E

commentWhich particle states are expected to be relativistic and which interacting when the temperature $T$ of the early universe satisfies (i) $10^{10} \mathrm{~K}<T<5 \times 10^{10} \mathrm{~K}$, (ii) $5 \times 10^{9} \mathrm{~K}<T<10^{10} \mathrm{~K}$, (iii) $T<5 \times 10^{9} \mathrm{~K}$ ?

Calculate the total spin weight factor, $g_{*}$, of the relativistic particles and the total spin weight factor, $g_{I}$, of the interacting particles, in each of the three temperature intervals.

What happens when the temperature falls below $5 \times 10^{9} \mathrm{~K} ?$ Calculate the ratio of the temperatures of neutrinos and photons. Find the effective value of $g_{*}$ after the universe cools below this temperature. [Note that the equilibrium entropy density is given by $s=\left(\rho c^{2}+P\right) / T$, where $\rho$ is the density and $P$ is the pressure.]

Paper 1, Section II, E

commentWhat are the cosmological flatness and horizon problems? Explain what form of time evolution of the cosmological expansion scale factor $a(t)$ must occur during a period of inflationary expansion in a Friedmann universe. How can inflation solve the horizon and flatness problems? [You may assume an equation of state where pressure $P$ is proportional to density $\rho$.]

The universe has Hubble expansion rate $H=\dot{a} / a$ and contains only a scalar field $\phi$ with self-interaction potential $V(\phi)>0$. The density and pressure are given by

$\begin{aligned} \rho &=\frac{1}{2} \dot{\phi}^{2}+V(\phi) \\ P &=\frac{1}{2} \dot{\phi}^{2}-V(\phi) \end{aligned}$

in units where $c=\hbar=1$. Show that the conservation equation

$\dot{\rho}+3 H(\rho+P)=0$

requires

$\ddot{\phi}+3 H \dot{\phi}+d V / d \phi=0 .$

If the Friedmann equation has the form

$3 H^{2}=8 \pi G \rho$

and the scalar-field potential has the form

$V(\phi)=V_{0} e^{-\lambda \phi}$

where $V_{0}$ and $\lambda$ are positive constants, show that there is an exact cosmological solution with

$\begin{aligned} &a(t) \propto t^{16 \pi G / \lambda^{2}} \\ &\phi(t)=\phi_{0}+\frac{2}{\lambda} \ln (t), \end{aligned}$

where $\phi_{0}$ is a constant. Find the algebraic relation between $\lambda, V_{0}$ and $\phi_{0}$. Show that a solution only exists when $0<\lambda^{2}<48 \pi G$. For what range of values of $\lambda^{2}$ does inflation occur? Comment on what happens when $\lambda \rightarrow 0$.

Paper 2, Section I, E

commentA self-gravitating fluid with density $\rho$, pressure $P(\rho)$ and velocity $\mathbf{v}$ in a gravitational potential $\Phi$ obeys the equations

$\begin{aligned} \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v}) &=0 \\ \frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot \nabla) \mathbf{v}+\frac{\nabla P}{\rho}+\nabla \Phi &=\mathbf{0} \\ \nabla^{2} \Phi &=4 \pi G \rho \end{aligned}$

Assume that there exists a static constant solution of these equations with $\mathbf{v}=\mathbf{0}, \rho=\rho_{0}$ and $\Phi=\Phi_{0}$, for which $\nabla \Phi_{0}$ can be neglected. This solution is perturbed. Show that, to first order in the perturbed quantities, the density perturbations satisfy

$\frac{\partial^{2} \rho_{1}}{\partial t^{2}}=c_{s}^{2} \nabla^{2} \rho_{1}+4 \pi G \rho_{0} \rho_{1}$

where $\rho=\rho_{0}+\rho_{1}(\mathbf{x}, t)$ and $c_{s}^{2}=d P / d \rho$. Show that there are solutions to this equation of the form

$\rho_{1}(\mathbf{x}, t)=A \exp [-i \mathbf{k} \cdot \mathbf{x}+i \omega t]$

where $A, \omega$ and $\mathbf{k}$ are constants and

$\omega^{2}=c_{s}^{2} \mathbf{k} \cdot \mathbf{k}-4 \pi G \rho_{0}$

Interpret these solutions physically in the limits of small and large $|\mathbf{k}|$, explaining what happens to density perturbations on large and small scales, and determine the critical wavenumber that divides the two distinct behaviours of the perturbation.

Paper 3, Section I, E

commentConsider a finite sphere of zero-pressure material of uniform density $\rho(t)$ which expands with radius $r(t)=a(t) r_{0}$, where $r_{0}$ is an arbitary constant, due to the evolution of the expansion scale factor $a(t)$. The sphere has constant total mass $M$ and its radius satisfies

$\ddot{r}=-\frac{d \Phi}{d r}$

where

$\Phi(r)=-\frac{G M}{r}-\frac{1}{6} \Lambda r^{2} c^{2},$

with $\Lambda$ constant. Show that the scale factor obeys the equation

$\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{K c^{2}}{a^{2}}+\frac{1}{3} \Lambda c^{2},$

where $K$ is a constant. Explain why the sign, but not the magnitude, of $K$ is important. Find exact solutions of this equation for $a(t)$ when

(i) $K=\Lambda=0$ and $\rho(t) \neq 0$,

(ii) $\rho=K=0$ and $\Lambda>0$,

(iii) $\rho=\Lambda=0$ and $K \neq 0$.

