Core modules
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. 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.
Year One
Mathematical Analysis I/II
Analysis is the rigorous study of calculus. In this module, there will be a 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. The module will allow you to deal carefully with limits and infinite summations, approximations to pi and e, and the Taylor series. The module also covers construction of the integral and the Fundamental Theorem of Calculus.
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
Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector spaces, linear mappings and their representation by matrices. 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.
Methods of Mathematical Modelling I and II
Methods of Mathematical Modelling I introduces you to the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of 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 Methods of Mathematical Modelling II, 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.
Physics Foundations
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 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.
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 Special 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.
Physics Programming Workshop
You will be introduced to scientific programming with the help of the Python programming language, a language widely used by physicists. 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.
Year Two
Mathematical Analysis III
In the first half of this module, you will investigate some applications of year one analysis: integrals of limits and series; differentiation under an integral sign; a first look at Fourier series. In the second half you will study analysis of complex functions of a complex variable: contour integration and Cauchy’s theorem, and its application to Taylor and Laurent series and the evaluation of real integrals.
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.
Multivariable Calculus
There are many situations in pure and applied mathematics where the continuity and differentiability of a function f: Rn. → Rm 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.
Norms, Metrics and Topologies
Roughly speaking, a metric space is any set provided with a sensible notion of the “distance” between points. The ways in which distance is measured and the sets involved may be very diverse. For example, the set could be the sphere, and we could measure distance either along great circles or along straight lines through the globe; or the set could be New York and we could measure distance “as the crow flies” or by counting blocks. This module examines how the important concepts introduced in first-year Mathematical Analysis, such as convergence of sequences and continuity of functions, can be extended to general metric spaces. Applying these ideas, we will be able to prove some powerful and important results, used in many parts of mathematics.
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.
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.
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 discusses how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars.
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 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 (radios, telephones, computers, the lot...).
Year Three
Fluid Dynamics
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.
Electrodynamics
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
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.
Year Four
Physics Project
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.
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
