# Teaching

Current undergraduate course schedules are found here, and lecture lists here. Some supplemental information about lecturers and major changes to courses in the last few years is given on my page here. My term calendars can be found on this page.

## Supervising

For information on my supervising in Lent 2021, see this page

I have supervised the following courses (“sheets” links to the (current version of) the example sheets, “handouts” to the corresponding section below, although you may wish to look at the introductory section first):

Lent 2021
Analysis I (sheets) (handouts)
Probability (sheets) (handouts)
Complex Analysis (sheets) (handouts)
Michaelmas 2020
Vectors and Matrices (sheets) (handouts)
Methods (sheets) (notes) (handouts)
Lent 2020
Analysis I (sheets) (handouts)
Probability (sheets) (handouts)
Michaelmas 2019
Vectors and Matrices (sheets) (handouts)
Lent 2019
Analysis I (sheets) (handouts)
Probability (sheets) (handouts)
Asymptotic Methods (sheets) (notes) (handout)
Michaelmas 2018
Differential Equations (sheets) (handouts)
Variational Principles (sheets) (handouts)
Lent 2018
Asymptotic Methods (sheets) (notes) (handout)
Michaelmas 2016
Quantum Mechanics (sheets) (handouts)
Lent 2016
Further Complex Methods (sheets) (handouts)
Michaelmas 2015
Principles of Quantum Mechanics (sheets) (handouts)
Lent 2015
Complex Analysis (sheets) (handouts)
Michaelmas 2014
Variational Principles (sheets) (handouts)
Methods (sheets) (notes) (handouts)
Lent 2014
Complex Analysis (sheets) (handouts)

## Lecture Notes

### Asymptotic Methods

My lecture notes for the course may be downloaded here; they cover the course precisely, apart from at present the optional section on applications of the Airy function to rainbows. Please send corrections, comments, suggestions, &c to my email address (which may be found on the back page of the PDF, but you should be able to guess it from my web address!). Last updated: 23/04/2016 (all that has changed in the Schedules as of 2017–18 is the Recommended Books). Will be updated at some point in 2020.

### Methods

As requested by some, I have archived here (pdf) the lecture notes from the 2009 Methods course as given by Dr Caulfield. Please direct any comments to him. (As of writing, the schedules are identical. This document also includes at the end some notes on material from IB Variational Principles, from before it was separated into the VP course, quite a few years ago now (2009).)

## Handouts

I have written quite a lot of handouts for courses I have supervised. As may be evident from the titles, they vary between summaries of formulae, more detailed explanations of things from the course, and the other category of Random Interesting Junk Related to the Course (and occasionally more than one of these). I have marked the latter, just in case.

Please note that these handouts are at least nominally a work in progress. This means that there may be typos (should you spot any, please let me know!), and the formatting and style may be somewhat inconsistent between handouts (and possibly also between what is here and what I actually hand out, which is more like Version 0 than Version 1). General feedback is also appreciated.

Any references to question numbers refer to the example sheets from last time I taught the course. I have indicated these on this page, although I may have missed some.

New items are marked with a “*”.

### Vectors and Matrices

1. Examples of things that are, and are not, vector spaces (Lots of examples of things that are vector spaces, including important ones like the trivial vector space, the space of column vectors, and spaces of linear maps between vector spaces. Also includes some examples that are not vector spaces, through violating at least one of the axioms. The main methods of constructing larger vector spaces out of smaller ones are also discussed, although this (starred) section is almost entirely [RIJRttC])
2. Never be confused by a change of basis again (maybe) (One of the parts of this course that people often struggle with is how changing the basis of a vector space changes the components of vectors and linear maps, also known as matrices. This is an attempt to explain what is going on a bit more clearly by using “commutative” diagrams to distinguish the same vector space having different bases.)
3. Dr Stephen Cowley’s comprehensive (and very comprehensible) lecture notes are available on his website here. (The “Teaching” link in the left frame.)
4. A general discussion of topics that people tend to find difficult, in this course and in IB Linear Algebra, is given by David Mond here. It covers general vector spaces vs. $$\mathbb{R}^n$$, some examples, bases and change-of-basis, determinant, and several multivariable calculus topics.

