# Part II, 2006

### Jump to course

1.II.21H

commentCompute the homology groups of the "pinched torus" obtained by identifying a meridian circle $S^{1} \times\{p\}$ on the torus $S^{1} \times S^{1}$ to a point, for some point $p \in S^{1}$.

2.II.21H

commentState the simplicial approximation theorem. Compute the number of 0 -simplices (vertices) in the barycentric subdivision of an $n$-simplex and also compute the number of $n$-simplices. Finally, show that there are at most countably many homotopy classes of continuous maps from the 2-sphere to itself.

3.II.20H

commentLet $X$ be the union of two circles identified at a point: the "figure eight". Classify all the connected double covering spaces of $X$. If we view these double coverings just as topological spaces, determine which of them are homeomorphic to each other and which are not.

4.II.21H

commentFix a point $p$ in the torus $S^{1} \times S^{1}$. Let $G$ be the group of homeomorphisms $f$ from the torus $S^{1} \times S^{1}$ to itself such that $f(p)=p$. Determine a non-trivial homomorphism $\phi$ from $G$ to the group $\operatorname{GL}(2, \mathbb{Z})$.

[The group $\mathrm{GL}(2, \mathbb{Z})$ consists of $2 \times 2$ matrices with integer coefficients that have an inverse which also has integer coefficients.]

Establish whether each $f$ in the kernel of $\phi$ is homotopic to the identity map.

1.II.33A

commentConsider a particle of mass $m$ and momentum $\hbar k$ moving under the influence of a spherically symmetric potential $V(r)$ such that $V(r)=0$ for $r \geqslant a$. Define the scattering amplitude $f(\theta)$ and the phase shift $\delta_{\ell}(k)$. Here $\theta$ is the scattering angle. How is $f(\theta)$ related to the differential cross section?

Obtain the partial-wave expansion

$f(\theta)=\frac{1}{k} \sum_{\ell=0}^{\infty}(2 \ell+1) e^{i \delta_{\ell}} \sin \delta_{\ell} P_{\ell}(\cos \theta) .$

Let $R_{\ell}(r)$ be a solution of the radial Schrödinger equation, regular at $r=0$, for energy $\hbar^{2} k^{2} / 2 m$ and angular momentum $\ell$. Let

$Q_{\ell}(k)=a \frac{R_{\ell}^{\prime}(a)}{R_{\ell}(a)}-k a \frac{j_{\ell}^{\prime}(k a)}{j_{\ell}(k a)}$

Obtain the relation

$\tan \delta_{\ell}=\frac{Q_{\ell}(k) j_{\ell}^{2}(k a) k a}{Q_{\ell}(k) n_{\ell}(k a) j_{\ell}(k a) k a-1} .$

Suppose that

$\tan \delta_{\ell} \approx \frac{\gamma}{k_{0}-k},$

for some $\ell$, with all other $\delta_{\ell}$ small for $k \approx k_{0}$. What does this imply for the differential cross section when $k \approx k_{0}$ ?

[For $V=0$, the two independent solutions of the radial Schrödinger equation are $j_{\ell}(k r)$ and $n_{\ell}(k r)$ with

$\begin{aligned} j_{\ell}(\rho) & \sim \frac{1}{\rho} \sin \left(\rho-\frac{1}{2} \ell \pi\right), \quad n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos \left(\rho-\frac{1}{2} \ell \pi\right) \quad \text { as } \quad \rho \rightarrow \infty \\ e^{i \rho \cos \theta} &=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(\rho) P_{\ell}(\cos \theta) \end{aligned}$

Note that the Wronskian $\rho^{2}\left(j_{\ell}(\rho) n_{\ell}^{\prime}(\rho)-j_{\ell}^{\prime}(\rho) n_{\ell}(\rho)\right)$ is independent of $\left.\rho .\right]$

2.II.33D

commentState and prove Bloch's theorem for the electron wave functions for a periodic potential $V(\mathbf{r})=V(\mathbf{r}+\mathbf{l})$ where $\mathbf{l}=\sum_{i} n_{i} \mathbf{a}_{i}$ is a lattice vector.

What is the reciprocal lattice? Explain why the Bloch wave-vector $\mathbf{k}$ is arbitrary up to $\mathbf{k} \rightarrow \mathbf{k}+\mathbf{g}$, where $\mathbf{g}$ is a reciprocal lattice vector.

Describe in outline why one can expect energy bands $E_{n}(\mathbf{k})=E_{n}(\mathbf{k}+\mathbf{g})$. Explain how $\mathbf{k}$ may be restricted to a Brillouin zone $B$ and show that the number of states in volume $d^{3} k$ is

$\frac{2}{(2 \pi)^{3}} \mathrm{~d}^{3} k$

Assuming that the velocity of an electron in the energy band with Bloch wave-vector $\mathbf{k}$ is

$\mathbf{v}(\mathbf{k})=\frac{1}{\hbar} \frac{\partial}{\partial \mathbf{k}} E_{n}(\mathbf{k})$

show that the contribution to the electric current from a full energy band is zero. Given that $n(\mathbf{k})=1$ for each occupied energy level, show that the contribution to the current density is then

$\mathbf{j}=-e \frac{2}{(2 \pi)^{3}} \int_{B} \mathrm{~d}^{3} k n(\mathbf{k}) \mathbf{v}(\mathbf{k})$

where $-e$ is the electron charge.

