Mathematics and Statistics tutorials

Students will learn how to encode a geometric scenario into vector equations and meet the vector algebra needed to manipulate such equations. They will also meet the benefits of choosing sensible co-ordinate systems and appreciate what geometry is invariant of such choices.

The course presents the tools needed to be able to solve a range of ordinary differential equations (ODEs) and introduces the notion of partial derivatives for use in a variety of applications.

This course introduces the general notions of a vector space, a subspace, linear independence and dependence, spanning sets and bases. It will also develop an understanding of matrices and their applications to systems of linear equations and linear maps between vector spaces.

This course is split into two parts: metric spaces (10 lectures) and complex analysis (22 lectures). It begins with an introduction to point-set topology and the central importance of complex variables in analysis, before moving onto differentiation and integration in this setting. It then presents the tools and results of complex analysis including Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Laurent’s expansion and the theory of residues.

Students will learn a range of different techniques and results used in the study of ordinary differential equations (ODEs) and partial differential equations (PDEs), such as: Picard’s theorem proved both by successive approximation and the contraction mapping theorem; Gronwall’s inequality; phase plane analysis; method of characteristics for first order semi-linear PDEs; classification of second order semi-linear PDEs and their reduction to normal form using characteristic variables; well-posedness; the maximum principle and some of its consequences.

In this course students will be introduced to multi-dimensional vector calculus. They will be shown how to evaluate volume, surface and line integrals in three dimensions and how they are related via the divergence theorem and Stokes’ theorem. Students will also learn how to perform calculations involving div, grad and curl, including appreciating their meanings physically and proving important identities.

The course begins by introducing students to Fourier series, concentrating on their practical application and the conditions required for their convergence. They will then be shown how to derive the heat, wave and Laplace’s equations in several independent variables and to solve them. Finally, they will begin the study of uniqueness of solution of these important partial differential equations (PDEs).

Students will acquire a range of techniques for solving second order ODE’s and boundary value problems, including Frobenius solutions, Fredholm alternative and Bessel’s functions. The course concludes with an introduction to asymptotic theory and how the presence of a small parameter can affect solution construction and form.

Students will gain a range of techniques employing the Laplace and Fourier transforms in the solution of ordinary and partial differential equations. They will also have an appreciation of generalized functions and their calculus and applications.

This course introduces the concept of an algorithm and explains how to construct simple algorithms for the solution of certain elementary problems. Students will make and run simple procedures in Matlab to demonstrate examples of applications in real analysis and implementation.

Students will develop a sound knowledge and appreciation of the ideas and concepts related to modelling biological and ecological systems using both discrete- and continuous-time non-spatial models. Techniques used include linear stability analysis and phase planes.

In this course it is shown that such variational problems give rise to a system of differential equations, the Euler–Lagrange equations. Furthermore, the minimizing principle that underlies these equations leads to direct methods for analysing the solutions to these equations. These methods have far reaching applications and will help develop students’ technique.