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):

• Mathematical Analysis 1Link opens in a new window
• Mathematical Analysis 2Link opens in a new window

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):

• Mathematical Methods and Modelling 1Link opens in a new window
• Mathematical Methods and Modelling 2Link opens in a new window

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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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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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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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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Scholarships and bursaries

Learn about scholarships and bursaries available to undergraduate students.

Further information

Find out more about Warwick undergraduate bursaries and scholarships.

Eligibility for student loans

Your eligibility for student finance will depend on certain criteria, such as your nationality and residency status, your course, and previous study at higher education level.

Check if you're eligible for student finance.

Help with living costs

If you struggle to meet your essential living costs, our Student Funding team Link opens in a new windowwill be on hand to offer advice and support.

There are a number of options that may be available to you including government, bursary and/or hardship support. Warwick also has a number of bursaries and scholarships Link opens in a new windowthat you may be eligible to apply for.

Repaying your loans

You will repay your loan or loans gradually once you are working and earning above a certain amount. For students starting their course after 1 August 2023, the repayment threshold is £25,000. Repayments will be taken directly from your salary if you are an employee. If your income falls below the earnings threshold, your repayments will stop until your income goes back up above this figure.

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Life at Warwick

Within a close-knit community of staff and students from all over the world, discover a campus alive with possibilities. A place where all the elements of your student experience come together in one place. Our supportive, energising, welcoming space creates the ideal environment for forging new connections, having fun and finding inspiration.

• Accommodation
• Arts, Culture and Events
• Campus
• Campus map
• Clubs and societies
• Food and drink
• Library
• Sports and Fitness
• Wellbeing support

Keep exploring life at Warwick

Find out how to apply to us, ask your questions, and find out more.

Warwick Accommodation

Finding the right accommodation is key to helping you settle in quickly.

We have a range of residences for undergraduate students on campus.

Explore Warwick Accommodation

Our campus

You won't be short of ways to spend your time on campus - whether it's visiting Warwick Arts Centre, using our incredible new sports facilities, socialising in our bars, nightclub and cafés, or enjoying an open-air event. Or if you need some peace and quiet, you can explore lakes, woodland and green spaces just a few minutes’ walk from central campus.

Explore our campus

Food and drink

We have lots of cafés, restaurants and shops on campus. You can enjoy great quality food and drink, with plenty of choice for all tastes and budgets. There is a convenience store on central campus, as well as two supermarkets and a small shopping centre in the nearby Cannon Park Retail Park. Several of them offer delivery services to help you stay stocked up.

And don't miss our regular food market day on the Piazza with tempting, fresh and delicious street food. Soak up the atmosphere and try something new, with mouth-watering food for all tastes.

Explore food and shops

Explore Students' Union venues

Clubs and societies

We currently have more than 300 student-run societies.

So whether you’re into films, martial arts, astronomy, gaming or musical theatre, you can instantly connect with people with similar interests.

Or you could try something new, or even form your own society.

Explore our societies

Sports and fitness

Staying active at Warwick is no sweat, thanks to our amazing new Sports and Wellness Hub, indoor and outdoor tennis centre, 60 acres of sports pitches, and more than 60 sports clubs.

Whether you want to compete, relax or just have fun, you can achieve your fitness goals.

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Studying on campus

Our campus is designed to cater for all of your learning needs.

You will benefit from a variety of flexible, well-equipped study spaces and teaching facilities across the University.

• The Oculus, our outstanding learning hub, houses state-of-the-art lecture theatres and innovative social learning and network areas.
• The University Library provides access to over one million printed works and tens of thousands of electronic journals
• Different study spaces offering you flexible individual and group study spaces.

Studying at Warwick

Travel and local area

Our campus is in Coventry, a modern city with high street shops, restaurants, nightclubs and bars sitting alongside medieval monuments. The Warwickshire towns of Leamington Spa and Kenilworth are also nearby.

The University is close to major road, rail and air links. London is just an hour by direct train from Coventry, with Birmingham a 20-minute trip. Birmingham International Airport is nearby (a 20-minute drive).

Travelling from campus

Wellbeing support and faith provision

Our continuous support network is here to help you adjust to student life and to ensure you can easily access advice on many different issues. These may include managing your finances and workload, and settling into shared accommodation. We also have specialist disability and mental health support teams.

Our Chaplaincy is home to Chaplains from the Christian, Jewish and Muslim faiths. We provide regular services for all Christian denominations and a Shabbat meal every Friday for our Jewish students. There is also an Islamic prayer hall, halal kitchen and ablution facilities.

Student support

Chaplaincy

Talk to us

Join us at a live event. You can ask about courses, applying to Warwick, life at Warwick, visas and immigration, and more.

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Warwick Experience

Take a virtual, student-led campus tour. Then join an interactive panel session, where you can hear from and chat to our current students and staff.

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Student blogs

Explore our student blogs in Unibuddy. You can read about campus life from students themselves, and register to post questions directly to students.

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Explore campus with our virtual tour

Our 360 tour lets you:

• Watch student videos
• View 360 photography and drone footage
• Learn about facilities and landmarks

Explore our campus virtually through our 360 campus tour now

Come to an Open Day

Don’t just take it from us, come and see for yourself what Warwick is all about. Whether it's a virtual visit or in-person, our University Open Days give you the chance to meet staff and students, visit academic departments, tour the campus and get a real feel for life at Warwick.

Open Days at Warwick