3.II.33A

commentConsider a one-dimensional crystal of lattice space $b$, with atoms having positions $x_{s}$ and momenta $p_{s}, s=0,1,2, \ldots, N-1$, such that the classical Hamiltonian is

$H=\sum_{s=0}^{N-1}\left(\frac{p_{s}^{2}}{2 m}+\frac{1}{2} m \lambda^{2}\left(x_{s+1}-x_{s}-b\right)^{2}\right)$

where we identify $x_{N}=x_{0}$. Show how this may be quantized to give the energy eigenstates consisting of a ground state $|0\rangle$ together with free phonons with energy $\hbar \omega\left(k_{r}\right)$ where $k_{r}=2 \pi r / N b$ for suitable integers $r$. Obtain the following expression for the quantum operator $x_{s}$

$x_{s}=s b+\left(\frac{\hbar}{2 m N}\right)^{\frac{1}{2}} \sum_{r} \frac{1}{\sqrt{\omega\left(k_{r}\right)}}\left(a_{r} e^{i k_{r} s b}+a_{r}^{\dagger} e^{-i k_{r} s b}\right)$

where $a_{r}, a_{r}^{\dagger}$ are annihilation and creation operators, respectively.

An interaction involves the matrix element

$M=\sum_{s=0}^{N-1}\left\langle 0\left|e^{i q x_{s}}\right| 0\right\rangle .$

Calculate this and show that $|M|^{2}$ has its largest value when $q=2 \pi n / b$ for integer $n$.

Disregard the case $\omega\left(k_{r}\right)=0$.

[You may use the relations

$\sum_{s=0}^{N-1} e^{i k_{r} s b}= \begin{cases}N, & r=N b \\ 0 & \text { otherwise }\end{cases}$

and $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ if $[A, B]$ commutes with $A$ and with $\left.B .\right]$

4.II.33D

commentFor the one-dimensional potential

$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{n} \delta(x-n a)$

solve the Schrödinger equation for negative energy and obtain an equation that determines possible energy bands. Show that the results agree with the tight-binding model in appropriate limits.

[It may be useful to note that $\left.V(x)=-\frac{\hbar^{2} \lambda}{m a} \sum_{n} e^{2 \pi i n x / a} .\right]$

1.II.26J

comment(a) What is a $Q$-matrix? What is the relationship between the transition matrix $P(t)$ of a continuous time Markov process and its generator $Q$ ?

(b) A pond has three lily pads, labelled 1, 2, and 3. The pond is also the home of a frog that hops from pad to pad in a random fashion. The position of the frog is a continuous time Markov process on $\{1,2,3\}$ with generator

$Q=\left(\begin{array}{ccc} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 1 & 0 & -1 \end{array}\right)$

Sketch an arrow diagram corresponding to $Q$ and determine the communicating classes. Find the probability that the frog is on pad 2 in equilibrium. Find the probability that the frog is on pad 2 at time $t$ given that the frog is on pad 1 at time 0 .

2.II.26J

comment(a) Define a renewal process $\left(X_{t}\right)$ with independent, identically-distributed holding times $S_{1}, S_{2}, \ldots .$ State without proof the strong law of large numbers for $\left(X_{t}\right)$. State without proof the elementary renewal theorem for the mean value $m(t)=\mathbb{E} X_{t}$.

(b) A circular bus route consists of ten bus stops. At exactly 5am, the bus starts letting passengers in at the main bus station (stop 1). It then proceeds to stop 2 where it stops to let passengers in and out. It continues in this fashion, stopping at stops 3 to 10 in sequence. After leaving stop 10, the bus heads to stop 1 and the cycle repeats. The travel times between stops are exponentially distributed with mean 4 minutes, and the time required to let passengers in and out at each stop are exponentially distributed with mean 1 minute. Calculate approximately the average number of times the bus has gone round its route by $1 \mathrm{pm}$.

When the driver's shift finishes, at exactly $1 \mathrm{pm}$, he immediately throws all the passengers off the bus if the bus is already stopped, or otherwise, he drives to the next stop and then throws the passengers off. He then drives as fast as he can round the rest of the route to the main bus station. Giving reasons but not proofs, calculate approximately the average number of stops he will drive past at the end of his shift while on his way back to the main bus station, not including either the stop at which he throws off the passengers or the station itself.

3.II.25J

commentA passenger plane with $N$ numbered seats is about to take off; $N-1$ seats have already been taken, and now the last passenger enters the cabin. The first $N-1$ passengers were advised by the crew, rather imprudently, to take their seats completely at random, but the last passenger is determined to sit in the place indicated on his ticket. If his place is free, he takes it, and the plane is ready to fly. However, if his seat is taken, he insists that the occupier vacates it. In this case the occupier decides to follow the same rule: if the free seat is his, he takes it, otherwise he insists on his place being vacated. The same policy is then adopted by the next unfortunate passenger, and so on. Each move takes a random time which is exponentially distributed with mean $\mu^{-1}$. What is the expected duration of the plane delay caused by these displacements?

4.II.26J

comment(a) Let $\left(N_{t}\right)_{t \geqslant 0}$ be a Poisson process of rate $\lambda>0$. Let $p$ be a number between 0 and 1 and suppose that each jump in $\left(N_{t}\right)$ is counted as type one with probability $p$ and type two with probability $1-p$, independently for different jumps and independently of the Poisson process. Let $M_{t}^{(1)}$ be the number of type-one jumps and $M_{t}^{(2)}=N_{t}-M_{t}^{(1)}$ the number of type-two jumps by time $t$. What can you say about the pair of processes $\left(M_{t}^{(1)}\right)_{t \geqslant 0}$ and $\left(M_{t}^{(2)}\right)_{t \geqslant 0}$ ? What if we fix probabilities $p_{1}, \ldots, p_{m}$ with $p_{1}+\ldots+p_{m}=1$ and consider $m$ types instead of two?

(b) A person collects coupons one at a time, at jump times of a Poisson process $\left(N_{t}\right)_{t \geqslant 0}$ of rate $\lambda$. There are $m$ types of coupons, and each time a coupon of type $j$ is obtained with probability $p_{j}$, independently of the previously collected coupons and independently of the Poisson process. Let $T$ be the first time when a complete set of coupon types is collected. Show that

$\mathbb{P}(T<t)=\prod_{j=1}^{m}\left(1-e^{-p_{j} \lambda t}\right)$

Let $L=N_{T}$ be the total number of coupons collected by the time the complete set of coupon types is obtained. Show that $\lambda \mathbb{E} T=\mathbb{E} L$. Hence, or otherwise, deduce that $\mathbb{E} L$ does not depend on $\lambda$.

