Part IB, 2008

Analysis II

• $3 . \mathrm{I} . 3 \mathrm{~F} \quad$

  Analysis II | Part IB, 2008

  Explain what it means for a function $f(x, y)$ of two variables to be differentiable at a point $\left(x_{0}, y_{0}\right)$. If $f$ is differentiable at $\left(x_{0}, y_{0}\right)$, show that for any $\alpha$ the function $g_{\alpha}$ defined by

  $$g_{\alpha}(t)=f\left(x_{0}+t \cos \alpha, y_{0}+t \sin \alpha\right)$$

  is differentiable at $t=0$, and find its derivative in terms of the partial derivatives of $f$ at $\left(x_{0}, y_{0}\right)$.

  Consider the function $f$ defined by

  

  Is $f$ differentiable at $(0,0)$ ? Justify your answer.

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• 1.II.11F

  Analysis II | Part IB, 2008

  State and prove the Contraction Mapping Theorem.

  Let $(X, d)$ be a nonempty complete metric space and $f: X \rightarrow X$ a mapping such that, for some $k>0$, the $k$ th iterate $f^{k}$ of $f$ (that is, $f$ composed with itself $k$ times) is a contraction mapping. Show that $f$ has a unique fixed point.

  Now let $X$ be the space of all continuous real-valued functions on $[0,1]$, equipped with the uniform norm $\|h\|_{\infty}=\sup \{|h(t)|: t \in[0,1]\}$, and let $\phi: \mathbb{R} \times[0,1] \rightarrow \mathbb{R}$ be a continuous function satisfying the Lipschitz condition

  $$|\phi(x, t)-\phi(y, t)| \leqslant M|x-y|$$

  for all $t \in[0,1]$ and all $x, y \in \mathbb{R}$, where $M$ is a constant. Let $F: X \rightarrow X$ be defined by

  $$F(h)(t)=g(t)+\int_{0}^{t} \phi(h(s), s) d s$$

  where $g$ is a fixed continuous function on $[0,1]$. Show by induction on $n$ that

  $$\left|F^{n}(h)(t)-F^{n}(k)(t)\right| \leqslant \frac{M^{n} t^{n}}{n !}\|h-k\|_{\infty}$$

  for all $h, k \in X$ and all $t \in[0,1]$. Deduce that the integral equation

  $$f(t)=g(t)+\int_{0}^{t} \phi(f(s), s) d s$$

  has a unique continuous solution $f$ on $[0,1]$.

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• 2.I.3F

  Analysis II | Part IB, 2008

  Explain what is meant by the statement that a sequence $\left(f_{n}\right)$ of functions defined on an interval $[a, b]$ converges uniformly to a function $f$. If $\left(f_{n}\right)$ converges uniformly to $f$, and each $f_{n}$ is continuous on $[a, b]$, prove that $f$ is continuous on $[a, b]$.

  Now suppose additionally that $\left(x_{n}\right)$ is a sequence of points of $[a, b]$ converging to a limit $x$. Prove that $f_{n}\left(x_{n}\right) \rightarrow f(x)$.

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• 2.II.13F

  Analysis II | Part IB, 2008

  Let $\left(u_{n}(x): n=0,1,2, \ldots\right)$ be a sequence of real-valued functions defined on a subset $E$ of $\mathbb{R}$. Suppose that for all $n$ and all $x \in E$ we have $\left|u_{n}(x)\right| \leqslant M_{n}$, where $\sum_{n=0}^{\infty} M_{n}$ converges. Prove that $\sum_{n=0}^{\infty} u_{n}(x)$ converges uniformly on $E$.

  Now let $E=\mathbb{R} \backslash \mathbb{Z}$, and consider the series $\sum_{n=0}^{\infty} u_{n}(x)$, where $u_{0}(x)=1 / x^{2}$ and

  $$u_{n}(x)=1 /(x-n)^{2}+1 /(x+n)^{2}$$

  for $n>0$. Show that the series converges uniformly on $E_{R}=\{x \in E:|x|<R\}$ for any real number $R$. Deduce that $f(x)=\sum_{n=0}^{\infty} u_{n}(x)$ is a continuous function on $E$. Does the series converge uniformly on $E$ ? Justify your answer.

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• 3.II.13F

  Analysis II | Part IB, 2008

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function, and $\left(x_{0}, y_{0}\right)$ a point of $\mathbb{R}^{2}$. Prove that if the partial derivatives of $f$ exist in some open disc around $\left(x_{0}, y_{0}\right)$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.

  Now let $X$ denote the vector space of all $(n \times n)$ real matrices, and let $f: X \rightarrow \mathbb{R}$ be the function assigning to each matrix its determinant. Show that $f$ is differentiable at the identity matrix $I$, and that $\left.D f\right|_{I}$ is the linear map $H \mapsto \operatorname{tr} H$. Deduce that $f$ is differentiable at any invertible matrix $A$, and that $\left.D f\right|_{A}$ is the linear map $H \mapsto \operatorname{det} A \operatorname{tr}\left(A^{-1} H\right) .$

  Show also that if $K$ is a matrix with $\|K\|<1$, then $(I+K)$ is invertible. Deduce that $f$ is twice differentiable at $I$, and find $\left.D^{2} f\right|_{I}$ as a bilinear map $X \times X \rightarrow \mathbb{R}$.

