Master of Mathematics and Physics (MMathPhys)
4 years full-time
27 September 2021
Department of Study
Department of Physics
Location of Study
University of Warwick
Mathematics and Physics are complementary disciplines, making them a natural combination for university study. Mathematicians and physicists often address common questions and challenges, resulting in exciting unexpected discoveries at the intersection of the two subjects.
Mathematics and Physics are complementary disciplines, making them a natural combination for university study. Mathematicians and physicists often address common questions and challenges, resulting in exciting discoveries at the intersection of the two subjects. Ideas developed in particle physics have led to advances in geometry; learning from chaos theory is being applied increasingly in the modelling of complex physical systems such as the atmosphere and lasers. You’ll be jointly taught by the Institute of Mathematics and Department of Physics, both of which have a reputation for excellence. In addition to core modules, you’ll have flexibility in your second and third years to choose modules to explore areas of interest in more depth. You may also choose to develop breadth of learning by selecting from approved modules outside the two departments, such as the interdisciplinary module Challenges of Climate Change or learning a modern language.
Our four-year course provides further opportunities to explore the breadth of the two subjects, and provides a good foundation for a career related directly to one or both subjects, or further research.
The Warwick joint degree course is among the best established in the country and the course includes a number of modules from both contributing departments designed specifically for joint degree students.
In the first year you take essential (core) modules in both mathematics and physics. You also take at least one additional module chosen from a list of options. At the end of the first year it is possible to change to either of the single honours courses, providing you satisfy certain requirements in the end of year examinations.
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.
You should expect to attend around 14 lectures a week, supported by weekly supervision meetings, problems classes and personal tutorials. For each 1 hour lecture, you should expect to put in a further 1-2 hours of private study.
Lecture size will naturally vary from module to module. The first year core modules may have up to 350 students in a session, whilst the more specialist modules in the later years will have fewer than 100. The core physics modules in the first year are supported by weekly classes, at which you and your fellow students meet in small groups with a member of the research staff or a postgraduate student. Tutorials with your personal tutor and weekly supervision sessions are normally with a group of 5 students.
How will I be assessed?
Most lecture modules are assessed by 15% coursework and 85% final examinations or by 100% exam, with almost all exams taken in the third term. Essays and projects, such as the final-year project, are assessed by coursework and an oral presentation.
The weighting for each year's contribution to your final mark is 10:20:30:40 for the MPhys and MMathPhys courses.
We support student mobility through study abroad programmes. BSc students have the opportunity to apply for an intercalated year abroad at one of our partner universities.
The Study Abroad Team based in the Office for Global Engagement offers support for these activities. The Department's Study Abroad Co-ordinator can provide more specific information and assistance.
All students can apply for research vacation projects - small research projects supervised by a member of academic staff. BSc students can register for the Intercalated Year Scheme, which involves spending a year in scientific employment or UK industry between their second and final year.
General entry requirements
- A*AA to include A* in Mathematics, A in Further Mathematics and A in Physics
- For students not taking A level Further Mathematics, the typical offer is A* (Mathematics), A* (Physics) and A in a third subject at A level
- 38 to include 7 in Higher Level Mathematics (‘Analysis and Approaches’ only) and 6 in Higher Level Physics
- We will consider applications from students taking a BTEC in a relevant Science/Engineering alongside A level Maths and Further Maths on an individual basis
We welcome applications from students with other internationally recognised qualifications.
Contextual data and differential offers
Warwick may make differential offers to students in a number of circumstances. These include students participating in the Realising Opportunities programme, or who meet two of the contextual data criteria. Differential offers will be one or two grades below Warwick’s standard offer (to a minimum of BBB).
Warwick International Foundation Programme (IFP)
All students who successfully complete the Warwick IFP and apply to Warwick through UCAS will receive a guaranteed conditional offer for a related undergraduate programme (selected courses only).
Taking a gap year
Applications for deferred entry welcomed.