1.II $. 30 \mathrm{~B}$

commentTwo real functions $p(t), q(t)$ of a real variable $t$ are given on an interval $[0, b]$, where $b>0$. Suppose that $q(t)$ attains its minimum precisely at $t=0$, with $q^{\prime}(0)=0$, and that $q^{\prime \prime}(0)>0$. For a real argument $x$, define

$I(x)=\int_{0}^{b} p(t) e^{-x q(t)} d t$

Explain how to obtain the leading asymptotic behaviour of $I(x)$ as $x \rightarrow+\infty$ (Laplace's method).

The modified Bessel function $I_{\nu}(x)$ is defined for $x>0$ by:

$I_{\nu}(x)=\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos \theta} \cos (\nu \theta) d \theta-\frac{\sin (\nu \pi)}{\pi} \int_{0}^{\infty} e^{-x(\cosh t)-\nu t} d t .$

Show that

$I_{\nu}(x) \sim \frac{e^{x}}{\sqrt{2 \pi x}}$

as $x \rightarrow \infty$ with $\nu$ fixed.

3.II $. 30$ B

commentThe Airy function $\operatorname{Ai}(z)$ is defined by

$\operatorname{Ai}(z)=\frac{1}{2 \pi i} \int_{C} \exp \left(-\frac{1}{3} t^{3}+z t\right) d t$

where the contour $C$ begins at infinity along the ray $\arg (t)=4 \pi / 3$ and ends at infinity along the ray $\arg (t)=2 \pi / 3$. Restricting attention to the case where $z$ is real and positive, use the method of steepest descent to obtain the leading term in the asymptotic expansion for $\operatorname{Ai}(z)$ as $z \rightarrow \infty$ :

$\operatorname{Ai}(z) \sim \frac{\exp \left(-\frac{2}{3} z^{3 / 2}\right)}{2 \pi^{1 / 2} z^{1 / 4}}$

$\left[\right.$ Hint: put $\left.t=z^{1 / 2} \tau .\right]$

4.II $. 31 \mathrm{~B}$

comment(a) Outline the Liouville-Green approximation to solutions $w(z)$ of the ordinary differential equation

$\frac{d^{2} w}{d z^{2}}=f(z) w$

in a neighbourhood of infinity, in the case that, near infinity, $f(z)$ has the convergent series expansion

$f(z)=\sum_{s=0}^{\infty} \frac{f_{s}}{z^{s}}$

with $f_{0} \neq 0$.

In the case

$f(z)=1+\frac{1}{z}+\frac{2}{z^{2}},$

explain why you expect a basis of two asymptotic solutions $w_{1}(z), w_{2}(z)$, with

$\begin{aligned} &w_{1}(z) \sim z^{\frac{1}{2}} e^{z}\left(1+\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\cdots\right), \\ &w_{2}(z) \sim z^{-\frac{1}{2}} e^{-z}\left(1+\frac{b_{1}}{z}+\frac{b_{2}}{z^{2}}+\cdots\right), \end{aligned}$

as $z \rightarrow+\infty$, and show that $a_{1}=-\frac{9}{8}$.

(b) Determine, at leading order in the large positive real parameter $\lambda$, an approximation to the solution $u(x)$ of the eigenvalue problem:

$u^{\prime \prime}(x)+\lambda^{2} g(x) u(x)=0 ; \quad u(0)=u(1)=0$

where $g(x)$ is greater than a positive constant for $x \in[0,1]$.

1.I.9C

commentHamilton's equations for a system with $n$ degrees of freedom can be written in vector form as

$\dot{\mathbf{x}}=J \frac{\partial H}{\partial \mathbf{x}}$

where $\mathbf{x}=\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)^{T}$ is a $2 n$-vector and the $2 n \times 2 n$ matrix $J$ takes the form

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

where 1 is the $n \times n$ identity matrix. Derive the condition for a transformation of the form $x_{i} \rightarrow y_{i}(\mathbf{x})$ to be canonical. For a system with a single degree of freedom, show that the following transformation is canonical for all nonzero values of $\alpha$ :

$Q=\tan ^{-1}\left(\frac{\alpha q}{p}\right), \quad P=\frac{1}{2}\left(\alpha q^{2}+\frac{p^{2}}{\alpha}\right)$

1.II.15C

comment(a) In the Hamiltonian framework, the action is defined as

$S=\int\left(p_{a} \dot{q}_{a}-H\left(q_{a}, p_{a}, t\right)\right) d t$

Derive Hamilton's equations from the principle of least action. Briefly explain how the functional variations in this derivation differ from those in the derivation of Lagrange's equations from the principle of least action. Show that $H$ is a constant of the motion whenever $\partial H / \partial t=0$.

(b) What is the invariant quantity arising in Liouville's theorem? Does the theorem depend on assuming $\partial H / \partial t=0$ ? State and prove Liouville's theorem for a system with a single degree of freedom.

(c) A particle of mass $m$ bounces elastically along a perpendicular between two parallel walls a distance $b$ apart. Sketch the path of a single cycle in phase space, assuming that the velocity changes discontinuously at the wall. Compute the action $I=\oint p d q$ as a function of the energy $E$ and the constants $m, b$. Verify that the period of oscillation $T$ is given by $T=d I / d E$. Suppose now that the distance $b$ changes slowly. What is the relevant adiabatic invariant? How does $E$ change as a function of $b$ ?

2.I.9C

commentTwo point masses, each of mass $m$, are constrained to lie on a straight line and are connected to each other by a spring of force constant $k$. The left-hand mass is also connected to a wall on the left by a spring of force constant $j$. The right-hand mass is similarly connected to a wall on the right, by a spring of force constant $\ell$, so that the potential energy is

$V=\frac{1}{2} k\left(\eta_{1}-\eta_{2}\right)^{2}+\frac{1}{2} j \eta_{1}^{2}+\frac{1}{2} \ell \eta_{2}^{2}$

where $\eta_{i}$ is the distance from equilibrium of the $i^{\text {th }}$mass. Derive the equations of motion. Find the frequencies of the normal modes.

