Mathematics and Statistics

The following modules are available to incoming Study Abroad students interested in Mathematics and Statistics.

Alternatively you may return to the complete list of Study Abroad Subject Areas.

MATH101: Calculus

  • Terms Taught: Michaelmas Term Only
  • US Credits: 2 semester credits
  • ECTS Credits: 4 ECTS
  • Pre-requisites: A year of general mathematics, including basic calculus.

Course Description

The course covers: complex numbers, functions and graphs; limits of sequences and sums of infinite series; differentiation, product and chain rules; Taylor series; integration: fundamental theorem of calculus; integration by parts and substitution.

Educational Aims

This course aims to provide the student with an understanding of functions, limits, and series, and a knowledge of the basic techniques of differentiation and integration. The purpose of this course is to study functions of a single real variable. Some of the topics will be familiar, others will be studied more thoroughly in subsequent courses.

The module begins by introducing examples of functions and their graphs, and techniques for building new functions from old. We consider rational functions and the exponential function. We then consider the notion of a limit, sequences and series and then introduce the main tools of calculus. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The rules for calculating derivatives are obtained from the definition of the derivative as a rate of change. Taylor series are calculated for functions such as sin, cos and the hyperbolic functions.

We then introduce the integral and review techniques for calculating integrals. We learn how to add, multiply and divide polynomials and introduce rational functions and their partial fractions. Rational functions are important in calculations, and we learn how to integrate rational functions systematically. The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parameterize geometrical curves.

Outline Syllabus

  • Arithmetic of complex numbers;
  • Polynomials;
  • Rational functions and partial fractions;
  • Exponential and hyperbolic functions;
  • Compositions and inverses;
  • Induction;
  • Sequences and limits;
  • Differentiation;
  • Product and Chain rules;
  • Maxima and minima;
  • Taylor series;
  • Complex exponentials and trigonometric functions;
  • Definite integral as areas;
  • Fundamental theorem of calculus;
  • Integration by parts and by substitution;

Assessment Proportions

  • Coursework: 50%
  • Exam: 50%

MATH102: Further Calculus

  • Terms Taught: Michaelmas Term Only
  • US Credits: 2 semester credits
  • ECTS Credits: 4 ECTS
  • Pre-requisites: A year of general mathematics, including calculus.

Course Description

The course covers: improper integrals; integration over infinite ranges; Simpson’s rule; functions of two or more real variables; partial derivatives; curves in the plane; implicit functions; the chain rule for differentiating along a curve; stationary points for functions of two real variables; double and repeated integrals; Cavalieri’s slicing principle; volumes.

Educational Aims

The first part of this course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.

Outline Syllabus

  • Complex polynomials and complex roots;
  • Integration of rational functions;
  • Improper integrals
  • Integration over infinite ranges;
  • Simpson's rule;
  • Functions of two or more real variables;
  • Partial derivatives;
  • Curves in the plane;
  • Implicit functions;
  • The chain rule for differentiating along a curve;
  • Stationary points for functions of two real variables;
  • Double and repeated integrals;
  • Cavalieri's slicing principle;
  • Volumes

Assessment Proportions

  • Coursework: 50%
  • Exam: 50%

MATH103: Probability

  • Terms Taught:  Lent and Summer Terms Only
  • US Credits: 2 Semester credits
  • ECTS Credits: 4 ECTS credits
  • Pre-requisites: A year of general mathematics

Course Description

Probability theory is the study of chance phenomena; the concepts of probability are fundamental to the study of statistics. The course will emphasise the role of probability models which characterise the outcomes of different types of experiment that involve a chance or random component. The course will cover the ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models. No previous exposure to the subject will be assumed.

Educational Aims

  • To provide an introduction to probability theory for discrete distributions.
  • To introduce students to some simple combinatorics, set theory and the axioms of probability.
  • To make students aware of the different probability models used to model varied practical situations.

Outline Syllabus

  • The axioms of probability
  • Conditional probability
  • Independence
  • Discrete Random variables
  • Expectation, mean and variance.
  • The binomial, Poisson and geometric distributions

Assessment Proportions

  • Coursework: 50%
  • Exam: 50%