Part IB, 2003, Paper 3

# Part IB, 2003, Paper 3

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3.I.1F

commentLet $V$ be the vector space of continuous real-valued functions on $[-1,1]$. Show that the function

$\|f\|=\int_{-1}^{1}|f(x)| d x$

defines a norm on $V$.

Let $f_{n}(x)=x^{n}$. Show that $\left(f_{n}\right)$ is a Cauchy sequence in $V$. Is $\left(f_{n}\right)$ convergent? Justify your answer.

3.I.8C

commentShow that the velocity field

$\mathbf{u}=\mathbf{U}+\frac{\boldsymbol{\Gamma} \times \mathbf{r}}{2 \pi r^{2}},$

where $\mathbf{U}=(U, 0,0), \mathbf{\Gamma}=(0,0, \Gamma)$ and $\mathbf{r}=(x, y, 0)$ in Cartesian coordinates $(x, y, z)$, represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.

Show that

$\oint_{C_{R}}(\mathbf{u} \cdot \mathbf{n}) \mathbf{u} d l \rightarrow \frac{1}{2} \boldsymbol{\Gamma} \times \mathbf{U} \quad \text { as } \quad R \rightarrow \infty$

where $C_{R}$ is a circle $x^{2}+y^{2}=R^{2}, z=$ const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.

3.II.18C

commentState the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.

Water of density $\rho$ is driven through a tube of length $L$ and internal radius $a$ by the pressure exerted by a spherical, water-filled balloon of radius $R(t)$ attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at $2 \gamma / R$ in excess of atmospheric pressure, where $\gamma$ is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that

$R^{3} \ddot{R}+2 R^{2} \dot{R}^{2}=-\frac{\gamma a^{2}}{2 \rho L} .$

Solve equation ( $\dagger$ ), by multiplying through by $2 R \dot{R}$ or otherwise, to obtain

$t=R_{0}^{2}\left(\frac{2 \rho L}{\gamma a^{2}}\right)^{1 / 2}\left[\frac{\pi}{4}-\frac{\theta}{2}+\frac{1}{4} \sin 2 \theta\right]$

where $\theta=\sin ^{-1}\left(R / R_{0}\right)$ and $R_{0}$ is the initial radius of the balloon. Hence find the time when $R=0$.

3.II.13E

comment(a) State Taylor's Theorem.

(b) Let $f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}$ and $g(z)=\sum_{n=0}^{\infty} b_{n}\left(z-z_{0}\right)^{n}$ be defined whenever $\left|z-z_{0}\right|<r$. Suppose that $z_{k} \rightarrow z_{0}$ as $k \rightarrow \infty$, that no $z_{k}$ equals $z_{0}$ and that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $a_{n}=b_{n}$ for every $n \geqslant 0$.

(c) Let $D$ be a domain, let $z_{0} \in D$ and let $\left(z_{k}\right)$ be a sequence of points in $D$ that converges to $z_{0}$, but such that no $z_{k}$ equals $z_{0}$. Let $f: D \rightarrow \mathbb{C}$ and $g: D \rightarrow \mathbb{C}$ be analytic functions such that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $f(z)=g(z)$ for every $z \in D$.

(d) Let $D$ be the domain $\mathbb{C} \backslash\{0\}$. Give an example of an analytic function $f: D \rightarrow \mathbb{C}$ such that $f\left(n^{-1}\right)=0$ for every positive integer $n$ but $f$ is not identically 0 .

(e) Show that any function with the property described in (d) must have an essential singularity at the origin.

3.I.4F

commentShow that any isometry of Euclidean 3 -space which fixes the origin can be written as a composite of at most three reflections in planes through the origin, and give (with justification) an example of an isometry for which three reflections are necessary.

3.II.14F

commentState and prove the Gauss-Bonnet formula for the area of a spherical triangle. Deduce a formula for the area of a spherical $n$-gon with angles $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$. For what range of values of $\alpha$ does there exist a (convex) regular spherical $n$-gon with angle $\alpha$ ?

Let $\Delta$ be a spherical triangle with angles $\pi / p, \pi / q$ and $\pi / r$ where $p, q, r$ are integers, and let $G$ be the group of isometries of the sphere generated by reflections in the three sides of $\Delta$. List the possible values of $(p, q, r)$, and in each case calculate the order of the corresponding group $G$. If $(p, q, r)=(2,3,5)$, show how to construct a regular dodecahedron whose group of symmetries is $G$.

[You may assume that the images of $\Delta$ under the elements of $G$ form a tessellation of the sphere.]