3.I.9C

commentA pendulum of length $\ell$ oscillates in the $x y$ plane, making an angle $\theta(t)$ with the vertical $y$ axis. The pivot is attached to a moving lift that descends with constant acceleration $a$, so that the position of the bob is

$x=\ell \sin \theta, \quad y=\frac{1}{2} a t^{2}+\ell \cos \theta .$

Given that the Lagrangian for an unconstrained particle is

$L=\frac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)+m g y,$

determine the Lagrangian for the pendulum in terms of the generalized coordinate $\theta$. Derive the equation of motion in terms of $\theta$. What is the motion when $a=g$ ?

Find the equilibrium configurations for arbitrary $a$. Determine which configuration is stable when

$\text { (i) } a<g$

and when

$\text { (ii) } a>g \text {. }$

3.II.15C

commentA particle of mass $m$ is constrained to move on the surface of a sphere of radius $\ell$.

The Lagrangian is given in spherical polar coordinates by

$L=\frac{1}{2} m \ell^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+m g \ell \cos \theta,$

where gravity $g$ is constant. Find the two constants of the motion.

The particle is projected horizontally with velocity $v$ from a point whose depth below the centre is $\ell \cos \theta=D$. Find $v$ such that the particle trajectory

(i) just grazes the horizontal equatorial plane $\theta=\pi / 2$;

(ii) remains at depth $D$ for all time $t$.

4.I.9C

commentCalculate the principal moments of inertia for a uniform cylinder, of mass $M$, radius $R$ and height $2 h$, about its centre of mass. For what height-to-radius ratio does the cylinder spin like a sphere?

1.I.4G

commentDefine a linear feedback shift register. Explain the Berlekamp-Massey method for "breaking" a key stream produced by a linear feedback shift register of unknown length. Use it to find the feedback polynomial of a linear feedback shift register with output sequence

$010111100010 \ldots$

2.I.4G

commentLet $\Sigma_{1}$ and $\Sigma_{2}$ be alphabets of sizes $m$ and $a$. What does it mean to say that an $a$-ary code $f: \Sigma_{1} \rightarrow \Sigma_{2}^{*}$ is decipherable? Show that if $f$ is decipherable then the word lengths $s_{1}, \ldots, s_{m}$ satisfy

$\sum_{i=1}^{m} a^{-s_{i}} \leqslant 1$

Find a decipherable binary code consisting of codewords 011, 0111, 01111, 11111, and three further codewords of length 2. How do you know the example you have given is decipherable?

2.II.12G

commentDefine a cyclic code. Show that there is a bijection between the cyclic codes of length $n$, and the factors of $X^{n}-1$ in $\mathbb{F}_{2}[X]$.

If $n$ is an odd integer then we can find a finite extension $K$ of $\mathbb{F}_{2}$ that contains a primitive $n$th root of unity $\alpha$. Show that a cyclic code of length $n$ with defining set $\left\{\alpha, \alpha^{2}, \ldots, \alpha^{\delta-1}\right\}$ has minimum distance at least $\delta$. Show that if $n=7$ and $\delta=3$ then we obtain Hamming's original code.

[You may quote a formula for the Vandermonde determinant without proof.]

3.I.4G

commentWhat does it mean to say that a binary code $C$ has length $n$, size $m$ and minimum distance $d$ ? Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V\left(n,\left\lfloor\frac{1}{2}(d-1)\right\rfloor\right)}$

where $V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l}n \\ j\end{array}\right)$.

3.II.12G

commentDescribe the RSA system with public key $(N, e)$ and private key $(N, d)$. Briefly discuss the possible advantages or disadvantages of taking (i) $e=2^{16}+1$ or (ii) $d=2^{16}+1$.

Explain how to factor $N$ when both the private key and public key are known.

Describe the bit commitment problem, and briefly indicate how RSA can be used to solve it.

4.I.4G

commentA binary erasure channel with erasure probability $p$ is a discrete memoryless channel with channel matrix

$\left(\begin{array}{ccc} 1-p & p & 0 \\ 0 & p & 1-p \end{array}\right)$

State Shannon's second coding theorem, and use it to compute the capacity of this channel.

1.I.10D

comment(a) Introduce the concept of comoving co-ordinates in a homogeneous and isotropic universe and explain how the velocity of a galaxy is determined by the scale factor $a$. Express the Hubble parameter $H_{0}$ today in terms of the scale factor.

(b) The Raychaudhuri equation states that the acceleration of the universe is determined by the mass density $\rho$ and the pressure $P$ as

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

Now assume that the matter constituents of the universe satisfy $\rho+3 P / c^{2} \geqslant 0$. In this case explain clearly why the Hubble time $H_{0}^{-1}$ sets an upper limit on the age of the universe; equivalently, that the scale factor must vanish $\left(a\left(t_{i}\right)=0\right)$ at some time $t_{i}<t_{0}$ with $t_{0}-t_{i} \leqslant H_{0}^{-1}$.

The observed Hubble time is $H_{0}^{-1}=1 \times 10^{10}$ years. Discuss two reasons why the above upper limit does not seem to apply to our universe.

2.I.10D

commentThe total energy of a gas can be expressed in terms of a momentum integral

$E=\int_{0}^{\infty} \mathcal{E}(p) \bar{n}(p) d p$

where $p$ is the particle momentum, $\mathcal{E}(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the particle energy and $\bar{n}(p) d p$ is the average number of particles in the momentum range $p \rightarrow p+d p$. Consider particles in a cubic box of side $L$ with $p \propto L^{-1}$. Explain why the momentum varies as

$\frac{d p}{d V}=-\frac{p}{3 V}$

Consider the overall change in energy $d E$ due to the volume change $d V$. Given that the volume varies slowly, use the thermodynamic result $d E=-P d V$ (at fixed particle number $N$ and entropy $S$ ) to find the pressure

$P=\frac{1}{3 V} \int_{0}^{\infty} p \mathcal{E}^{\prime}(p) \bar{n}(p) d p .$

Use this expression to derive the equation of state for an ultrarelativistic gas.

