Part IB, 2003, Paper 3

Analysis II

• 3.I.1F

  Analysis II | Part IB, 2003

  Let $V$ 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.

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Fluid Dynamics

• 3.I.8C

  Fluid Dynamics | Part IB, 2003

  Show 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.

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• 3.II.18C

  Fluid Dynamics | Part IB, 2003

  State 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 $R=0$.

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Further Analysis

• 3.II.13E

  Further Analysis | Part IB, 2003

  (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.

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Geometry

• 3.I.4F

  Geometry | Part IB, 2003

  Show 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.

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• 3.II.14F

  Geometry | Part IB, 2003

  State 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.]

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Linear Mathematics

• 3.I.7G

  Linear Mathematics | Part IB, 2003

  Let $\alpha$ 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 $\alpha$.

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• 3.II.17G

  Linear Mathematics | Part IB, 2003

  Define 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 $Q^{-1} A Q=B$.

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Methods

• 3.I.2D

  Methods | Part IB, 2003

  Consider 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.

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• 3.II.12D

  Methods | Part IB, 2003

  The 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?

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Numerical Analysis

• 3.I.6B

  Numerical Analysis | Part IB, 2003

  Given $(n+1)$ 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.

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• 3.II.16B

  Numerical Analysis | Part IB, 2003

  The 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.]

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Optimization

• 3.II.15H

  Optimization | Part IB, 2003

  Explain 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.

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Quadratic Mathematics

• 3.I.9G

  Quadratic Mathematics | Part IB, 2003

  Let $f(x, y)=a x^{2}+b x y+c y^{2}$ 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 $x^{2}+3 y^{2}$.

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• 3.II.19G

  Quadratic Mathematics | Part IB, 2003

  Let $U$ 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.

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Quantum Mechanics

• 3.II.20A

  Quantum Mechanics | Part IB, 2003

  The 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?

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Special Relativity

• 3.I.10A

  Special Relativity | Part IB, 2003

  What 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.

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