*4 points *

*Semester 1*

This unit deals with multivariable calculus including the theory and applications of the differential and integral calculus of vector functions and of functions of several variables. Topics covered include infinite sequences and series, power series; curves, tangents, normals and curvature; mean-value theorem and Taylor's theorem for functions of several variables; vector functions of vector variables; Jacobian matrix; chain rule; implicit and inverse functions; and multiple integration.

**Lectures/workshops:**- 3 per week
**Prerequisite:**- Mathematics 102
**Incompatible:**- 2MC: Multivariable Calculus

*4 points *

*Semester 1*

This unit reviews and extends the study of linear algebra from the first year. Topics
include general vector spaces over** R **and **C**; linear transformations and matrix representations; determinants and inverses of matrices;
change of basis and similarity; eigenvalues and eigenvectors; matrix diagonalisation; inner product spaces and
orthonormal bases; Gram-Schmidt process; orthogonal complement; symmetric and Hermitian matrices; and
quadratic forms.

**Lectures/workshops:**- 3 per week
**Prerequisite:**- Mathematics 102
**Incompatible:**- 2MA2: Matrix Algebra 2

*4 points *

*Semester 2*

This unit continues the study of multivariable calculus including the theory and applications of differential and integral calculus of vector fields. Topics covered include optimisation; Lagrange multipliers; scalar and vector fields; line and surface integrals; conservative forces, divergence, curl; the theorems of Green, Gauss and Stokes; Fourier series; and partial differential equations.

**Lectures/workshops:**- 3 per week
**Prerequisite:**- 2C1: Calculus 1

*4 points*

*Semester 2*

This unit deals with applications of linear algebra and calculus to the development of techniques for solving a wide variety of mathematical problems with special emphasis on methods for solving differential equations. Topics include systems of linear differential equations; phase plane, equilibrium and stability of second-order systems; Laplace transforms; numerical methods for solving non-linear equations; numerical methods for solving initial-value problems; examples of simple boundary-value problems and numerical solutions of boundary-value problems using finite differences and shooting methods.

**Lectures/workshops:**- 3 per week
**Prerequisites:**- 2C1: Calculus 1 and 2LA: Linear Algebra

*4 points*

*Semester 1*

This unit describes the use of mathematics in understanding and solving problems of organisation and planning such as those found in planning telecommunications networks, rostering airline staff, optimising the design and operation of a mine, shipment and warehousing of goods, scheduling machines, competitive and co-operative games, and many others.

Mathematically, these are all examples of linear programmes and many are examples of network optimisation. The unit introduces the theory and practice of linear programming and of linear network optimisation. The theory has very powerful consequences, some of which appear in this unit and some in later units.

**Lectures/workshops:**- 3 per week
**Prerequisites:**-
Mathematics 102,
*or*Mathematics 176 with 2MA1: Matrix Algebra 1 taken concurrently

*4 points*

*Semester 2*

This unit shows how to use mathematics to solve real world problems. The emphasis is on the process of modelling and not on the mathematical techniques. The unit is suitable for biological scientists, computer scientists, economists, physical scientists and social scientists, but is also suitable for a Mathematics major as an introduction to Applied Mathematics.

Applications are drawn from physics, biology, medicine, engineering, business, finance, and others, but no prior knowledge of these subjects is assumed. This unit is useful to anyone who may need to analyse systems and data to understand, predict or control the system. Topics include planning and organising for optimal efficiency in a construction project, factory or business; using game theory to make strategic business decisions; using difference equations and differential equations to model population growth, ozone depletion and global warming, or to control industrial processes; a discussion of stability of systems and chaos; and using power spectra and other tools to analyse data, for example periodic fluctuations in infant breathing patterns.

**Lectures/workshops:**- 3 per week
**Prerequisite:**-
Mathematics 101, 155, 175,
*or*176

*4 points*

*Semester 1*

This unit introduces a wide range of analytical concepts and shows how these fit into mathematics and related disciplines. Topics may include metric spaces, completeness, Banach and Hilbert spaces with applications in physics; calculus of curves and surfaces in Euclidean 3-space, curvatures, tensor fields, with applications in engineering and physics.

**Lectures/workshops:**- 3 per week
**Prerequisite:**- Mathematics 102

*4 points*

*Semester 2*

This unit is an introduction to discrete mathematics and geometry, with applications in computer vision. Many of the basic concepts and structures of discrete mathematics are covered, especially graphs and posets. An introduction is given to Euclidean 3-space and projective geometry. Applications are made to 3-dimensional computer vision. This unit is suitable for majors in any stream of mathematics, computer science or information technology.

