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
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 ends with the 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 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.
Methods of Mathematical Modelling I and II
In this module you will learn the modelling cycle and learn to analyse simple models, using scaling, non-dimensionalisation and linear stability analysis to understand the main dynamics. 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 their solution and numerical approximation.
In the second term 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 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 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.
Quantum Phenomena
This module explains how classical physics is unable to explain the properties of light, electrons and atoms. (Theories in physics, which make no reference to quantum theory, are usually called classical theories.) It covers the most important contributions to the development of quantum physics including: wave-particle 'duality', de Broglie's relation and the Schrodinger equation. It also looks at applications of quantum theory to describe elementary particles: their classification by symmetry, how this allows us to interpret simple reactions between particles and how elementary particles interact with 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.
Year Two
Analysis III
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.
Multivariable Calculus
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.
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
Communicating Science
Employers look for many things in would-be employees. Sometimes they will be looking for specific knowledge, but often they will be more interested in general skills, frequently referred to as transferable skills. One such transferable skill is the ability to communicate effectively, both orally and in writing. Over the past two years you may have had experience in writing for an academic audience in the form of your laboratory reports. The aim of this module is to introduce you to the different approaches required to write for other audiences. This module will provide you with experience in presenting technical material in different formats to a variety of audiences.
Optional modules
- Measure Theory
- Topics in Mathematical Biology
- Complex Analysis
- Fluid Dynamics
- Theory of PDEs
- Physics Project
- Quantum Physics of Atoms
- Electrodynamics
- Kinetic Theory
- Condensed Matter Physics
- Scientific Computing
- The Earth and Its Atmosphere
- Plasma Physics and Fusion
- The Standard Model
- Galaxies and Cosmology
- Statistical Physics
- Physics of Life and Medicine
- Black Holes, White Dwarfs and Neutron Stars