  [You may assume that the norm $\|-\|$ on $X$ is complete, and that it satisfies the inequality $\|A B\| \leqslant\|A\| \cdot\|B\|$ for any two matrices $A$ and $B .]$

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• 4.I.3F

  Analysis II | Part IB, 2008

  Let $X$ be the vector space of all continuous real-valued functions on the unit interval $[0,1]$. Show that the functions

  $$\|f\|_{1}=\int_{0}^{1}|f(t)| d t \quad \text { and } \quad\|f\|_{\infty}=\sup \{|f(t)|: 0 \leqslant t \leqslant 1\}$$

  both define norms on $X$.

  Consider the sequence $\left(f_{n}\right)$ defined by $f_{n}(t)=n t^{n}(1-t)$. Does $\left(f_{n}\right)$ converge in the norm $\|-\|_{1}$ ? Does it converge in the norm $\|-\|_{\infty}$ ? Justify your answers.

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• 4.II.13F

  Analysis II | Part IB, 2008

  Explain what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms are Lipschitz equivalent, then one is complete if and only if the other is.

  Let $\|-\|$ be an arbitrary norm on the finite-dimensional space $\mathbb{R}^{n}$, and let $\|-\|_{2}$ denote the standard (Euclidean) norm. Show that for every $\mathbf{x} \in \mathbb{R}^{n}$ with $\|\mathbf{x}\|_{2}=1$, we have

  $$\|\mathbf{x}\| \leqslant\left\|\mathbf{e}_{1}\right\|+\left\|\mathbf{e}_{2}\right\|+\cdots+\left\|\mathbf{e}_{n}\right\|$$

  where $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right)$ is the standard basis for $\mathbb{R}^{n}$, and deduce that the function $\|-\|$ is continuous with respect to $\|-\|_{2}$. Hence show that there exists a constant $m>0$ such that $\|\mathbf{x}\| \geqslant m$ for all $\mathbf{x}$ with $\|\mathbf{x}\|_{2}=1$, and deduce that $\|-\|$ and $\|-\|_{2}$ are Lipschitz equivalent.

  [You may assume the Bolzano-Weierstrass Theorem.]

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Complex Analysis

• 3.II.14E

  Complex Analysis | Part IB, 2008

  State and prove Rouché's theorem, and use it to count the number of zeros of $3 z^{9}+8 z^{6}+z^{5}+2 z^{3}+1$ inside the annulus $\{z: 1<|z|<2\}$.

  Let $\left(p_{n}\right)_{n=1}^{\infty}$ be a sequence of polynomials of degree at most $d$ with the property that $p_{n}(z)$ converges uniformly on compact subsets of $\mathbb{C}$ as $n \rightarrow \infty$. Prove that there is a polynomial $p$ of degree at most $d$ such that $p_{n} \rightarrow p$ uniformly on compact subsets of $\mathbb{C}$. [If you use any results about uniform convergence of analytic functions, you should prove them.]

  Suppose that $p$ has $d$ distinct roots $z_{1}, \ldots, z_{d}$. Using Rouché's theorem, or otherwise, show that for each $i$ there is a sequence $\left(z_{i, n}\right)_{n=1}^{\infty}$ such that $p_{n}\left(z_{i, n}\right)=0$ and $z_{i, n} \rightarrow z_{i}$ as $n \rightarrow \infty$.

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• 4.I.4E

  Complex Analysis | Part IB, 2008

  Suppose that $f$ and $g$ are two functions which are analytic on the whole complex plane $\mathbb{C}$. Suppose that there is a sequence of distinct points $z_{1}, z_{2}, \ldots$ with $\left|z_{i}\right| \leqslant 1$ such that $f\left(z_{i}\right)=g\left(z_{i}\right)$. Show that $f(z)=g(z)$ for all $z \in \mathbb{C}$. [You may assume any results on Taylor expansions you need, provided they are clearly stated.]

  What happens if the assumption that $\left|z_{i}\right| \leqslant 1$ is dropped?

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Complex Analysis or Complex Methods

• 1.I.3C

  Complex Analysis or Complex Methods | Part IB, 2008

  Given that $f(z)$ is an analytic function, show that the mapping $w=f(z)$

  (a) preserves angles between smooth curves intersecting at $z$ if $f^{\prime}(z) \neq 0$;

  (b) has Jacobian given by $\left|f^{\prime}(z)\right|^{2}$.

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• 1.II.13C

  Complex Analysis or Complex Methods | Part IB, 2008

  By a suitable choice of contour show the following:

  (a)

  $$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$$

  where $n>1$,

  (b)

  $$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x=\frac{\pi^{2}}{2 \sqrt{2}}$$

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• 2.II.14C

  Complex Analysis or Complex Methods | Part IB, 2008

  Let $f(z)=1 /\left(e^{z}-1\right)$. Find the first three terms in the Laurent expansion for $f(z)$ valid for $0<|z|<2 \pi$.

  Now let $n$ be a positive integer, and define

  $$\begin{aligned} &f_{1}(z)=\frac{1}{z}+\sum_{r=1}^{n} \frac{2 z}{z^{2}+4 \pi^{2} r^{2}} \\ &f_{2}(z)=f(z)-f_{1}(z) \end{aligned}$$

  Show that the singularities of $f_{2}$ in $\{z:|z|<2(n+1) \pi\}$ are all removable. By expanding $f_{1}$ as a Laurent series valid for $|z|>2 n \pi$, and $f_{2}$ as a Taylor series valid for $|z|<2(n+1) \pi$, find the coefficients of $z^{j}$ for $-1 \leq j \leq 1$ in the Laurent series for $f$ valid for $2 n \pi<|z|<2(n+1) \pi$.