### Differential Equations

1. The integral $$\int_0^{\infty} \frac{\sin{x}}{x} \, dx$$, also known as the Dirichlet integral, is very important in applications (as you will learn in IB Methods). G. H. Hardy wrote two papers about it, as a basic example of a definite integral that can be evaluated exactly (despite its integrand not having an elementary primitive) to determine whether it is reasonable to teach such definite integrals at school. Despite the age of the papers, the curriculum still agrees with his conclusion. [RIJRttC]
2. Proof of Differentiation Under the Integral Sign (A proper proof of Leibniz's rule for changing the order of a derivative and an integral. Skim it now, read it properly after you have studied integration rigorously in Analysis I.) [RIJRttC]
3. The Versatile Wronskian (Every use of the Wronskian you’ll see in undergraduate mathematics.) [Starred sections are RIJRttC]
4. Amusingly, only a couple of pages away from Hardy's second article is the following note from H. Piaggio, one general way of justifying the “multiply by another $$x$$” rule for repeated roots of the auxiliary equation:
(The same applies equally well to finding the particular integral.)
5. The phase portrait diagram for 2D linear matrix differential equations I hand out is from TeXample.net, by Gernot Salzer.
6. Series Solutions: The Method of Frobenius (Derivation of the Method, the algorithm that we actually use, and some examples of the various cases to which it applies.)
7. Euler’s Trigonometrical Products and Partial Fraction Formulae (Derivation of Euler’s product formula $$\sin{\pi x} = \pi x \prod_{k=1}^{\infty} ( 1- x^2/k^2 )$$, “partial fractions” for the cotangent, and the other trigonometric and hyperbolic formulae of the same family.) [RIJRttC]

### Analysis I

1. Introduction to Analysis I (What this course is for, reviews of recommended books, other book and resource recommendations.)
2. Summary of Convergence Tests (A rather telegraphic summary of the conditions and conclusions of essentially every convergence test for series of real numbers known to man. (If you know any I've missed, I'll be happy to add them.)) [Mostly, but not entirely, RIJRttC]
3. * The Root Test (One of the simplest convergence tests for series, that is not listed in the Schedules, and so may not be covered in lectures.)
4. A number of contributions to the early work on rigorous convergence tests made by Abel. An historical article that describes one of these, arising out of a misunderstanding about the nature of the test
If $$\sum_n a_n$$ converges, then $$n a_n \to 0$$.
is
5. A beautiful proof of the twin results
• If $$\sum_n a_n$$ converges, then we can find a sequence $$b_n$$ so that $$a_n/b_n \to \infty$$ and $$\sum_n b_n$$ converges
• If $$\sum_n a_n$$ diverges, then we can find a sequence $$b_n$$ so that $$a_n/b_n \to 0$$ and $$\sum_n b_n$$ converges
(i.e., there is no slowest-converging or slowest-diverging series) is given in
6. Cantor's original proof of the uncountability of $$\mathbb{R}$$ (It's not the diagonal argument! It also applies to any linear continuum, which can be used to prove other facts about $$\mathbb{R}$$.)
7. The Fundamental Theorem of Calculus (Statements and proofs of the four (!) most common versions of the Fundamental Theorem of Calculus for the Darboux integral (that is, the definition used in this course, which is occasionally also called the Riemann integral). Beware [RIJRttC] that there is no consensus for which of these are called the Fundamental Theorem: the last section of this handout contains an extensive examination of the terminology used in “modern” Analysis I–level textbooks (roughly the last 50 years).)

### Probability

1. More so than most other courses, various recent past lecturers have made their notes available online. Their course pages are listed below.
2. * Table of some probability distributions (All the ones you need for this course, and some more that you may need in IB Statistics)
3. The Galton–Watson process as known today began life as a model of the propagation and extinction of surnames: the traditional story traced its origin to the following question, posed by Francis Galton[1] in the Educational Times, Vol. 25, Issue 143 (Mar., 1873), Question 4001:
A large nation of whom we will only concern ourselves with the adult males, N in number, and who each bear separate surnames, colonise a district. Their law of population is such that, in each generation, $$a_0$$ per cent. of the adult males have no male children who reach adult life; $$a_1$$ have one such male child; $$a_2$$ have two; and so on, up to $$a_5$$ who have five. Find (1) what proportion of the surnames will have become extinct after $$r$$ generations; and (2) how many instances there will be of the same surname being held by $$m$$ persons.
[The Proposer remarks that a general solution of this problem would be of much aid in certain rather important statistical enquiries, and that he finds it a laborious matter to work it out numerically, in even the simplest special cases, and to only a few generations. In reality, the generations would overlap and mix, but it is not necessary to suppose them otherwise than as occurring in successive steps.]
This was answered by Reverend Henry William Watson in Educational Times Vol. 26, Issue 148, (Aug., 1874), pp. 115f. The two then presented a joint paper to the Journal of the Royal Anthropological Institute:
Alas, it turns out that not only does this article contain errors, it is also not the first discovery of this process, which was discussed (correctly) in 1845 by the obscure French statistician Irénée-Jules Bienaymé:
4. A handout (by Sinho Chewi) containing a discussion and more general proof of Wald’s identity may be found here. The nice feature of this particular handout is that it only introduces one concept from outside the scope of this course (stopping time), and then the proof itself requires only ideas from this course.
5. * Inequalities in IA Probability (Statements, proof, consequences and some pictures for the two most important inequalities in the course, Markov and Jensen.) [Everything after the AM–GM inequality is RIJRttC]
6. Relating to Jensen’s inequality and convexity:
• Jensen’s original paper is It contains much of the origins of the theory of convex functions (and is based on an earlier paper in Danish, which is theoretically available on JSTOR here).
• A good visual explanation is given by Needham in
7. Why the Cumulative Distribution Function Is Enough (Specifically, enough to determine the entire probability distribution of a random variable. Also explains what a probability distribution actually is.) [RIJRttC, described as “unsuitable for first years” by a fellow(!)]
8. Unfortunately a lot of the interesting topics in this course are postponed to II Probability and Measure. A partial list is
• What a (probability) measure actually is, and why the axioms are the sensible ones.
• The correct way to define an integral on a probability space (Lebesgue integral), and its awesome convergence theorems.
• The better alternative to the MGF (the characteristic function, which exists for any probability distribution).
• The more general way to deal with random variables, including both discrete and continuous (Lebesgue–Stieltjes measure).
• More general conditions for Wald’s identity to hold, and random processes (martingales, also in II Stochastic Financial Models).
• Stronger and more useful Central Limit Theorems.
If you do plan to go to this course, be aware that it is not easy!

