Module Outlines
Here are the outlines of all the modules that may be taken by current 1st year Mathematics and Physics students.
Term 1
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How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.
However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a 'pure' subject like group theory can be part of the daily armoury of the 'applied' mathematician, so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.
In this module we will cover relatively simple examples, first order equations (dy/dx=f(x,y)) and linear second order equations, (d2x(t)/dx2 + p(t)dx/dt +q(t)x = g(t)) and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, ie. its 'qualitative' properties. This approach is invaluable for more complex equations for which we cannot find an explicit solution.
We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.
The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.
Lecturer: Andreas Dedner
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At the beginning of the nineteenth century the familiar tools of calculus, differentiation and integration, began to run into problems. Mathematicians were unsure of how to apply these tools to sums of infinitely many functions. The origins of Analysis lie in their attempt to formalize the ideas of calculus purely in the language of arithmetic and to resolve these problems.
You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others, concerning sequences and series in term one, continuity and differentiability in term two and integration in term one of your second year.
By the end of the year you will be able to answer many interesting questions, such as: What do we mean by 'infinity'? How do you compute the value of pi or e? How can you add up infinitely many numbers, or infinitely many functions? Can all functions be approximated by polynomials?
There will be considerable emphasis throughout the module 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. You will also be expected to question the concepts underlying your solutions, and understand why a particular method is meaningful and another not so. In other words, your mathematical focus should shift from methods to concepts and clarity of thought.
LECTURERS: Mario Micallef (Term 1), Richard Gratwick/Gavin Brown (Term 2)
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University mathematics introduces progressively more and more abstract ideas and structures, and demands more and more in the way of proof, until by the end of a mathematics degree most of the student's time is occupied with understanding proofs and creating his or her own. This is not because university mathematicians are more pedantic than schoolteachers, but because proof is how one knows things in mathematics, and it is in its proofs that the strength and richness of mathematics is to be found.
But learning to deal with abstraction and with proofs takes time. This module aims to bridge the gap between school and university mathematics, by beginning with some rather concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
Lecturer: Damiano Testa
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This module begins with a quick tour through elementary plane Euclidean geometry. We emphasise proof, and the careful use of diagrams as an aid to understanding problems and finding proofs. Plane geometry then provides the setting for an introduction to the geometry of the sphere and of polyhedra.
Lecturer: Gavin Brown
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This module looks 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. Loosely speaking this is the requirement that "apples can only equal apples". Examples we will look at include the size of an atom, the length scale on which a theory of gravity has to take account of quantum effects and the speed of a wave in a shallow channel. We will also study the concept of heat. Studying heat transfers and how they can lead to useful work is the basis of thermodynamics. It is intuitively obvious that not all heat can be turned into work and, hence, some other quantity is needed. This is the quantity called entropy and actually controls how much work can be done by a heat engine. Even though the results are universal, the simplest way to introduce this topic is via the ideal gas, whose properties we will discuss and derive in some detail.
The second half of the module introduces the language and concepts used to describe waves. Waves are time-dependent variations about some some time-independent (often equilibrium) state. For example, they can be variations in pressure (sound waves), variations in electric and magnetic fields (light waves) or variations in the height of water above the sea-bed (water waves). They carry energy, momentum and information and much of their behaviour is similar whatever their nature. We will revise the relation between the wavelength, frequency and velocity and the definition of the amplitude and phase of a wave. The module will also cover 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.
LECTURER: Neil Wilson
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By 1905, there was a successful theory (Newton's laws) describing the motion of massive bodies and there was a successful theory of light waves (Maxwell's equations of electromagnetism). But the two theories are inconsistent: in mechanics objects only move relative to each other, whereas light appears to move relative to nothing at all (the vacuum). Physicists (including Maxwell himself) had therefore assumed that there had to be some background 'ether', through which light propagated. The problem was that all attempts to detect this ether had failed.
Einstein realised that there was nothing wrong with Maxwell's equations and that there was no need for an ether. Newtonian mechanics itself was the problem. He proposed that the laws of classical mechanics had to be consistent with just two postulates, namely that the speed of light is a constant and that all frames of reference are equivalent. These postulates forced Einstein to reject previous ideas of space and time and led directly to the special theory of relativity.
In this module, we 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). We will also look at the classical mechanics of oscillations and of rotating bodies. We will then explain why the failure to find the ether was such an important experimental result and explain how Einstein constructed his theory of special relativity. The module will cover some of the consequences of the theory for classical mechanics and some of the counter-intuitive predictions it makes, including: the relation between mass and energy, length-contraction, time-dilation and the twin paradox.