  By estimating an appropriate integral around the contour $|z|=(2 n+1) \pi$, show that

  $$\sum_{r=1}^{\infty} \frac{1}{r^{2}}=\frac{\pi^{2}}{6}$$

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Complex Methods

• 3.I.5C

  Complex Methods | Part IB, 2008

  Using the contour integration formula for the inversion of Laplace transforms find the inverse Laplace transforms of the following functions: (a) $\frac{s}{s^{2}+a^{2}} \quad(a$ real and non-zero $)$, (b) $\frac{1}{\sqrt{s}}$.

  [You may use the fact that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\pi / b}$.]

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• 4.II.15C

  Complex Methods | Part IB, 2008

  Let $H$ be the domain $\mathbb{C}-\{x+i y: x \leq 0, y=0\}$ (i.e., $\mathbb{C}$ cut along the negative $x$-axis). Show, by a suitable choice of branch, that the mapping

  $$z \mapsto w=-i \log z$$

  maps $H$ onto the strip $S=\{z=x+i y,-\pi<x<\pi\}$.

  How would a different choice of branch change the result?

  Let $G$ be the domain $\{z \in \mathbb{C}:|z|<1,|z+i|>\sqrt{2}\}$. Find an analytic transformation that maps $G$ to $S$, where $S$ is the strip defined above.

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Electromagnetism

• 1.II.16B

  Electromagnetism | Part IB, 2008

  Suppose that the current density $\mathbf{J}(\mathbf{r})$ is constant in time but the charge density $\rho(\mathbf{r}, t)$ is not.

  (i) Show that $\rho$ is a linear function of time:

  $$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0)+\dot{\rho}(\mathbf{r}, 0) t$$

  where $\dot{\rho}(\mathbf{r}, 0)$ is the time derivative of $\rho$ at time $t=0$.

  (ii) The magnetic induction due to a current density $\mathbf{J}(\mathbf{r})$ can be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d V^{\prime}$$

  Show that this can also be written as

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \nabla \times \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  (iii) Assuming that $\mathbf{J}$ vanishes at infinity, show that the curl of the magnetic field in (1) can be written as

  $$\nabla \times \mathbf{B}(\mathbf{r})=\mu_{0} \mathbf{J}(\mathbf{r})+\frac{\mu_{0}}{4 \pi} \nabla \int \frac{\nabla^{\prime} \cdot \mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d V^{\prime}$$

  [You may find useful the identities $\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A}$ and also $\left.\nabla^{2}\left(1 /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)=-4 \pi \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) .\right]$

  (iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that $\mathbf{B}$ itself obeys Ampère's law with Maxwell's displacement current term, i.e. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \epsilon_{0} \partial \mathbf{E} / \partial t$.

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• 2.I.6B

  Electromagnetism | Part IB, 2008

  Given the electric potential of a dipole

  $$\phi(r, \theta)=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}},$$

  where $p$ is the magnitude of the dipole moment, calculate the corresponding electric field and show that it can be written as

  $$\mathbf{E}(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[3\left(\mathbf{p} \cdot \hat{\mathbf{e}}_{r}\right) \hat{\mathbf{e}}_{r}-\mathbf{p}\right]$$

  where $\hat{\mathbf{e}}_{r}$ is the unit vector in the radial direction.

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• 2.II.17B

  Electromagnetism | Part IB, 2008

  Two perfectly conducting rails are placed on the $x y$-plane, one coincident with the $x$-axis, starting at $(0,0)$, the other parallel to the first rail a distance $\ell$ apart, starting at $(0, \ell)$. A resistor $R$ is connected across the rails between $(0,0)$ and $(0, \ell)$, and a uniform magnetic field $\mathbf{B}=B \hat{\mathbf{e}}_{z}$, where $\hat{\mathbf{e}}_{z}$ is the unit vector along the $z$-axis and $B>0$, fills the entire region of space. A metal bar of negligible resistance and mass $m$ slides without friction on the two rails, lying perpendicular to both of them in such a way that it closes the circuit formed by the rails and the resistor. The bar moves with speed $v$ to the right such that the area of the loop becomes larger with time.

  (i) Calculate the current in the resistor and indicate its direction of flow in a diagram of the system.

  (ii) Show that the magnetic force on the bar is

  $$\mathbf{F}=-\frac{B^{2} \ell^{2} v}{R} \hat{\mathbf{e}}_{x}$$

  (iii) Assume that the bar starts moving with initial speed $v_{0}$ at time $t=0$, and is then left to slide freely. Using your result from part (ii) and Newton's laws show that its velocity at the time $t$ is

  $$v(t)=v_{0} e^{-\left(B^{2} \ell^{2} / m R\right) t} .$$

  (iv) By calculating the total energy delivered to the resistor, verify that energy is conserved.

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• 3.II.17B

  Electromagnetism | Part IB, 2008

  (i) From Maxwell's equations in vacuum,

  $$\begin{array}{ll} \nabla \cdot \mathbf{E}=0 & \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0 & \nabla \times \mathbf{B}=\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t} \end{array}$$

  obtain the wave equation for the electric field E. [You may find the following identity useful: $\left.\nabla \times(\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^{2} \mathbf{A} .\right]$

  (ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are

  $$\mathbf{E}(z, t)=\mathbf{E}_{0} \mathrm{e}^{i(k z-\omega t)} \text { and } \mathbf{B}(z, t)=\mathbf{B}_{0} \mathrm{e}^{i(k z-\omega t)}$$

  show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).