During the radiation-dominated era, photons remain in equilibrium with energy density $\epsilon_{\gamma} \propto T^{4}$ and number density $n_{\gamma} \propto T^{3}$. Briefly explain why the photon temperature falls inversely with the scale factor, $T \propto a^{-1}$. Discuss the implications for photon number and entropy conservation.

2.II.15D

comment(a) Consider a homogeneous and isotropic universe filled with relativistic matter of mass density $\rho(t)$ and scale factor $a(t)$. Consider the energy $E(t) \equiv \rho(t) c^{2} V(t)$ of a small fluid element in a comoving volume $V_{0}$ where $V(t)=a^{3}(t) V_{0}$. Show that for slow (adiabatic) changes in volume, the density will satisfy the fluid conservation equation

$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right)$

where $P$ is the pressure.

(b) Suppose that a flat $(k=0)$ universe is filled with two matter components:

(i) radiation with an equation of state $P_{\mathrm{r}}=\frac{1}{3} \rho_{\mathrm{r}} c^{2}$.

(ii) a gas of cosmic strings with an equation of state $P_{\mathrm{s}}=-\frac{1}{3} \rho_{\mathrm{s}} c^{2}$.

Use the fluid conservation equation to show that the total relativistic mass density behaves as

$\rho=\frac{\rho_{\mathrm{r} 0}}{a^{4}}+\frac{\rho_{\mathrm{s} 0}}{a^{2}},$

where $\rho_{\mathrm{r} 0}$ and $\rho_{\mathrm{s} 0}$ are respectively the radiation and string densities today (that is, at $t=t_{0}$ when $a\left(t_{0}\right)=1$ ). Assuming that both the Hubble parameter today $H_{0}$ and the ratio $\beta \equiv \rho_{\mathrm{r} 0} / \rho_{\mathrm{s} 0}$ are known, show that the Friedmann equation can be rewritten as

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{H_{0}^{2}}{a^{4}}\left(\frac{a^{2}+\beta}{1+\beta}\right) .$

Solve this equation to find the following solution for the scale factor

$a(t)=\frac{\left(H_{0} t\right)^{1 / 2}}{(1+\beta)^{1 / 2}}\left[H_{0} t+2 \beta^{1 / 2}(1+\beta)^{1 / 2}\right]^{1 / 2} .$

Show that the scale factor has the expected asymptotic behaviour at early times $t \rightarrow 0$.

Hence show that the age of this universe today is

$t_{0}=H_{0}^{-1}(1+\beta)^{1 / 2}\left[(1+\beta)^{1 / 2}-\beta^{1 / 2}\right]$

and that the time $t_{\mathrm{eq}}$ of equal radiation and string densities $\left(\rho_{\mathrm{r}}=\rho_{\mathrm{s}}\right)$ is

$t_{\mathrm{eq}}=H_{0}^{-1}(\sqrt{2}-1) \beta^{1 / 2}(1+\beta)^{1 / 2}$

3.I.10D

comment(a) Consider a spherically symmetric star with outer radius $R$, density $\rho(r)$ and pressure $P(r)$. By balancing the gravitational force on a shell at radius $r$ against the force due to the pressure gradient, derive the pressure support equation

$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$

where $m(r)=\int_{0}^{r} \rho\left(r^{\prime}\right) 4 \pi r^{\prime 2} d r^{\prime}$. Show that this implies

$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho$

Suggest appropriate boundary conditions at $r=0$ and $r=R$, together with a brief justification.

(b) Describe qualitatively the endpoint of stellar evolution for our sun when all its nuclear fuel is spent. Your discussion should briefly cover electron degeneracy pressure and the relevance of stability against inverse beta-decay.

[Note that $m_{n}-m_{p} \approx 2.6 m_{e}$, where $m_{n}, m_{p}, m_{e}$ are the masses of the neutron, proton and electron respectively.]

4.I.10D

commentThe number density of fermions of mass $m$ at equilibrium in the early universe with temperature $T$, is given by the integral

$n=\frac{4 \pi}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(\mathcal{E}(p)-\mu) / k T]+1}$

where $\mathcal{E}(p)=c \sqrt{p^{2}+m^{2} c^{2}}$, and $\mu$ is the chemical potential. Assuming that the fermions remain in equilibrium when they become non-relativistic $\left(k T, \mu \ll m c^{2}\right)$, show that the number density can be expressed as

$n=\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$

[Hint: You may assume $\left.\int_{0}^{\infty} d x e^{-\sigma^{2} x^{2}}=\sqrt{\pi} /(2 \sigma), \quad(\sigma>0) .\right]$

Suppose that the fermions decouple at a temperature given by $k T=m c^{2} / \alpha$ where $\alpha \gg 1$. Assume also that $\mu=0$. By comparing with the photon number density at $n_{\gamma}=16 \pi \zeta(3)(k T / h c)^{3}$, where $\zeta(3)=\sum_{n=1}^{\infty} n^{-3}=1.202 \ldots$, show that the ratio of number densities at decoupling is given by

$\frac{n}{n_{\gamma}}=\frac{\sqrt{2 \pi}}{8 \zeta(3)} \alpha^{3 / 2} e^{-\alpha}$

Now assume that $\alpha \approx 20$, (which implies $n / n_{\gamma} \approx 5 \times 10^{-8}$ ), and that the fermion mass $m=m_{p} / 20$, where $m_{p}$ is the proton mass. Explain clearly why this new fermion would be a good candidate for solving the dark matter problem of the standard cosmology.