LECTURER: Tom Marsh
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The tutor's mark is made up from your answers to the assessed weekly problems (50%) and from your work associated with five worksheets (50%). The worksheets cover some background mathematical material assumed by other modules. The material covered includes complex numbers, vectors, matrices, multiple integration and integration along surfaces and contours.
ORGANIZER: Michael Pounds
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This module introduces groups and rings. You are familiar with examples of both rings and groups. The set of integers together with the operation of addition form a group: Each application of the operation generates another member of the set, each element has its inverse and there is an identity (zero). The integers, together with the two operations addition and multiplication, form a ring. Certain matrices under multiplication form groups and certain matrices with the canonical operations of multiplication and addition also form rings.
Groups and rings are classified by their properties, such as what sub-groups or sub-rings do they have, are they commutative, etc. The properties and classification are very general. The generality is the idea behind abstract algebra. We can study the properties of these groups and rings independently of the actual examples which suggested them in the first place. The module looks at groups, their definition, the role of subgroups. It then looks at examples including the groups based on the integers, arithmetic modulo n and permutations and explains the role of normal subgroups and quotient groups. The second part introduces rings, and studies examples based on Z (integers), Q (rationals), R (reals), C (complex nos) and Zn (integers modulo n).
Lecturer: Samir Siksek
Term 2
The modules Mathematical Analysis, Electricity and Magnetism and Tutorial continue from term 1.
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Many problems in maths and science are solved by reduction to a system of simultaneous linear equations in a number of variables. Even for problems which cannot be solved in this way, it is often possible to obtain an approximate solution by solving a system of simultaneous linear equations, giving the "best possible linear approximation''.
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.
Lecturer: Derek Holt
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Aspects of software specification, design, implementation and testing will be introduced in the context of the Java language. The description of basic elements of Java will include data types, expressions, assignment and compound, alternative and repetitive statements. Program structuring and object oriented development will be introduced and illustrated in terms of Java's method, class and interface. This will enable the development of software that reads data in a variety of contexts, performs computations on that data and displays results in text and graphical form. Examples of iterative and recursive algorithms will be given. The importance of Java and Java Virtual Machine in networked computing will be described. The majority of examples will be standard applications but the development of Java Applets to be delivered by web browsers will also be covered.
Lecturer: Adam Chester
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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 explored in some detail. The module also deals with both d.c. and a.c. circuit theory including the use of the notion of the complex impedance.
LECTURER: Erwin Verwichte
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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. However it also 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. The module will also look at visualisation of data.
Lecturers: Yorck Ramachers and Richard West
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The elementary constituents of matter are classified into three generations of quarks and leptons (electrons and neutrinos), which interact with each other through the electromagnetic, the weak and the strong forces. An account of how to classify the elementary particles and their interactions and a description of some of the experimental tools used to probe their properties will be the subject of this introductory module.
The module will start by discussing the relationship between conservation laws and the symmetry of the families of elementary particles. Understanding this relationship turns out to be the key to understanding how elementary particles behave. We will look at which quantities are conserved by which interactions and how this allows us to interpret simple reactions between particles. We will discuss how particles are studied experimentally and describe the operation of some standard pieces of equipment including cathode ray tubes, mass spectrometers and particle accelerators. Finally we will look how elementary particles interact with matter. One example we will look at is that of neutrinos in cosmic rays and their interaction with the earth's atmosphere.
LECTURER: Sinead Farrington
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This module is about the mathematics of random patterns. It is essential to all subjects concerned with uncertainty. It lays the foundation for statistical inference, prediction, and decision making. Consequently it is vital to real world problems.
This module aims to introduce the concept of probability as quantified uncertainty, to give a critique of the frequentist interpretation of probability and to provide the basic knowledge necessary to pursue further study in probability and statistics.
Lecturer: Sigurd Assing
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This module is a continuation of the course Probability A.
This module is about the mathematics of random patterns. It is essential to all subjects concerned with uncertainty. It lays the foundation for statistical inference, prediction, and decision making. Consequently it is vital to real world problems.
This module aims to introduce the concept of probability as quantified uncertainty, to give a critique of the frequentist interpretation of probability and to provide the basic knowledge necessary to pursue further study in probability and statistics.
Lecturer: Sigurd Assing
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The Universe contains a huge variety of objects, ranging from black-holes, red giants and white dwarfs to brown dwarfs, gamma-ray bursts and globular clusters, to name but a few. The module will introduce these objects, and show how, from the application of physics, we have come to know much about their distances, sizes, masses and natures. The module will start with a discussion of the Sun and planets and move on to discuss the Universe as a whole.
LECTURER: Pier-Emmanuel Tremblay
Term 3
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These lectures begin by showing 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.) They 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 Schrodinger who wrote down the first wave-equations to describe matter.
LECTURER: Oleg Petrenko
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