  (iii) Use Faraday's law for these fields to show that

  $$\mathbf{B}_{0}=\frac{k}{\omega}\left(\hat{\mathbf{e}}_{z} \times \mathbf{E}_{0}\right)$$

  (iv) Explain with symmetry arguments how these results generalise to

  $$\mathbf{E}(\mathbf{r}, t)=E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} \hat{\mathbf{n}} \quad \text { and } \quad \mathbf{B}(\mathbf{r}, t)=\frac{1}{c} E_{0} \mathrm{e}^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}(\hat{\mathbf{k}} \times \hat{\mathbf{n}})$$

  where $\hat{\mathbf{n}}$ is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and $\hat{\mathbf{k}}$ is the unit vector in the direction of propagation of the wave.

  (v) Using Maxwell's equations in vacuum prove that:

  $$\oint_{\mathcal{A}}\left(1 / \mu_{0}\right)(\mathbf{E} \times \mathbf{B}) \cdot d \mathcal{A}=-\frac{\partial}{\partial t} \int_{\mathcal{V}}\left(\frac{\epsilon_{0} E^{2}}{2}+\frac{B^{2}}{2 \mu_{0}}\right) d V$$

  where $\mathcal{V}$ is the closed volume and $\mathcal{A}$ is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.

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• 4.I.7B

  Electromagnetism | Part IB, 2008

  The energy stored in a static electric field $\mathbf{E}$ is

  $$U=\frac{1}{2} \int \rho \phi d V,$$

  where $\phi$ is the associated electric potential, $\mathbf{E}=-\nabla \phi$, and $\rho$ is the volume charge density.

  (i) Assuming that the energy is calculated over all space and that $\mathbf{E}$ vanishes at infinity, show that the energy can be written as

  $$U=\frac{\epsilon_{0}}{2} \int|\mathbf{E}|^{2} d V$$

  (ii) Find the electric field produced by a spherical shell with total charge $Q$ and radius $R$, assuming it to vanish inside the shell. Find the energy stored in the electric field.

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Fluid Dynamics

• 1.I.5B

  Fluid Dynamics | Part IB, 2008

  Verify that the two-dimensional flow given in Cartesian coordinates by

  $$\mathbf{u}=\left(\mathrm{e}^{y} \sinh x,-\mathrm{e}^{y} \cosh x\right)$$

  satisfies $\nabla \cdot \mathbf{u}=0$. Find the stream function $\psi(x, y)$. Sketch the streamlines.

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• 1.II.17B

  Fluid Dynamics | Part IB, 2008

  Two incompressible fluids flow in infinite horizontal streams, the plane of contact being $z=0$, with $z$ positive upwards. The flow is given by

  $$\mathbf{U}(\mathbf{r})= \begin{cases}U_{2} \hat{\mathbf{e}}_{x}, & z>0 \\ U_{1} \hat{\mathbf{e}}_{x}, & z<0\end{cases}$$

  where $\hat{\mathbf{e}}_{x}$ is the unit vector in the positive $x$ direction. The upper fluid has density $\rho_{2}$ and pressure $p_{0}-g \rho_{2} z$, the lower has density $\rho_{1}$ and pressure $p_{0}-g \rho_{1} z$, where $p_{0}$ is a constant and $g$ is the acceleration due to gravity.

  (i) Consider a perturbation to the flat surface $z=0$ of the form

  $$z \equiv \zeta(x, y, t)=\zeta_{0} e^{i(k x+\ell y)+s t} .$$

  State the kinematic boundary conditions on the velocity potentials $\phi_{i}$ that hold on the interface in the two domains, and show by linearising in $\zeta$ that they reduce to

  $$\frac{\partial \phi_{i}}{\partial z}=\frac{\partial \zeta}{\partial t}+U_{i} \frac{\partial \zeta}{\partial x} \quad(z=0, i=1,2) .$$

  (ii) State the dynamic boundary condition on the perturbed interface, and show by linearising in $\zeta$ that it reduces to

  $$\rho_{1}\left(U_{1} \frac{\partial \phi_{1}}{\partial x}+\frac{\partial \phi_{1}}{\partial t}+g \zeta\right)=\rho_{2}\left(U_{2} \frac{\partial \phi_{2}}{\partial x}+\frac{\partial \phi_{2}}{\partial t}+g \zeta\right) \quad(z=0)$$

  (iii) Use the velocity potentials

  $$\phi_{1}=U_{1} x+A_{1} e^{q z} e^{i(k x+\ell y)+s t}, \quad \phi_{2}=U_{2} x+A_{2} e^{-q z} e^{i(k x+\ell y)+s t},$$

  where $q=\sqrt{k^{2}+\ell^{2}}$, and the conditions in (i) and (ii) to perform a stability analysis. Show that the relation between $s, k$ and $\ell$ is

  $$s=-i k \frac{\rho_{1} U_{1}+\rho_{2} U_{2}}{\rho_{1}+\rho_{2}} \pm\left[\frac{k^{2} \rho_{1} \rho_{2}\left(U_{1}-U_{2}\right)^{2}}{\left(\rho_{1}+\rho_{2}\right)^{2}}-\frac{q g\left(\rho_{1}-\rho_{2}\right)}{\rho_{1}+\rho_{2}}\right]^{1 / 2} .$$

  Find the criterion for instability.

