Core modules
In the first year you take essential (core) modules in both mathematics and physics.
In the second and third years, there is considerable freedom to choose modules. By then you will have a good idea of your main interests and be well placed to decide which areas of mathematics and physics to study in greater depth.
Year One
Mathematical Analysis 1/2
Mathematical Analysis is the rigorous study of calculus. In this module, there will be a considerable emphasis throughout on the need to lay out mathematical arguments with much greater precision and care than you had to at school. With the support of your fellow students, lecturers, and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to, say, differentiate and integrate a function, to the point where you can develop your own rigorous proofs of calculus results that you may have taken for granted. The module will allow you to deal carefully with limits and infinite summations, approximations to pi and e, and the Taylor series. The module ends with the construction of the integral and the Fundamental Theorem of Calculus.
Read more about these modules, including the methods of teaching and assessment (content applies to 2024/25 year of study):
Sets and Numbers
Mathematics can be described as the science of logical deduction - if we assume such and such as given, what can we deduce with absolute certainty? Consequently, mathematics has a very high standard of truth - the only way to establish a mathematical claim is to give a complete, rigorous proof. Sets and Numbers aims to show students what can be achieved through abstract mathematical reasoning.
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Linear Algebra
The branch of maths treating simultaneous linear equations is called linear algebra. The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another. It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix. These theoretical ideas have many applications, which will be discussed in the module. These applications include: Solutions of simultaneous linear equations. Properties of vectors. Properties of matrices, such as rank, row reduction, eigenvalues and eigenvectors. Properties of determinants and ways of calculating them.
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Mathematical Methods and Modelling 1 and 2
Introduces the fundamentals of mathematical modelling, before discussing and analysing difference and differential equations in physics, chemistry, engineering as well as the life and social sciences. This will require the basic theory of ordinary differential equations (ODEs), the cornerstone of all applied mathematics. ODE theory later proves invaluable in branches of pure mathematics, such as geometry and topology. You will be introduced to simple differential and difference equations, methods for obtaining their solutions and numerical approximation.
In the second term for Mathematical Methods and Modelling 2, you will study the differential geometry of curves, calculus of functions of several variables, multi-dimensional integrals, calculus of vector functions of several variables (divergence and circulation), and their uses in line and surface integrals.
Read more about these modules, including the methods of teaching and assessment (content applies to 2024/25 year of study):
Physics Foundations
You will learn about dimensional analysis, thermodynamics and waves. Often the qualitative features of systems can be understood (at least partially) by thinking about which quantities in a problem are allowed to depend on each other on dimensional grounds. Thermodynamics is the study of heat flow and how it can lead to useful work. Even though the results are universal, the simplest way to introduce this topic is via the ideal gas, whose properties are discussed and derived in some detail. Finally, waves are time-dependent variations about some time-independent (often equilibrium) state. You will look at phenomena like the Doppler effect (this is the effect that the frequency of a wave changes as a function of the relative velocity of the source and observer), the reflection and transmission of waves at boundaries and some elementary ideas about diffraction and interference patterns.
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Electricity and Magnetism
This module is largely concerned with the great developments in electricity and magnetism, which took place during the nineteenth century. The origins and properties of electric and magnetic fields in free space, and in materials, are tested in some detail and all the basic levels up to, but not including, Maxwell's equations are considered. In addition, the module deals with both dc and ac circuit theory including the use of complex impedance. You will be introduced to the properties of electrostatic and magnetic fields, and their interaction with dielectrics, conductors and magnetic materials.
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Classical Mechanics and Special Relativity
You will study Newtonian mechanics emphasising the conservation laws inherent in the theory. These have a wider domain of applicability than classical mechanics (for example they also apply in quantum mechanics). You will also look at the classical mechanics of oscillations and of rotating bodies. The module then explains why the failure to find the ether was such an important experimental result and how Einstein constructed his theory of special relativity. You will cover some of the consequences of the theory for classical mechanics and some of the predictions it makes, including: the relation between mass and energy, length-contraction, time-dilation and the twin paradox.
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Physics Programming Workshop
This module introduces the Python programming language. It is quick to learn and encourages good programming style. Python is an interpreted language, which makes it flexible and easy to share. It allows easy interfacing with modules that have been compiled from faster C or Fortran code. It is widely used throughout physics and there are many downloadable, free-to-use codes available. The module also looks at the visualisation of data.
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Year Two
Analysis 3
This module lays the basis for many subsequent mathematically-inclined modules, and it is concerned with further study of the notions of convergence and calculus seen in Analysis 1 and 2. Creating a consistent theoretical framework for these concepts has kept many great mathematicians busy for many centuries, and in this module you walk in their footsteps.
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Methods of Mathematical Physics
The module covers the theory of Fourier transforms and the Dirac delta function. The module also introduces Lagrange multipliers, co-ordinate transformations and cartesian tensors illustrating them with examples of their use in physics. Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. A Fourier transform will turn a linear differential equation with constant coefficients into a nice algebraic equation which is in general easier to solve. The module explains why diffraction patterns in the far-field limit are the Fourier transforms of the "diffracting" object. The case of a repeated pattern of motifs illustrates beautifully one of the most important theorems in the business - the convolution theorem. The diffraction pattern is the product of the Fourier transform of repeated delta functions and the Fourier transform for a single copy of the motif.