We do not typically interview applicants. Offers are made based on your UCAS form which includes predicted and actual grades, your personal statement and school reference.
Analysis is the rigorous study of calculus. In this module there will be considerable emphasis throughout on the need to argue 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 solve each kind of problem, to the point where you can develop your own methods for solving problems. By the end of the year you will be able to answer interesting questions like, what do we mean by `infinity'?
Sets and Numbers
It is in its proofs that the strength and richness of mathematics is to be found. University mathematics introduces progressively more abstract ideas and structures, and demands more in the way of proof, until most of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector space, linear mapping and its representation by a matrix. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.
Can you predict the trajectory of a tennis ball? In this module you cover the basic theory of ordinary differential equations (ODEs), the cornerstone of all applied mathematics. ODE theory proves invaluable in branches of pure mathematics, such as geometry and topology. You will be introduced to simple differential and difference equations and methods for their solution. You will cover first-order equations, linear second-order equations and coupled first-order linear systems with constant coefficients, and solutions to differential equations with one-and two-dimensional systems. We will discuss why in three dimensions we see new phenomena, and have a first glimpse of chaotic solutions.
You will look at dimensional analysis, matter 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 transfers and how they can lead to useful work. Even though the results are universal, the simplest way to introduce this topic to you is via the ideal gas, whose properties are discussed and derived in some detail. You will also cover waves. Waves are time-dependent variations about some time-independent (often equilibrium) state. You will revise the relation between the wavelength, frequency and velocity and the definition of the amplitude and phase of a wave.
Electricity and Magnetism
You will largely be 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.
Classical Mechanics and Relativity
You will study Newtonian mechanics emphasizing 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. It 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.
This module begins by showing you how classical physics is unable to explain some of the properties of light, electrons and atoms. (Theories in physics, which make no reference to quantum theory, are usually called classical theories.) You will then deal with some of the key contributions to the development of quantum physics including those of: Planck, who first suggested that the energy in a light wave comes in discrete units or 'quanta'; Einstein, whose theory of the photoelectric effect implied a 'duality' between particles and waves; Bohr, who suggested a theory of the atom that assumed that not only energy, but also angular momentum, was quantised; and Schrödinger who wrote down the first wave-equations to describe matter.
Physics Programming Workshop
You will be introduced to the Python programming language in this module. 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, which have been compiled from C or Fortran sources. It is widely used throughout physics and there are many downloadable free-to-user codes available. You will also look at the visualisation of data. You will be introduced to scientific programming with the help of the Python programming language, a language widely used by physicists.
In this module, you will learn methods to prove that every continuous function can be integrated, and prove the fundamental theorem of calculus. You will discuss how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties. You will develop a working knowledge of the Riemann integral; understand uniform and pointwise convergence of functions; study complex differentiability (Cauchy-Riemann equations) and complex power series; study contour integrals, Cauchy's integral formulas and applications.
Methods of Mathematical Physics
On this module, you will learn the mathematical techniques required by second-, third- and fourth-year physics students. Starting with the theory of Fourier transforms and the Dirac delta function, you will learn why diffraction patterns in the far-field limit are the Fourier transforms of the ‘diffracting’ object before moving to diffraction more generally, including in the light of the convolution theorem. You will also be introduced to Lagrange multipliers, co-ordinate transformations and Cartesian tensors, which will be illustrated with examples of their use in physics.
There are many situations in pure and applied mathematics where the continuity and differentiability of a function f: R n. → R m has to be considered. Yet, partial derivatives, while easy to calculate, are not robust enough to yield a satisfactory differentiation theory. In this module you will establish the basic properties of this derivative, which will generalise those of single-variable calculus. The module will review line and surface integrals, introduce div, grad and curl and establish the divergence theorem.
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 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.
This course will introduce you to the calculus of variations and to appreciate its centrality to the formulation and understanding of physical laws and problems in geometry. At its conclusion, you should be able to set up and solve minimisation problems with and without constraints, derive Euler-Lagrange equations and appreciate how the laws of mechanics and geometrical problems involving least length and least area fit into this framework.