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• 2.I.8B

  Fluid Dynamics | Part IB, 2008

  (i) Show that for a two-dimensional incompressible flow $(u(x, y), v(x, y), 0)$, the vorticity is given by $\boldsymbol{\omega} \equiv \omega_{z} \hat{\mathbf{e}}_{z}=\left(0,0,-\nabla^{2} \psi\right)$ where $\psi$ is the stream function.

  (ii) Express the $z$-component of the vorticity equation

  $$\frac{\partial \boldsymbol{\omega}}{\partial t}+(\mathbf{u} \cdot \nabla) \boldsymbol{\omega}=(\boldsymbol{\omega} \cdot \nabla) \mathbf{u}$$

  in terms of the stream function $\psi$.

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• 3.II.18B

  Fluid Dynamics | Part IB, 2008

  An ideal liquid contained within a closed circular cylinder of radius $a$ rotates about the axis of the cylinder (assume this axis to be in the vertical $z$-direction).

  (i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity $\mathbf{v}=\boldsymbol{\Omega} \times \mathbf{r}$, where $\boldsymbol{\Omega}=\Omega \hat{\mathbf{e}}_{z}$ is the angular velocity, with $\hat{\mathbf{e}}_{z}$ the unit vector in the $z$-direction, which depends only on time, and $\mathbf{r}$ is the position vector measured from a point on the axis of rotation.

  (ii) Calculate the angular momentum $\mathbf{M}=\rho \int(\mathbf{r} \times \mathbf{v}) d V$ per unit length of the cylinder.

  (iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass $\mathbf{f}=(\alpha x+\beta y, \gamma x+\delta y, 0)$. By taking the curl of the Euler equation, prove that

  $$\frac{d \Omega}{d t}=\frac{1}{2}(\gamma-\beta)$$

  (iv) Find the pressure.

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• 4.II.18B

  Fluid Dynamics | Part IB, 2008

  (i) Starting from Euler's equation for an incompressible fluid show that for potential flow with $\mathbf{u}=\nabla \phi$,

  $$\frac{\partial \phi}{\partial t}+\frac{1}{2} u^{2}+\chi=f(t)$$

  where $u=|\mathbf{u}|, \chi=p / \rho+V$, the body force per unit mass is $-\nabla V$ and $f(t)$ is an arbitrary function of time.

  (ii) Hence show that, for the steady flow of a liquid of density $\rho$ through a pipe of varying cross-section that is subject to a pressure difference $\Delta p=p_{1}-p_{2}$ between its two ends, the mass flow through the pipe per unit time is given by

  $$m \equiv \frac{d M}{d t}=S_{1} S_{2} \sqrt{\frac{2 \rho \Delta p}{S_{1}^{2}-S_{2}^{2}}}$$

  where $S_{1}$ and $S_{2}$ are the cross-sectional areas of the two ends.

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Geometry

• 1.I.2G

  Geometry | Part IB, 2008

  Show that any element of $S O(3, \mathbb{R})$ is a rotation, and that it can be written as the product of two reflections.

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• 2.II.12G

  Geometry | Part IB, 2008

  Show that the area of a spherical triangle with angles $\alpha, \beta, \gamma$ is $\alpha+\beta+\gamma-\pi$. Hence derive the formula for the area of a convex spherical $n$-gon.

  Deduce Euler's formula $F-E+V=2$ for a decomposition of a sphere into $F$ convex polygons with a total of $E$ edges and $V$ vertices.

  A sphere is decomposed into convex polygons, comprising $m$ quadrilaterals, $n$ pentagons and $p$ hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair $(m, n)$, and that at least 3 of these do occur.

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• 3.I.2G

  Geometry | Part IB, 2008

  A smooth surface in $\mathbb{R}^{3}$ has parametrization

  $$\sigma(u, v)=\left(u-\frac{u^{3}}{3}+u v^{2}, v-\frac{v^{3}}{3}+u^{2} v, u^{2}-v^{2}\right) .$$

  Show that a unit normal vector at the point $\sigma(u, v)$ is

  $$\left(\frac{-2 u}{1+u^{2}+v^{2}}, \frac{2 v}{1+u^{2}+v^{2}}, \frac{1-u^{2}-v^{2}}{1+u^{2}+v^{2}}\right)$$

  and that the curvature is $\frac{-4}{\left(1+u^{2}+v^{2}\right)^{4}}$.

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• 3.II.12G

  Geometry | Part IB, 2008

  Let $D$ be the unit disc model of the hyperbolic plane, with metric

  $$\frac{4|d \zeta|^{2}}{\left(1-|\zeta|^{2}\right)^{2}}$$

  (i) Show that the group of Möbius transformations mapping $D$ to itself is the group of transformations

  $$\zeta \mapsto \omega \frac{\zeta-\lambda}{\bar{\lambda} \zeta-1},$$

  where $|\lambda|<1$ and $|\omega|=1$.

  (ii) Assuming that the transformations in (i) are isometries of $D$, show that any hyperbolic circle in $D$ is a Euclidean circle.

  (iii) Let $P$ and $Q$ be points on the unit circle with $\angle P O Q=2 \alpha$. Show that the hyperbolic distance from $O$ to the hyperbolic line $P Q$ is given by

  $$2 \tanh ^{-1}\left(\frac{1-\sin \alpha}{\cos \alpha}\right)$$

  (iv) Deduce that if $a>2 \tanh ^{-1}(2-\sqrt{3})$ then no hyperbolic open disc of radius $a$ is contained in a hyperbolic triangle.