3.I.7G

commentLet $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ and let $\beta$ be another endomorphism of $U$ that commutes with $\alpha$. If $\lambda$ is an eigenvalue of $\alpha$, show that $\beta$ maps the kernel of $\alpha-\lambda \iota$ into itself, where $\iota$ is the identity map. Suppose now that $\alpha$ is diagonalizable with $n$ distinct real eigenvalues where $n=\operatorname{dim} U$. Prove that if there exists an endomorphism $\beta$ of $U$ such that $\alpha=\beta^{2}$, then $\lambda \geqslant 0$ for all eigenvalues $\lambda$ of $\alpha$.

3.II.17G

commentDefine the determinant $\operatorname{det}(A)$ of an $n \times n$ complex matrix A. Let $A_{1}, \ldots, A_{n}$ be the columns of $A$, let $\sigma$ be a permutation of $\{1, \ldots, n\}$ and let $A^{\sigma}$ be the matrix whose columns are $A_{\sigma(1)}, \ldots, A_{\sigma(n)}$. Prove from your definition of determinant that $\operatorname{det}\left(A^{\sigma}\right)=\epsilon(\sigma) \operatorname{det}(A)$, where $\epsilon(\sigma)$ is the sign of the permutation $\sigma$. Prove also that $\operatorname{det}(A)=\operatorname{det}\left(A^{t}\right) .$

Define the adjugate matrix $\operatorname{adj}(A)$ and prove from your $\operatorname{definitions}$ that $A \operatorname{adj}(A)=$ $\operatorname{adj}(A) A=\operatorname{det}(A) I$, where $I$ is the identity matrix. Hence or otherwise, prove that if $\operatorname{det}(A) \neq 0$, then $A$ is invertible.

Let $C$ and $D$ be real $n \times n$ matrices such that the complex matrix $C+i D$ is invertible. By considering $\operatorname{det}(C+\lambda D)$ as a function of $\lambda$ or otherwise, prove that there exists a real number $\lambda$ such that $C+\lambda D$ is invertible. [You may assume that if a matrix $A$ is invertible, then $\operatorname{det}(A) \neq 0$.]

Deduce that if two real matrices $A$ and $B$ are such that there exists an invertible complex matrix $P$ with $P^{-1} A P=B$, then there exists an invertible real matrix $Q$ such that $Q^{-1} A Q=B$.

3.I.2D

commentConsider the path between two arbitrary points on a cone of interior angle $2 \alpha$. Show that the arc-length of the path $r(\theta)$ is given by

$\int\left(r^{2}+r^{\prime 2} \operatorname{cosec}^{2} \alpha\right)^{1 / 2} d \theta$

where $r^{\prime}=\frac{d r}{d \theta}$. By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.

3.II.12D

commentThe transverse displacement $y(x, t)$ of a stretched string clamped at its ends $x=0, l$ satisfies the equation

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}-2 k \frac{\partial y}{\partial t}, \quad y(x, 0)=0, \frac{\partial y}{\partial t}(x, 0)=\delta(x-a)$

where $c>0$ is the wave velocity, and $k>0$ is the damping coefficient. The initial conditions correspond to a sharp blow at $x=a$ at time $t=0$.

(a) Show that the subsequent motion of the string is given by

$y(x, t)=\frac{1}{\sqrt{\alpha_{n}^{2}-k^{2}}} \sum_{n} 2 e^{-k t} \sin \frac{\alpha_{n} a}{c} \sin \frac{\alpha_{n} x}{c} \sin /\left(\sqrt{\alpha_{n}^{2}-k^{2}} t\right)$

where $\alpha_{n}=\pi c n / l$.

(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?

3.I.6B

commentGiven $(n+1)$ distinct points $x_{0}, x_{1}, \ldots, x_{n}$, let

$\ell_{i}(x)=\prod_{\substack{k=0 \\ k \neq i}}^{n} \frac{x-x_{k}}{x_{i}-x_{k}}$

be the fundamental Lagrange polynomials of degree $n$, let

$\omega(x)=\prod_{i=0}^{n}\left(x-x_{i}\right)$

and let $p$ be any polynomial of degree $\leq n$.

(a) Prove that $\sum_{i=0}^{n} p\left(x_{i}\right) \ell_{i}(x) \equiv p(x)$.