**Lectures/workshops:**- 3 per week
**Prerequisites:**-
Mathematics 102,
*or*Mathematics 155 with 2MA1: Matrix Algebra 1 taken concurrently*or*Mathematics 176 with 2MA1: Matrix Algebra 1 taken concurrently

*4 points*

*Semester 1*

This unit develops the techniques of probability modelling together with the distribution theory required for a study of statistical inference. Topics may include random variables and their distributions; joint and conditional distributions; a survey of common distributions and some of their applications; the Poisson process and related distributions; convergence of random variables and the central limit theorem; and an introduction to Markov chains. This unit is suitable for students interested in the fields of mathematics and statistics, computer science, information technology, genetics, quantitative finance, psychology and economics.

**Lectures/workshops:**- 3 per week
**Prerequisites:**-
Mathematics 101 and 102,
*or*Mathematics 155 and Statistics and Modelling 155 with 2MA1: Matrix Algebra 1 taken concurrently,*or*Mathematics 175 with 2MA1: Matrix Algebra 1 taken concurrently. Either 2C1: Calculus or 2MC: Multivariate Calculus must be taken concurrently.

*4 points*

*Semester 2*

This unit is an introduction to statistical inference based on the use of likelihood theory for estimation and hypothesis testing. Emphasis is placed both on the basic theory and the practical application of the methodology. Topics may include statistical models and general approaches to estimation; least squares; properties of estimators; testing statistical hypotheses; linear models; regression and analysis of variance; and contingency tables.

**Lectures/workshops:**- 3 per week
**Prerequisite:**- 2S1: Probability

*4 points*

*Semester 1*

This unit synthesises multiple regression, analysis of variance and the analysis of univariate data with covariates. It includes a revision of simple linear regression and one-way anova, then progresses to factorial designs with interaction, multiple regression, comparisons of regressions between levels of factors or strata and analysis of covariance. Principles of the design and analysis of experiments and clinical trials, and sample size selection, are stressed. The emphasis is on the statistical analysis of data and the interpretations of such an analysis.

**Lectures/workshops:**- 3 per week
**Prerequisites:**-
Mathematics 101,
*or*Mathematics 155 and Statistics and Modelling 155,*or*Mathematics 175

*4 points*

*Semester 2*

This unit provides an introduction to modelling and estimation for some simple random processes. In the first half of the unit some common probability distributions are discussed, together with elementary finite state Markov chains and first properties of the Poisson process. The use of these techniques for modelling real-life random processes is emphasised. The second half of the unit concentrates on maximum likelihood estimation for fitting probability models to data. The unit stresses the practical application of the material taught and includes examples from areas such as economics and finance; biological, behavioural and human sciences; information theory; queuing and system reliability.

**Lectures/workshops:**- 3 per week
**Prerequisites:**-
Mathematics 101,
*or*Mathematics 155 and Statistics and Modelling 155,*or*Mathematics 175

*4 points*

*Semester 1*

This unit comprises elementary matrix algebra and its applications to the solution of linear
systems, linear algebra in R^{n} including subspaces, independence and
bases, and an introduction to linear transformations. Students learn to use the computer algebra package Maple
for matrix computations.

**Lectures/workshops:**- 3 per week
**Prerequisite:**-
Mathematics 122, 155
*or*175 **Incompatible:**- Mathematics 102

*4 points*

*Semester 1*

This unit emphasises the techniques of multivariable calculus and the calculus of vector functions including curves, tangents, partial derivatives, use of the chain rule, Lagrange multipliers, multiple integrals and change of variables.

**Lectures/workshops:**- 3 per week
**Prerequisite:**-
Mathematics 101, 155
*or*175 **Incompatible:**- 2C1: Calculus 1

*4 points*

*Semester 2*

This unit provides further material on linear algebra and its applications, with emphasis on
R^{n} and its subspaces. Topics include matrix inverses; determinants;
inner products; change of basis; eigenvalues and eigenvectors; diagonalisation; and selected applications of
matrix theory.

**Lectures/workshops:**- 3 per week
**Prerequisite:**-
Mathematics 102
*or*2MA1: Matrix Algebra 1 **Incompatible:**- 2LA: Linear Algebra