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• 4.II.12G

  Geometry | Part IB, 2008

  Let $\gamma:[a, b] \rightarrow S$ be a curve on a smoothly embedded surface $S \subset \mathbf{R}^{3}$. Define the energy of $\gamma$. Show that if $\gamma$ is a stationary point for the energy for proper variations of $\gamma$, then $\gamma$ satisfies the geodesic equations

  $$\begin{aligned} \frac{d}{d t}\left(E \dot{\gamma}_{1}+F \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{u} \dot{\gamma}_{1}^{2}+2 F_{u} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{u} \dot{\gamma}_{2}^{2}\right) \\ \frac{d}{d t}\left(F \dot{\gamma}_{1}+G \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{v} \dot{\gamma}_{1}^{2}+2 F_{v} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{v} \dot{\gamma}_{2}^{2}\right) \end{aligned}$$

  where $\gamma=\left(\gamma_{1}, \gamma_{2}\right)$ in terms of a smooth parametrization $(u, v)$ for $S$, with first fundamental form $E d u^{2}+2 F d u d v+G d v^{2}$.

  Now suppose that for every $c, d$ the curves $u=c, v=d$ are geodesics.

  (i) Show that $(F / \sqrt{G})_{v}=(\sqrt{G})_{u}$ and $(F / \sqrt{E})_{u}=(\sqrt{E})_{v}$.

  (ii) Suppose moreover that the angle between the curves $u=c, v=d$ is independent of $c$ and $d$. Show that $E_{v}=0=G_{u}$.

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Groups, Rings and Modules

• $3 . \mathrm{II} . 11 \mathrm{G}$

  Groups, Rings and Modules | Part IB, 2008

  What is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.

  Show that $\mathbb{Z}[\sqrt{-7}]$ is not a Euclidean domain (for any choice of norm), but that the ring

  $$\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$$

  is Euclidean for the norm function $N(z)=z \bar{z}$.

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• 1.II.10G

  Groups, Rings and Modules | Part IB, 2008

  (i) Show that $A_{4}$ is not simple.

  (ii) Show that the group Rot $(D)$ of rotational symmetries of a regular dodecahedron is a simple group of order 60 .

  (iii) Show that $\operatorname{Rot}(D)$ is isomorphic to $A_{5}$.

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• 2.I.2G

  Groups, Rings and Modules | Part IB, 2008

  What does it means to say that a complex number $\alpha$ is algebraic over $\mathbb{Q}$ ? Define the minimal polynomial of $\alpha$.

  Suppose that $\alpha$ satisfies a nonconstant polynomial $f \in \mathbb{Z}[X]$ which is irreducible over $\mathbb{Z}$. Show that there is an isomorphism $\mathbb{Z}[X] /(f) \cong \mathbb{Z}[\alpha]$.

  [You may assume standard results about unique factorisation, including Gauss's lemma.]

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• 2.II.11G

  Groups, Rings and Modules | Part IB, 2008

  Let $F$ be a field. Prove that every ideal of the ring $F\left[X_{1}, \ldots, X_{n}\right]$ is finitely generated.

  Consider the set

  $$R=\left\{p(X, Y)=\sum c_{i j} X^{i} Y^{j} \in F[X, Y] \mid c_{0 j}=c_{j 0}=0 \text { whenever } j>0\right\}$$

  Show that $R$ is a subring of $F[X, Y]$ which is not Noetherian.

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• 3.I.1G

  Groups, Rings and Modules | Part IB, 2008

  Let $G$ be the abelian group generated by elements $a, b, c, d$ subject to the relations

  $$4 a-2 b+2 c+12 d=0, \quad-2 b+2 c=0, \quad 2 b+2 c=0, \quad 8 a+4 c+24 d=0$$

  Express $G$ as a product of cyclic groups, and find the number of elements of $G$ of order 2 .

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• 4.I.2G

  Groups, Rings and Modules | Part IB, 2008

  Let $n \geq 2$ be an integer. Show that the polynomial $\left(X^{n}-1\right) /(X-1)$ is irreducible over $\mathbb{Z}$ if and only if $n$ is prime.

  [You may use Eisenstein's criterion without proof.]

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• 4.II.11G

  Groups, Rings and Modules | Part IB, 2008

  Let $R$ be a ring and $M$ an $R$-module. What does it mean to say that $M$ is a free $R$-module? Show that $M$ is free if there exists a submodule $N \subseteq M$ such that both $N$ and $M / N$ are free.

  Let $M$ and $M^{\prime}$ be $R$-modules, and $N \subseteq M, N^{\prime} \subseteq M^{\prime}$ submodules. Suppose that $N \cong N^{\prime}$ and $M / N \cong M^{\prime} / N^{\prime}$. Determine (by proof or counterexample) which of the following statements holds:

  (1) If $N$ is free then $M \cong M^{\prime}$.

  (2) If $M / N$ is free then $M \cong M^{\prime}$.

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Linear Algebra

• 1.I.1E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

  (i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

  (ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

  (iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

  (iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.

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• 1.II.9E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

  Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

  Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

  Hence, or otherwise, prove that

  $$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$$

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• 2.I.1E

  Linear Algebra | Part IB, 2008

  Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{R}$. What does it mean to say that $\psi: V \rightarrow W$ is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map $\psi: V \rightarrow V$ is surjective if, and only if, it is injective.