[1] Wikipedia gives the comprehensive description an English Victorian era statistician, progressive, polymath, sociologist, psychologist, anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, and psychometrician. (One may also add Trinity mathmo to this list.)

### Vector Calculus

1. * A venerable, but appalling, pun: an application of the change-of-variables in multiple integrals to history.
2. * However, as appalling puns go, this must be the masterpiece.
3. * The example of iterated integrals not giving the same value, ${ \int_0^1 \bigg( \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \, dx \bigg) \, dy } \neq { \int_0^1 \bigg( \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \, dy \bigg) \, dx } ,$ goes back at least to Cauchy: see pp. 678–680 (pp. 394–396 of the reprint) of
(The integrals are improper, or a result about absolutely convergent double integrals would make the values the same. You may find it informative to consider the (improper) integrals with generic lower limits, to see how the value depends on the order in which the double limit is taken.)
G.H. Hardy gives more examples of iterated integrals with this property in
• Hardy, G.H. “Note on the inversion of a repeated integral”, Proceedings of the London Mathematical Society, Series 2, Volume 24, (1926) p.l–li. (Paywalled link) Reprinted in Collected Papers, Volume V.
4. The two most useful pages on Wikipedia for this course, and many other applied ones:
5. Summary of Concepts in Vector Calculus (A 10-page summary of all the material in the course)

### Variational Principles

1. The Legendre transform is perhaps the weirdest-looking part of the course. There are a few explanations available online to help fill in the gaps in the course’s treatment.
2. Example Sheet 1, Question 6 [2017–18] is made much easier by first proving that the only rectangular parallelepipeds inscribable in a general ellipsoid are those with sides parallel to the axes. Several proofs of this result are given in the short article Rectangular Parallelepipeds in Ellipsoids by Duncan, Khavinson and Shapiro, SIAM Review Vol. 38, No. 4 (Dec. 1996), pp. 655–657.
3. [2017–18] Sheet 1 Question 9: Minimal Surface of Revolution Bounded by Two Circles (the results obtained in this question are quite surprising; this is a summary, with some graphs and diagrams)
4. The extra question set on Sheet 2 (to be treated as the third part of Question 6 [2017–18]) is as follows:
$$I[u] = \int (\det{Du}) \, dx \, dy$$, where $$\det{Du}$$ is the Jacobian determinant of the function $$u \colon \mathbb{R}^2 \to \mathbb{R}^2$$. What is strange about this example? Why does this happen?
5. Summary of Concepts in Variational Principles (An 8-page summary of all the material in the course)