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Partial Differential Equations
The theory of partial differential equations (PDE) is important in both pure and applied mathematics. Since the pioneering work on surfaces and manifolds by Gauss and Riemann, PDEs have been at the centre of much of mathematics. PDEs are also used to describe many phenomena from the natural sciences (such as fluid flow and electromagnetism) and social sciences (such as financial markets). In this module you will learn how to classify the most important partial differential equations into three types: elliptic, parabolic, and hyperbolic. You will study the role of boundary conditions and look at various methods for solving PDEs.
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Hamiltonian and Fluid Mechanics
This module looks at the Hamiltonian and Lagrangian formulation of classical mechanics and introduces the mechanics of fluids. Lagrangian and Hamiltonian mechanics have provided the natural framework for several important developments in theoretical physics including quantum mechanics. The field of fluids is one of the richest and most easily appreciated in physics. Tidal waves, cloud formation and the weather generally are some of the more spectacular phenomena encountered in fluids. The module establishes the basic equations of motion for a fluid - the Navier-Stokes equations - and shows that in many cases they can yield simple and intuitively appealing explanations of fluid flows.
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Quantum Mechanics and its Applications
In the first part of this module you will use ideas, introduced in the first year module, to explore atomic structure. This includes the time-independent and the time-dependent Schrödinger equations for spherically symmetric and harmonic potentials, angular momentum and hydrogenic atoms. The second half of the module looks at many-particle systems and aspects of the Standard Model of particle physics. It introduces the quantum mechanics of free fermions and discusses how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars.
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Statistical Mechanics, Electromagnetic Theory and Optics
Any macroscopic object we meet contains a large number of particles, each of which moves according to the laws of mechanics (which can be classical or quantum). Yet we can often ignore the details of this microscopic motion and use a few average quantities such as temperature and pressure to describe and predict the behaviour of the object. Why we can do this, when we can do this and how to do it are discussed in the first half of this module.
We also develop the ideas of first year electricity and magnetism into Maxwell's theory of electromagnetism. Establishing a complete theory of electromagnetism has proved to be one of the greatest achievements of physics. It was the principal motivation for Einstein to develop special relativity, it has served as the model for subsequent theories of the forces of nature and it has been the basis for all of electronics and optics (radios, telephones, computers, the lot...).
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Year Three
Fluid Dynamics
Starting with a solid understanding of the underlying mathematical description of different fluid flows, you will find qualitative and quantitative solutions for particular fluid dynamics problems, ranging from simple laminar flows to fully developed turbulence, and use the concepts and techniques you have learned to analyse other partial differential equations, for example in plasma physics or nonlinear optics. An important aim of the module is to provide you with an appreciation of the complexities and beauty of fluid motion, which will be brought out in computer demonstrations and visualisations.
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Quantum Physics of Atoms
The basic principles of quantum mechanics are applied to a range of problems in atomic physics. The intrinsic property of spin is introduced and its relation to the indistinguishability of identical particles in quantum mechanics discussed. Perturbation theory and variational methods are described and applied to several problems. The hydrogen and helium atoms are analysed and the ideas that come out from this work are used to obtain a good qualitative understanding of the periodic table. In this module, you will develop the ideas of quantum theory and apply these to atomic physics.
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Electrodynamics
Einstein's 1905 paper on special relativity was called "On the electrodynamics of moving bodies". It derived the transformation of electric and magnetic fields when moving between inertial frames of reference. The module works through this transformation and looks at its implications. The module starts by covering the magnetic vector potential, A, which is defined so that the magnetic field B=curl A and which is a natural quantity to consider when looking at relativistic invariance.
The radiation (EM-waves) emitted by accelerating charges are described using retarded potentials, which are the time-dependent analogs of the usual electrostatic potential and the magnetic vector potential, and have the wave-like nature of light built in. The scattering of light by free electrons (Thomson scattering) and by bound electrons (Rayleigh scattering) will also be described. Understanding the bound electron problem led Rayleigh to his celebrated explanation of why the sky is blue and why sunlight appears redder at sunrise and sunset.
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Kinetic Theory
Kinetic Theory' is the theory of how distributions change and is therefore essentially about non-equilibrium phenomena. The description of such phenomena is statistical and is based on Boltzmann's equation, and on the related Fokker-Planck equation. These study the evolution in time of a distribution function, which gives the density of particles in the system's phase space. In this module you will establish the relations between conductivity, diffusion constants and viscosity in gases. You will look also at molecular simulation and applications to financial modelling.
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Laboratory for Mathematics and Physics Students
You will be introduced to collaborative, experimental and computational work and some advanced research techniques. It will give you the opportunity to plan and direct an experiment and to work within a team. It should acquaint you with issues associated with experimental work, including data acquisition and the analysis of errors and the health and safety regulatory environment within which all experimental work must be undertaken. It will also provide you experience of report writing and making an oral presentation to a group.
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Year Four
Physics Project
The project will provide you with experience of working in a research environment. You will work, normally in pairs, on an extended project which may be experimental, computational or theoretical (or indeed a combination of these). Through discussions with your supervisor you will establish a plan of work which you will frequently review as you progress. In general, the project will not be closely prescribed and will contain an investigative element.
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Optional modules
Optional modules can vary from year to year. Example optional modules may include:
- Topics in Mathematical Biology
- Dynamical Systems
- Fourier Analysis
- Quantum Mechanics: Basic Principles and Probabilistic Methods
- Statistical Mechanics
- Mathematical Acoustics
- Structure and Dynamics of Solids
- General Relativity
- Planets, Exoplanets and Life
- Quantum Computation and Simulation
- Advanced Quantum Theory
- Theoretical Particle Physics
- Solar and Space Physics
- High Performance Computing
- The Distant Universe