  Suppose that $\psi: V \rightarrow V$ is a linear map which has a right inverse, that is to say there is a linear map $\phi: V \rightarrow V$ such that $\psi \phi=\mathrm{id}_{V}$, the identity map. Show that $\phi \psi=\mathrm{id}_{V}$.

  Suppose that $A$ and $B$ are two $n \times n$ matrices over $\mathbb{R}$ such that $A B=I$. Prove that $B A=I$.

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• 2.II.10E

  Linear Algebra | Part IB, 2008

  Define the determinant $\operatorname{det}(A)$ of an $n \times n$ square matrix $A$ over the complex numbers. If $A$ and $B$ are two such matrices, show that $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.

  Write $p_{M}(\lambda)=\operatorname{det}(M-\lambda I)$ for the characteristic polynomial of a matrix $M$. Let $A, B, C$ be $n \times n$ matrices and suppose that $C$ is nonsingular. Show that $p_{B C}=p_{C B}$. Taking $C=A+t I$ for appropriate values of $t$, or otherwise, deduce that $p_{B A}=p_{A B}$.

  Show that if $p_{A}=p_{B}$ then $\operatorname{tr}(A)=\operatorname{tr}(B)$. Which of the following statements is true for all $n \times n$ matrices $A, B, C$ ? Justify your answers.

  (i) $p_{A B C}=p_{A C B}$;

  (ii) $p_{A B C}=p_{B C A}$.

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• 3.II.10E

  Linear Algebra | Part IB, 2008

  Let $k=\mathbb{R}$ or $\mathbb{C}$. What is meant by a quadratic form $q: k^{n} \rightarrow k$ ? Show that there is a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ for $k^{n}$ such that, writing $x=x_{1} v_{1}+\ldots+x_{n} v_{n}$, we have $q(x)=a_{1} x_{1}^{2}+\ldots+a_{n} x_{n}^{2}$ for some scalars $a_{1}, \ldots, a_{n} \in\{-1,0,1\} .$

  Suppose that $k=\mathbb{R}$. Define the rank and signature of $q$ and compute these quantities for the form $q: \mathbb{R}^{3} \rightarrow \mathbb{R}$ given by $q(x)=-3 x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$.

  Suppose now that $k=\mathbb{C}$ and that $q_{1}, \ldots, q_{d}: \mathbb{C}^{n} \rightarrow \mathbb{C}$ are quadratic forms. If $n \geqslant 2^{d}$, show that there is some nonzero $x \in \mathbb{C}^{n}$ such that $q_{1}(x)=\ldots=q_{d}(x)=0$.

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• 4.I.1E

  Linear Algebra | Part IB, 2008

  Describe (without proof) what it means to put an $n \times n$ matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.

  Put the following matrix in Jordan normal form:

  $$\left(\begin{array}{ccc} -7 & 3 & -5 \\ 7 & -1 & 5 \\ 17 & -6 & 12 \end{array}\right)$$

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• 4.II.10E

  Linear Algebra | Part IB, 2008

  What is meant by a Hermitian matrix? Show that if $A$ is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for $\mathbb{C}^{n}$ consisting of eigenvectors of $A$.

  A Hermitian matrix is said to be positive definite if $\langle A x, x\rangle>0$ for all $x \neq 0$. We write $A>0$ in this case. Show that $A$ is positive definite if, and only if, all of its eigenvalues are positive. Show that if $A>0$ then $A$ has a unique positive definite square root $\sqrt{A}$.

  Let $A, B$ be two positive definite Hermitian matrices with $A-B>0$. Writing $C=\sqrt{A}$ and $X=\sqrt{A}-\sqrt{B}$, show that $C X+X C>0$. By considering eigenvalues of $X$, or otherwise, show that $X>0$.

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Markov Chains

• 1.II.19H

  Markov Chains | Part IB, 2008

  The village green is ringed by a fence with $N$ fenceposts, labelled $0,1, \ldots, N-1$. The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0 , and then chooses one of the two nearest neighbours, 1 or $N-1$, with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.

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• 2.II.20H

  Markov Chains | Part IB, 2008

  A Markov chain with state-space $I=\mathbb{Z}^{+}$has non-zero transition probabilities $p_{00}=q_{0}$ and

  $$p_{i, i+1}=p_{i}, \quad p_{i+1, i}=q_{i+1} \quad(i \in I) .$$

  Prove that this chain is recurrent if and only if

  $$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{q_{r}}{p_{r}}=\infty$$

  Prove that this chain is positive-recurrent if and only if

  $$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{p_{r-1}}{q_{r}}<\infty$$

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• 3.I.9H

  Markov Chains | Part IB, 2008

  What does it mean to say that a Markov chain is recurrent?

  Stating clearly any general results to which you appeal, prove that the symmetric simple random walk on $\mathbb{Z}$ is recurrent.

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• 4.I.9H

  Markov Chains | Part IB, 2008

  A Markov chain on the state-space $I=\{1,2,3,4,5,6,7\}$ has transition matrix

  $$P=\left(\begin{array}{ccccccc} 0 & 1 / 2 & 1 / 4 & 0 & 1 / 4 & 0 & 0 \\ 1 / 3 & 0 & 1 / 2 & 0 & 0 & 1 / 6 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 / 2 & 0 & 1 / 2 \end{array}\right)$$

  Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.

  For each $i, j \in I$ say whether the $\operatorname{limit}^{-1} \lim _{n \rightarrow \infty} p_{i j}^{(n)}$ exists, and evaluate the limit when it does exist.