Which two of the solutions (i)-(iii) are relevant for describing the evolution of the universe after the radiation-dominated era?

Paper 3, Section II, E

commentThe luminosity distance to an astronomical light source is given by $d_{L}=\chi / a(t)$, where $a(t)$ is the expansion scale factor and $\chi$ is the comoving distance in the universe defined by $d t=a(t) d \chi$. A zero-curvature Friedmann universe containing pressure-free matter and a cosmological constant with density parameters $\Omega_{m}$ and $\Omega_{\Lambda} \equiv 1-\Omega_{m}$, respectively, obeys the Friedmann equation

$H^{2}=H_{0}^{2}\left(\frac{\Omega_{m 0}}{a^{3}}+\Omega_{\Lambda 0}\right)$

where $H=(d a / d t) / a$ is the Hubble expansion rate of the universe and the subscript 0 denotes present-day values, with $a_{0} \equiv 1$.

If $z$ is the redshift, show that

$d_{L}(z)=\frac{1+z}{H_{0}} \int_{0}^{z} \frac{d z^{\prime}}{\left[\left(1-\Omega_{\Lambda 0}\right)\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda 0}\right]^{1 / 2}}$

Find $d_{L}(z)$ when $\Omega_{\Lambda 0}=0$ and when $\Omega_{m 0}=0$. Roughly sketch the form of $d_{L}(z)$ for these two cases. What is the effect of a cosmological constant $\Lambda$ on the luminosity distance at a fixed value of $z$ ? Briefly describe how the relation between luminosity distance and redshift has been used to establish the acceleration of the expansion of the universe.

Paper 4, Section I, E

commentA homogeneous and isotropic universe, with cosmological constant $\Lambda$, has expansion scale factor $a(t)$ and Hubble expansion rate $H=\dot{a} / a$. The universe contains matter with density $\rho$ and pressure $P$ which satisfy the positive-energy condition $\rho+3 P / c^{2} \geqslant 0$. The acceleration equation is

$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)+\frac{1}{3} \Lambda c^{2}$

If $\Lambda \leqslant 0$, show that

$\frac{d}{d t}\left(H^{-1}\right) \geqslant 1$

Deduce that $H \rightarrow \infty$ and $a \rightarrow 0$ at a finite time in the past or the future. What property of $H$ distinguishes the two cases?

Give a simple counterexample with $\rho=P=0$ to show that this deduction fails to hold when $\Lambda>0$.

Paper 1, Section II, G

commentDefine the concepts of (smooth) manifold and manifold with boundary for subsets of $\mathbf{R}^{N}$.

Let $X \subset \mathbf{R}^{6}$ be the subset defined by the equations

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=1, \quad x_{4}^{2}-x_{5}^{2}-x_{6}^{2}=-1 .$

Prove that $X$ is a manifold of dimension four.

For $a>0$, let $B(a) \subset \mathbf{R}^{6}$ denote the spherical ball $x_{1}^{2}+\ldots+x_{6}^{2} \leqslant a$. Prove that $X \cap B(a)$ is empty if $a<2$, is a manifold diffeomorphic to $S^{2} \times S^{1}$ if $a=2$, and is a manifold with boundary if $a>2$, with each component of the boundary diffeomorphic to $S^{2} \times S^{1}$.

[You may quote without proof any general results from lectures that you may need.]

Paper 2, Section II, G

commentDefine the terms Gaussian curvature $K$ and mean curvature $H$ for a smooth embedded oriented surface $S \subset \mathbf{R}^{3}$. [You may assume the fact that the derivative of the Gauss map is self-adjoint.] If $K=H^{2}$ at all points of $S$, show that both $H$ and $K$ are locally constant. [Hint: Use the symmetry of second partial derivatives of the field of unit normal vectors.]

If $K=H^{2}=0$ at all points of $S$, show that the unit normal vector $\mathbf{N}$ to $S$ is locally constant and that $S$ is locally contained in a plane. If $K=H^{2}$ is a strictly positive constant on $S$ and $\phi: U \rightarrow S$ is a local parametrization (where $U$ is connected) on $S$ with unit normal vector $\mathbf{N}(u, v)$ for $(u, v) \in U$, show that $\phi(u, v)+\mathbf{N}(u, v) / H$ is constant on $U$. Deduce that $S$ is locally contained in a sphere of radius $1 /|H|$.

If $S$ is connected with $K=H^{2}$ at all points of $S$, deduce that $S$ is contained in either a plane or a sphere.

Paper 3, Section II, G

Let $\alpha: I \rightarrow S$ be a parametrized curve on a smooth embedded surface $S \subset \mathbf{R}^{3}$. Define what is meant by a vector field $V$ along $\alpha$ and the concept of such a vector field being parallel. If $V$ and $W$ are both parallel vector fields along $\alpha$, show that the inner product $\langle V(t), W(t)\rangle$ is constant.

Given a local parametrization $\phi: U \rightarrow S$, define the Christoffel symbols $\Gamma_{j k}^{i}$ on $U$. Given a vector $v_{0} \in T_{\alpha(0)} S$