4.II.15D

commentThe perturbed motion of cold dark matter particles (pressure-free, $P=0$ ) in an expanding universe can be parametrized by the trajectories

$\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\boldsymbol{\psi}(\mathbf{q}, t)]$

where $a(t)$ is the scale factor of the universe, $\mathbf{q}$ is the unperturbed comoving trajectory and $\boldsymbol{\psi}$ is the comoving displacement. The particle equation of motion is $\ddot{\mathbf{r}}=-\nabla \Phi$, where the Newtonian potential satisfies the Poisson equation $\nabla^{2} \Phi=4 \pi G \rho$ with mass density $\rho(\mathbf{r}, t)$.

(a) Discuss how matter conservation in a small volume $d^{3} \mathbf{r}$ ensures that the perturbed density $\rho(\mathbf{r}, t)$ and the unperturbed background density $\bar{\rho}(t)$ are related by

$\rho(\mathbf{r}, t) d^{3} \mathbf{r}=\bar{\rho}(t) a^{3}(t) d^{3} \mathbf{q}$

By changing co-ordinates with the Jacobian

$\left|\partial r_{i} / \partial q_{j}\right|^{-1}=\left|a \delta_{i j}+a \partial \psi_{i} / \partial q_{j}\right|^{-1} \approx a^{-3}\left(1-\nabla_{q} \cdot \psi\right),$

show that the fractional density perturbation $\delta(\mathbf{q}, t)$ can be written to leading order as

$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}=-\nabla_{q} \cdot \psi,$

where $\nabla_{q} \cdot \boldsymbol{\psi}=\sum_{i} \partial \psi_{i} / \partial q_{i}$.

Use this result to integrate the Poisson equation once. Hence, express the particle equation of motion in terms of the comoving displacement as

$\ddot{\boldsymbol{\psi}}+2 \frac{\dot{a}}{a} \dot{\boldsymbol{\psi}}-4 \pi G \bar{\rho} \boldsymbol{\psi}=0$

Infer that the density perturbation evolution equation is

$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-4 \pi G \bar{\rho} \delta=0$

[Hint: You may assume that the integral of $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ is $\nabla \Phi=-4 \pi G \bar{\rho} \mathbf{r} / 3$. Note also that the Raychaudhuri equation (for $P=0$ ) is $\ddot{a} / a=-4 \pi G \bar{\rho} / 3 .$.]

(b) Find the general solution of equation $(*)$ in a flat $(k=0)$ universe dominated by cold dark matter $(P=0)$. Discuss the effect of late-time $\Lambda$ or dark energy domination on the growth of density perturbations.

1.II.24H

comment(a) State and prove the inverse function theorem for a smooth map $f: X \rightarrow Y$ between manifolds without boundary.

[You may assume the inverse function theorem for functions in Euclidean space.]

(b) Let $p$ be a real polynomial in $k$ variables such that for some integer $m \geqslant 1$,

$p\left(t x_{1}, \ldots, t x_{k}\right)=t^{m} p\left(x_{1}, \ldots, x_{k}\right)$

for all real $t>0$ and all $y=\left(x_{1}, \ldots, x_{k}\right) \in \mathbb{R}^{k}$. Prove that the set $X_{a}$ of points $y$ where $p(y)=a$ is a $(k-1)$-dimensional submanifold of $\mathbb{R}^{k}$, provided it is not empty and $a \neq 0$.

[You may use the pre-image theorem provided that it is clearly stated.]

(c) Show that the manifolds $X_{a}$ with $a>0$ are all diffeomorphic. Is $X_{a}$ with $a>0$ necessarily diffeomorphic to $X_{b}$ with $b<0$ ?

2.II $24 \mathrm{H} \quad$

commentLet $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the exponential map $\exp _{p}$ at a point $p \in S$. Assuming that exp ${ }_{p}$ is smooth, show that $\exp _{p}$ is a diffeomorphism in a neighbourhood of the origin in $T_{p} S$.

(b) Given a parametrization around $p \in S$, define the Christoffel symbols and show that they only depend on the coefficients of the first fundamental form.

(c) Consider a system of normal co-ordinates centred at $p$, that is, Cartesian coordinates $(x, y)$ in $T_{p} S$ and parametrization given by $(x, y) \mapsto \exp _{p}\left(x e_{1}+y e_{2}\right)$, where $\left\{e_{1}, e_{2}\right\}$ is an orthonormal basis of $T_{p} S$. Show that all of the Christoffel symbols are zero at $p$.

3.II.23H

commentLet $S \subset \mathbb{R}^{3}$ be a connected oriented surface.

(a) Define the Gauss map $N: S \rightarrow S^{2}$ of $S$. Given $p \in S$, show that the derivative of $N$,

$d N_{p}: T_{p} S \rightarrow T_{N(p)} S^{2}=T_{p} S$

is self-adjoint.

(b) Show that if $N$ is a diffeomorphism, then the Gaussian curvature is positive everywhere. Is the converse true?

4.II.24H

comment(a) Let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\lambda$ be a real number. Given a point $p \in S$ and a vector $v \in T_{p} S$ with unit norm, show that there exist $\varepsilon>0$ and a unique curve $\gamma:(-\varepsilon, \varepsilon) \rightarrow S$ parametrized by arc-length and with constant geodesic curvature $\lambda$ such that $\gamma(0)=p$ and $\dot{\gamma}(0)=v$.

[You may use the theorem on existence and uniqueness of solutions of ordinary differential equations.]

(b) Let $S$ be an oriented surface with positive Gaussian curvature and diffeomorphic to $S^{2}$. Show that two simple closed geodesics in $S$ must intersect. Is it true that two smooth simple closed curves in $S$ with constant geodesic curvature $\lambda \neq 0$ must intersect?

1.I.7E

commentFind the fixed points of the system

$\begin{aligned} &\dot{x}=x(x+2 y-3) \\ &\dot{y}=y(3-2 x-y) \end{aligned}$

Local linearization shows that all the fixed points with $x y=0$ are saddle points. Why can you be certain that this remains true when nonlinear terms are taken into account? Classify the fixed point with $x y \neq 0$ by its local linearization. Show that the equation has Hamiltonian form, and thus that your classification is correct even when the nonlinear effects are included.