Physics of Fluids
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. In this module you will establish the basic equations of motion for a fluid - the Navier-Stokes equations - and show that in many cases they can yield simple and intuitively appealing explanations of fluid flows. You will be concentrating on incompressible fluids.
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. You will discuss 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 discussing how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars.
Thermal Physics II
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 the subject of this module. The most important idea in the field is due to Boltzmann, who identified the connection between entropy and disorder. The module shows you how the structure of equilibrium thermodynamics follows from Boltzmann's definition of the entropy and shows you how, in principle, any observable equilibrium quantity can be computed.
Electromagnetic Theory and Optics
You will develop the ideas of first year electricity and magnetism into Maxwell's theory of electromagnetism. Maxwell's equations pulled the various laws of electricity and magnetism (Faraday's law, Ampere's law, Lenz's law, Gauss's law) into one unified and elegant theory. The module shows you that Maxwell's equations in free space have time-dependent solutions, which turn out to be the familiar electromagnetic waves (light, radio waves, X-rays, etc.), and studies their behaviour at material boundaries (Fresnel Equations). You will also cover the basics of optical instruments and light sources.
Starting with a solid understanding of the underlying mathematical description of fluid in 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.
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.
You will revise the magnetic vector potential, A, which is defined so that the magnetic field B=curl A. We will see that this is the natural quantity to consider when exploring how electric and magnetic fields transform under Lorentz transformations (special relativity). The radiation (EM-waves) emitted by accelerating charges will be described using retarded potentials and have the wave-like nature of light built in. The scattering of light by free electrons (Thompson 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.
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.
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.
Mathematical Methods for Physicists III
Requiring functions of complex variables to be analytic (differentiable with respect to their complex argument in some domain) turns out to constrain such functions very strongly. As the module shows: only the constant function is differentiable everywhere, analytic functions are actually equal to their Taylor series and not just approximated by them, a function that is once differentiable is differentiable infinitely many times. Complex differentiable functions are clean, they are fun and they are important in physics. For example, response functions like the dielectric response function are analytic functions with the domain, in which the function is analytic, being related to causality.
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 and partner 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. The project will provide you an experience of working on an extended 'research-like' project in collaboration with a supervisor and partner.
Examples of optional modules/options for current students:
- Programming for Scientists
- Metric Spaces
- Functional Analysis
- Physics in Medicine
- The Solar System
- Hamiltonian Mechanics
Find out more about Tuition fees fees and funding
Additional course costs
There may be costs associated with other items or services such as academic texts, course notes, and trips associated with your course. Students who choose to complete a work placement will pay reduced tuition fees for their third year.
Graduates from this course have gone on to work for employers including:
- Deloitte Digital
- Brunei Shell Petroleum
- British Red Cross
- EDF Energy
- Civil Service
- Deutsche Bank
They have pursued roles such as:
- physical scientists
- finance and investment analysts
- programmers and software development professionals
- graphic designers
Helping you find the right career
Our department has a dedicated professionally qualified Senior Careers Consultant to support you. They offer impartial advice and guidance, together with workshops and events throughout the year. Previous examples of workshops and events include:
- Career options with a Physics Degree
- Careers in Science
- Warwick careers fairs throughout the year
- Physics Alumni Evening
- Careers and Employer networking event for Physics students
“The overlap – using cool maths to explain the phenomena we see around us – is what I am particularly interested in. I don’t just want to be told the answers though; being able to learn more deeply, and fully understand these problems is much more important to me. It has enabled me to develop so many useful skills, alongside learning more about my subjects.”
MMathPhys Mathematics and Physics graduate
This information is applicable for 2021 entry. Given the interval between the publication of courses and enrolment, some of the information may change. It is important to check our website before you apply. Please read our terms and conditions to find out more.