(b) Hence or otherwise derive the formula

$\frac{p(x)}{\omega(x)}=\sum_{i=0}^{n} \frac{A_{i}}{x-x_{i}}, \quad A_{i}=\frac{p\left(x_{i}\right)}{\omega^{\prime}\left(x_{i}\right)}$

which is the decomposition of $p(x) / \omega(x)$ into partial fractions.

3.II.16B

commentThe functions $H_{0}, H_{1}, \ldots$ are generated by the Rodrigues formula:

$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$

(a) Show that $H_{n}$ is a polynomial of degree $n$, and that the $H_{n}$ are orthogonal with respect to the scalar product

$(f, g)=\int_{-\infty}^{\infty} f(x) g(x) e^{-x^{2}} d x$

(b) By induction or otherwise, prove that the $H_{n}$ satisfy the three-term recurrence relation

$H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) .$

[Hint: you may need to prove the equality $H_{n}^{\prime}(x)=2 n H_{n-1}(x)$ as well.]

3.II.15H

commentExplain what is meant by a transportation problem where the total demand equals the total supply. Write the Lagrangian and describe an algorithm for solving such a problem. Starting from the north-west initial assignment, solve the problem with three sources and three destinations described by the table

\begin{tabular}{|rrr|r|} \hline 5 & 9 & 1 & 36 \ 3 & 10 & 6 & 84 \ 7 & 2 & 5 & 40 \ \hline 14 & 68 & 78 & \ \hline \end{tabular}

where the figures in the $3 \times 3$ box denote the transportation costs (per unit), the right-hand column denotes supplies, and the bottom row demands.

3.I.9G

commentLet $f(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Explain what is meant by the discriminant $d$ of $f$. State a necessary and sufficient condition for some form of discriminant $d$ to represent an odd prime number $p$. Using this result or otherwise, find the primes $p$ which can be represented by the form $x^{2}+3 y^{2}$.

3.II.19G

commentLet $U$ be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map $\tau: U \rightarrow U$ is said to be an orthogonal projection if $\tau$ is self-adjoint and $\tau^{2}=\tau$.

(a) Prove that for every orthogonal projection $\tau$ there is an orthogonal decomposition

$U=\operatorname{ker}(\tau) \oplus \operatorname{im}(\tau)$

(b) Let $\phi: U \rightarrow U$ be a linear map. Show that if $\phi^{2}=\phi$ and $\phi \phi^{*}=\phi^{*} \phi$, where $\phi^{*}$ is the adjoint of $\phi$, then $\phi$ is an orthogonal projection. [You may find it useful to prove first that if $\phi \phi^{*}=\phi^{*} \phi$, then $\phi$ and $\phi^{*}$ have the same kernel.]

(c) Show that given a subspace $W$ of $U$ there exists a unique orthogonal projection $\tau$ such that $\operatorname{im}(\tau)=W$. If $W_{1}$ and $W_{2}$ are two subspaces with corresponding orthogonal projections $\tau_{1}$ and $\tau_{2}$, show that $\tau_{2} \circ \tau_{1}=0$ if and only if $W_{1}$ is orthogonal to $W_{2}$.

(d) Let $\phi: U \rightarrow U$ be a linear map satisfying $\phi^{2}=\phi$. Prove that one can define a positive definite inner product on $U$ such that $\phi$ becomes an orthogonal projection.

3.II.20A

commentThe radial wavefunction for the hydrogen atom satisfies the equation

$\frac{-\hbar^{2}}{2 m} \frac{1}{r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r} R(r)\right)+\frac{\hbar^{2}}{2 m r^{2}} \ell(\ell+1) R(r)-\frac{e^{2}}{4 \pi \epsilon_{0} r} R(r)=E R(r) .$

Explain the origin of each term in this equation.

The wavefunctions for the ground state and first radially excited state, both with $\ell=0$, can be written as

$\begin{aligned} &R_{1}(r)=N_{1} \exp (-\alpha r) \\ &R_{2}(r)=N_{2}(r+b) \exp (-\beta r) \end{aligned}$

respectively, where $N_{1}$ and $N_{2}$ are normalization constants. Determine $\alpha, \beta, b$ and the corresponding energy eigenvalues $E_{1}$ and $E_{2}$.

A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon?

3.I.10A

commentWhat are the momentum and energy of a photon of wavelength $\lambda$ ?

A photon of wavelength $\lambda$ is incident on an electron. After the collision, the photon has wavelength $\lambda^{\prime}$. Show that

$\lambda^{\prime}-\lambda=\frac{h}{m c}(1-\cos \theta),$

where $\theta$ is the scattering angle and $m$ is the electron rest mass.