Sketch the phase plane.

1.II.14E

comment(a) An autonomous dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$ has a periodic orbit $\mathbf{x}=\mathbf{X}(t)$ with period $T$. The linearized evolution of a small perturbation $\mathbf{x}=\mathbf{X}(t)+\boldsymbol{\eta}(t)$ is given by $\eta_{i}(t)=\Phi_{i j}(t) \eta_{j}(0)$. Obtain the differential equation and initial condition satisfied by the matrix $\boldsymbol{\Phi}(t)$.

Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and show that the other is given by

$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\mathbf{X}(t)) d t\right)$

(b) Use the 'energy-balance' method for nearly Hamiltonian systems to find a leadingorder approximation to the amplitude of the limit cycle of the equation

$\ddot{x}+\epsilon\left(\alpha x^{2}+\beta \dot{x}^{2}-\gamma\right) \dot{x}+x=0,$

where $0<\epsilon \ll 1$ and $(\alpha+3 \beta) \gamma>0$.

Compute a leading-order approximation to the nontrivial Floquet multiplier of the limit cycle and hence determine its stability.

[You may assume that $\int_{0}^{2 \pi} \sin ^{2} \theta \cos ^{2} \theta d \theta=\pi / 4$ and $\int_{0}^{2 \pi} \cos ^{4} \theta d \theta=3 \pi / 4$.]

2.I.7E

commentExplain what is meant by a strict Lyapunov function on a domain $\mathcal{D}$ containing the origin for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$. Define the domain of stability of a fixed point $\mathbf{x}_{0}$.

By considering the function $V=\frac{1}{2}\left(x^{2}+y^{2}\right)$ show that the origin is an asymptotically stable fixed point of

$\begin{aligned} &\dot{x}=-2 x+y+x^{3}-x y^{2} \\ &\dot{y}=-x-2 y+6 x^{2} y+4 y^{3} \end{aligned}$

Show also that its domain of stability includes $x^{2}+y^{2}<\frac{1}{2}$ and is contained in $x^{2}+y^{2} \leqslant 2$.

2.II.14E

commentLet $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that $F$ has a horseshoe, (b) that $F$ is chaotic (Glendinning's definition).

Consider the tent map defined on the interval $[0,1]$ by

$F(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$

with $1<\mu \leqslant 2$.

Find the non-zero fixed point $x_{0}$ and the points $x_{-1}<\frac{1}{2}<x_{-2}$ that satisfy

$F^{2}\left(x_{-2}\right)=F\left(x_{-1}\right)=x_{0} .$

Sketch a graph of $F$ and $F^{2}$ showing the points corresponding to $x_{-2}, x_{-1}$ and $x_{0}$. Hence show that $F^{2}$ has a horseshoe if $\mu \geqslant 2^{1 / 2}$.

Explain briefly why $F^{4}$ has a horseshoe when $2^{1 / 4} \leqslant \mu<2^{1 / 2}$ and why there are periodic points arbitrarily close to $x_{0}$ for $\mu \geqslant 2^{1 / 2}$, but no such points for $2^{1 / 4} \leqslant \mu<2^{1 / 2}$.

3.I.7E

commentState the normal-form equations for (a) a saddle-node bifurcation, (b) a transcritical bifurcation, and (c) a pitchfork bifurcation, for a dynamical system $\dot{x}=f(x, \mu)$.

Consider the system

$\begin{aligned} &\dot{x}=\mu+y-x^{2}+2 x y+3 y^{2} \\ &\dot{y}=-y+2 x^{2}+3 x y \end{aligned}$

Compute the extended centre manifold near $x=y=\mu=0$, and the evolution equation on the centre manifold, both correct to second order in $x$ and $\mu$. Deduce the type of bifurcation and show that the equation can be put in normal form, to the same order, by a change of variables of the form $T=\alpha t, X=x-\beta \mu, \tilde{\mu}=\gamma(\mu)$ for suitably chosen $\alpha, \beta$ and $\gamma(\mu)$.

4.I.7E

commentConsider the logistic map $F(x)=\mu x(1-x)$ for $0 \leqslant x \leqslant 1,0 \leqslant \mu \leqslant 4$. Show that there is a period-doubling bifurcation of the nontrivial fixed point at $\mu=3$. Show further that the bifurcating 2 -cycle $\left(x_{1}, x_{2}\right)$ is given by the roots of

$\mu^{2} x^{2}-\mu(\mu+1) x+\mu+1=0 .$

Show that there is a second period-doubling bifurcation at $\mu=1+\sqrt{6}$.

$3 . \mathrm{II} . 35 \mathrm{E} \quad$

commentA particle of rest mass $m$ and charge $q$ is moving along a trajectory $x^{a}(s)$, where $s$ is the particle's proper time, in a given external electromagnetic field with 4-potential $A^{a}\left(x^{c}\right)$. Consider the action principle $\delta S=0$ where the action is $S=\int L d s$ and

$L\left(s, x^{a}, \dot{x}^{a}\right)=-m \sqrt{\eta_{a b} \dot{x}^{a} \dot{x}^{b}}-q A_{a}\left(x^{c}\right) \dot{x}^{a},$

and variations are taken with fixed endpoints.

Show first that the action is invariant both under reparametrizations $s \rightarrow \alpha s+\beta$ where $\alpha$ and $\beta$ are constants and also under a change of electromagnetic gauge. Next define the generalized momentum $P_{a}=\partial L / \partial \dot{x}^{a}$, and obtain the equation of motion

$m \ddot{x}^{a}=q F_{b}^{a} \dot{x}^{b},$

where the tensor $F^{a}{ }_{b}$ should be defined and you may assume that $d / d s\left(\eta_{a b} \dot{x}^{a} \dot{x}^{b}\right)=0$. Then verify from $(*)$ that indeed $d / d s\left(\eta_{a b} \dot{x}^{a} \dot{x}^{b}\right)=0$.