### Methods

1. Fourier Series: the Short Guide (a summary of the important formulae)
2. Fourier Series/The Sine Product Formula/A Cotangent Series (a way to obtain the expression of the cotangent as a sum of reciprocals using Fourier Series) [RIJRttC]
3. Separation of Variables (a description of the method, and the process for Laplace’s equation in the usual co-ordinate systems)
4. Bessel Functions and Some of Their Many Identities (a summary of the theory of Bessel functions, sufficient for the course)
5. The Fourier Transform (a summary of conventions, properties and a few of the common transform pairs) [Last section is RIJRttC]
6. The Riemann Zeta Function at Positive Even Integers (sequel to the cotangent handout, introduces the Bernoulli numbers and derives the expansions of the singular trigonometrical functions, then the promised zeta values) [RIJRttC]
7. Similarity Solution to the Diffusion Equation (a more careful discussion of the reduction of the diffusion equation to an ordinary differential equation using a similarity variable than is normally provided in lectures)
8. Some Green’s Functions (a mixture of derivations and discussions of the multivariable Green’s functions in the course: Laplace in two and three dimensions, and the wave and diffusion equations in $$1+1$$ dimensions)
9. The Method of Characteristics (derivation of the Method and its applications: solving first-order linear, semilinear and quasilinear PDEs, classification of second-order linear PDEs; includes diagrams)
10. Summary of Concepts in Methods (A 13-page summary of the course material.)

### Quantum Mechanics

1. Non-Degeneracy in One Dimension (a summary of how to show rigorously that bound states of the Schrödinger equation in one dimension are unique)
2. Scattering Problems in One Dimension (a discussion of various issues in one-dimensional scattering, including the calculation of the reflexion and transmission coefficients for a rectangular barrier, some limits thereof, and [RIJRttC] the $$S$$-matrix)
3. Operators in Quantum Mechanics (summary of pretty much everything this course does about operators, and some other useful facts)
4. Orthogonal Polynomials in Quantum Mechanics (Derivations [RIJRttC] of the properties of Hermite, Legendre, associated Legendre and Laguerre polynomials, with applications to the course content: harmonic oscillator, spherical harmonics, hydrogen atom. Last page has a table of the properties that are worth remembering.)
5. Summary of Formulae (A 4-page summary of the course material.)

### Complex Analysis

1. Automorphisms of the Unit Disc (a derivation of a couple of important automorphism groups, based on some of the example sheet questions; includes an explanation of the link between the group of Möbius transformations and the general linear group of $$2 \times 2$$ matrices)
2. Harmonic Functions and Conformal Maps (A fuller explanation of the proceedure for obtaining a new harmonic function from a known one on a conformally equivalent domain, containing the surprisingly complicated diagram explaining the process)
3. Hardy’s proof of Hardy’s theorem (a detailed expansion of Hardy's paper showing that the Riemann zeta function has infinitely many zeros with real part $$\frac{1}{2}$$: a testament to the power of complex analysis, that such a result actually requires only a little more work to understand after a first course)

### Principles of Quantum Mechanics

I have written some notes on a couple of the harder questions on the example sheets.
1. Sheet 3, Questions 7&8 (including a full inductive proof of the (anti)symmetry result for identical particle states)
2. Sheet 4, Questions 3&4 (including a [RIJRttC] summary of some of the strange effects that parity has in the Standard Model)

### Further Complex Methods

1. Analytic Functions from Their Real Part (a way of obtaining analytic functions with a given real part, without resorting to solving partial differential equations)
2. Sheet 2, Questions 4&10 (two different proofs of the multiplication formula for the $$\Gamma$$-function, and evaluating terms in the series expansion of the Riemann $$\zeta$$-function using integrals)
3. Sheet 3, Questions 7&8 (solutions to the last two questions on Sheet 3, concerning the Laplace Transform; the last part is particularly nasty, and for once the branch of $$\sqrt{s}$$ that we choose matters)
4. Elliptic Functions (notes on the theory of general elliptic functions, and a detailed summary of the Weierstrass elliptic functions and their uses)
5. The Hypergeometric Function and the Papperitz Equation (notes covering the last part of the course, on the general differential equation with three regular singular points, its description with the Riemann $$P$$-symbol, and its solution using hypergeometric functions; summary of hypergeometric functions, including definitions, integral representations, differential equation and Kummer's connexion formulae)
6. Modular Functions and Picard's Theorem (a summary of the derivation and properties of the modular $$\lambda$$-function, and its rôle in the proof of Picard's little theorem on entire functions which omit two points; also a short explanation of the monodromy part of the proof, but no details)

### Asymptotic Methods

1. The Method of Steepest Descent (a summary of how to use the Method of Steepest Descent to approximate an integral, and a nontrivial, original example) [Updated 2019-05-16 (v2.1)]

On the other hand, those looking for the History of Mathematics example sheet may find an archived copy here.

### Acknowledgements

I would like to thank all of my students who have read these handouts over the years, and especially those who provide feedback. Special thanks go to Dr Thomas Forster and Daan van de Weem for their frequent help with proofreading, suggestions, and occasionally keeping me grounded with what students will find interesting and/or useful. Naturally any remaining errors remain my own!