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Methods

• $2 . \mathrm{I} . 5 \mathrm{D} \quad$

  Methods | Part IB, 2008

  Describe briefly the method of Lagrange multipliers for finding the stationary values of a function $f(x, y)$ subject to a constraint $g(x, y)=0$.

  Use the method to find the largest possible volume of a circular cylinder that has surface area $A$ (including both ends).

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• 1.II.14D

  Methods | Part IB, 2008

  Write down the Euler-Lagrange equation for the variational problem for $y(x)$ that extremizes the integral $I$ defined as

  $$I=\int_{x_{1}}^{x_{2}} f\left(x, y, y^{\prime}\right) d x$$

  with boundary conditions $y\left(x_{1}\right)=y_{1}, y\left(x_{2}\right)=y_{2}$, where $y_{1}$ and $y_{2}$ are positive constants such that $y_{2}>y_{1}$, with $x_{2}>x_{1}$. Find a first integral of the equation when $f$ is independent of $y$, i.e. $f=f\left(x, y^{\prime}\right)$.

  A light ray moves in the $(x, y)$ plane from $\left(x_{1}, y_{1}\right)$ to $\left(x_{2}, y_{2}\right)$ with speed $c(x)$ taking a time $T$. Show that the equation of the path that makes $T$ an extremum satisfies

  $$\frac{d y}{d x}=\frac{c(x)}{\sqrt{k^{2}-c^{2}(x)}}$$

  where $k$ is a constant and write down an integral relating $k, x_{1}, x_{2}, y_{1}$ and $y_{2}$.

  When $c(x)=a x$ where $a$ is a constant and $k=a x_{2}$, show that the path is given by

  $$\left(y_{2}-y\right)^{2}=x_{2}^{2}-x^{2} .$$

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• 2.II.15D

  Methods | Part IB, 2008

  (a) Legendre's equation may be written in the form

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

  Show that there is a series solution for $y$ of the form

  $$y=\sum_{k=0}^{\infty} a_{k} x^{k},$$

  where the $a_{k}$ satisfy the recurrence relation

  $$\frac{a_{k+2}}{a_{k}}=-\frac{(\lambda-k(k+1))}{(k+1)(k+2)} .$$

  Hence deduce that there are solutions for $y(x)=P_{n}(x)$ that are polynomials of degree $n$, provided that $\lambda=n(n+1)$. Given that $a_{0}$ is then chosen so that $P_{n}(1)=1$, find the explicit form for $P_{2}(x)$.

  (b) Laplace's equation for $\Phi(r, \theta)$ in spherical polar coordinates $(r, \theta, \phi)$ may be written in the axisymmetric case as

  $$\frac{\partial^{2} \Phi}{\partial r^{2}}+\frac{2}{r} \frac{\partial \Phi}{\partial r}+\frac{1}{r^{2}} \frac{\partial}{\partial x}\left(\left(1-x^{2}\right) \frac{\partial \Phi}{\partial x}\right)=0$$

  where $x=\cos \theta$.

  Write down without proof the general form of the solution obtained by the method of separation of variables. Use it to find the form of $\Phi$ exterior to the sphere $r=a$ that satisfies the boundary conditions, $\Phi(a, x)=1+x^{2}$, and $\lim _{r \rightarrow \infty} \Phi(r, x)=0$.

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• 3.I.6D

  Methods | Part IB, 2008

  Let $\mathcal{L}$ be the operator

  $$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}-k^{2} y$$

  on functions $y(x)$ satisfying $\lim _{x \rightarrow-\infty} \quad y(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$.

  Given that the Green's function $G(x ; \xi)$ for $\mathcal{L}$ satisfies

  $$\mathcal{L} G=\delta(x-\xi)$$

  show that a solution of

  $$\mathcal{L} y=S(x)$$

  for a given function $S(x)$, is given by

  $$y(x)=\int_{-\infty}^{\infty} G(x ; \xi) S(\xi) d \xi$$

  Indicate why this solution is unique.

  Show further that the Green's function is given by

  $$G(x ; \xi)=-\frac{1}{2|k|} \exp (-|k||x-\xi|)$$

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• 3.II.15D

  Methods | Part IB, 2008

  Let $\lambda_{1}<\lambda_{2}<\ldots \lambda_{n} \ldots$ and $y_{1}(x), y_{2}(x), \ldots y_{n}(x) \ldots$ be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system

  $$\mathcal{L} y_{n}=\lambda_{n} w(x) y_{n},$$

  where

  $$\mathcal{L} y \equiv \frac{d}{d x}\left(-p(x) \frac{d y}{d x}\right)+q(x) y,$$

  with $p(x)>0$ and $w(x)>0$. The boundary conditions on $y$ are that $y(0)=y(1)=0$.

  Show that two distinct eigenfunctions are orthogonal in the sense that

  $$\int_{0}^{1} w y_{n} y_{m} d x=\delta_{n m} \int_{0}^{1} w y_{n}^{2} d x .$$

  Show also that if $y$ has the form

  $$y=\sum_{n=1}^{\infty} a_{n} y_{n},$$

  with $a_{n}$ being independent of $x$, then

  $$\frac{\int_{0}^{1} y \mathcal{L} y d x}{\int_{0}^{1} w y^{2} d x} \geq \lambda_{1}$$

  Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,

  $$\frac{\mathrm{\partial }y}{\mathrm{\partial }t}=-\frac{1}{w}\mathcal{L}$$