How does $P_{a}$ differ from the momentum $p_{a}$ of an uncharged particle? Comment briefly on the principle of minimal coupling.

1.II $. 34 \mathrm{E} \quad$

comment$\mathcal{S}$ and $\mathcal{S}^{\prime}$ are two reference frames with $\mathcal{S}^{\prime}$ moving with constant speed $v$ in the $x$-direction relative to $\mathcal{S}$. The co-ordinates $x^{a}$ and ${x^{\prime}}^{a}$ are related by $d x^{\prime a}=L^{a}{ }_{b} d x^{b}$ where

$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

and $\gamma=\left(1-v^{2}\right)^{-1 / 2}$. What is the transformation rule for the scalar potential $\varphi$ and vector potential A between the two frames?

As seen in $\mathcal{S}^{\prime}$ there is an infinite uniform stationary distribution of charge along the $x$-axis with uniform line density $\sigma$. Determine the electric and magnetic fields $\mathbf{E}$ and B both in $\mathcal{S}^{\prime}$ and $\mathcal{S}$. Check your answer by verifying explicitly the invariance of the two quadratic Lorentz invariants.

Comment briefly on the limit $|v| \ll 1$.

4.II $. 35 \mathrm{E} \quad$

commentThe retarded scalar potential produced by a charge distribution $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$ is

$\varphi(t, \mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \int d^{3} x^{\prime} \frac{\rho\left(t-R, \mathbf{x}^{\prime}\right)}{R},$

where $R=|\mathbf{R}|$ and $\mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}$. By use of an appropriate delta function rewrite the integral as an integral over both $d^{3} x^{\prime}$ and $d t^{\prime}$ involving $\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)$.

Now specialize to a point charge $q$ moving on a path $\mathbf{x}^{\prime}=\mathbf{x}_{0}\left(t^{\prime}\right)$ so that we may set

$\rho\left(t^{\prime}, \mathbf{x}^{\prime}\right)=q \delta^{(3)}\left(\mathbf{x}^{\prime}-\mathbf{x}_{0}\left(t^{\prime}\right)\right) .$

By performing the volume integral first obtain the Liénard-Wiechert potential

$\varphi(t, \mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{1}{\left(R^{*}-\mathbf{v} \cdot \mathbf{R}^{*}\right)},$

where $\mathbf{R}^{*}$ and $\mathbf{v}$ should be specified.

Obtain the corresponding result for the magnetic potential.

1.II.36B

commentWrite down the boundary conditions that are satisfied at the interface between two viscous fluids in motion. Briefly discuss the physical meaning of these boundary conditions.

A layer of incompressible fluid of density $\rho$ and viscosity $\mu$ flows steadily down a plane inclined at an angle $\theta$ to the horizontal. The layer is of uniform thickness $h$ measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using co-ordinates parallel and perpendicular to the plane, write down the equations of motion, and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields. Show that the volume flux down the plane is $\frac{1}{3} \rho g h^{3} \sin \theta / \mu$ per unit cross-slope width.

Consider now the case where a second layer of fluid, of uniform thickness $\alpha h$, viscosity $\beta \mu$, and density $\rho$ flows steadily on top of the first layer. Determine the pressure and velocity fields in each layer. Why does the velocity profile in the bottom layer depend on $\alpha$ but not on $\beta$ ?

2.II.36B

commentA very long cylinder of radius a translates steadily at speed $V$ in a direction perpendicular to its axis and parallel to a plane boundary. The centre of the cylinder remains a distance $a+b$ above the plane, where $b \ll a$, and the motion takes place through an incompressible fluid of viscosity $\mu$.

Consider the force $F$ per unit length parallel to the plane that must be applied to the cylinder to maintain the motion. Explain why $F$ scales according to $F \propto \mu V(a / b)^{1 / 2}$.

Approximating the lower cylindrical surface by a parabola, or otherwise, determine the velocity and pressure gradient fields in the space between the cylinder and the plane. Hence, by considering the shear stress on the plane, or otherwise, calculate $F$ explicitly.

[You may use

$\int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-1} d x=\pi, \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-2} d x=\frac{1}{2} \pi \quad \text { and } \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-3} d x=\frac{3}{8} \pi$

3.II.36B

commentDefine the rate of strain tensor $e_{i j}$ in terms of the velocity components $u_{i}$.

Write down the relation between $e_{i j}$, the pressure $p$ and the stress tensor $\sigma_{i j}$ in an incompressible Newtonian fluid of viscosity $\mu$.

Prove that $2 \mu e_{i j} e_{i j}$ is the local rate of dissipation per unit volume in the fluid.

Incompressible fluid of density $\rho$ and viscosity $\mu$ occupies the semi-infinite domain $y>0$ above a rigid plane boundary $y=0$ that oscillates with velocity $(V \cos \omega t, 0,0)$, where $V$ and $\omega$ are constants. The fluid is at rest at $y=\infty$. Determine the velocity field produced by the boundary motion after any transients have decayed.

Evaluate the time-averaged rate of dissipation in the fluid, per unit area of boundary.

4.II.37B

A line force of magnitude $F$ is applied in the positive $x$-direction to an unbounded fluid, generating a thin two-dimensional jet along the positive $x$-axis. The fluid is at rest at $y=\pm \infty$ and there is negligible motion in $x<0$. Write down the pressure gradient within the boundary layer. Deduce that the function $M(x)$ defined by

$M(x)=\int_{-\infty}^{\infty} \rho u^{2}(x, y) d y$

is independent of $x$ for $x>0$. Interpret this result, and explain why $M=F$. Use scaling arguments to deduce that there is a similarity solution having stream function

$\psi=(F \nu x / \rho)^{1 / 3} f(\eta) \quad \text { where } \quad \eta=y\left(F / \rho \nu^{2} x^{2}\right)^{1 / 3}$

Hence show that $f$ satisfies

$3 f^{\prime \prime \prime}+f^{\prime 2}+f f^{\prime \prime}=0 .$