# School of Mathematics and Statistics

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2013/2014 Sem. 1

## Aims

This module is designed to introduce students to the ideas, methods and techniques which they will need for applying mathematics in the physical sciences or for taking the study of mathematics further.
It aims to extend and enhance their skills in algebraic manipulation and in the differential and integral calculus; to develop their geometric insight and their understanding of limiting processes and to introduce them to complex numbers and matrices.

## Objectives

By the end of the course the student should be able to:

- understand the notion of proof;

- apply the method of proof by induction;

- add and subtract vectors;

- handle scalar and vector products with confidence;

- apply vector methods to simple geometric problems in 2 and 3 dimensions;

- apply tests of convergence (ratio test, comparison test) to series;

- determine intervals of convergence for real power series using the ratio test;

- recognise the Maclaurin series for sine, cosine, exp and log;

- plot complex numbers on the Argand diagram;

- convert complex numbers to and from polar form;

- apply De Moivre's theorem and evaluate the roots of unity;

- carry out the basic operations of matrix algebra;

- invert a 3x3 non-singular matrix using row operations;

- determine the nature of the solution(s) of a system of linear equations;

- use row and column operations in the evaluation of determinants;

- define, and work with, hyperbolic and inverse hyperbolic functions;

- use L'Hopital's rule to evaluate limits;

- solve separable and linear differential equations of the first order;

- solve second order linear differential equations with constant coefficients.

De L'HOPITAL

## Textbooks

Advanced Engineering Mathematics E Kreyszig; Wiley; 2001

Calculus, R A Adams; Pearson; 2002.

Calculus (6th Edition) E W Swokowski, M Olinick, D Pence; PWS; 1994.
[This book is out of print, but may be available second-hand.]

K E Hirst Numbers, Sequences and Series, Edward Arnold, London, 1995

K E Hirst Vectors in 2 and 3 Dimensions, Edward Arnold, London, 1995
[The textbooks by Hirst are recommended, but are not essential.]

## Lectures, practicals and tutorials

The average load, in hours per week, is as follows:
Lectures: 5
Practicals: 1
Tutorials: 1
These figures do not include the revision and exam period at the end of each semester.

## Assessment

30% of the assessment mark is from continuous assessment during the semester. The remainder is from a 2 hour exam at the end of the semester.

Re-assessment is entirely by a 2 hour exam in August

## Prerequisites

MT1001 or B at Advanced Higher (including units 1 and 2) or B at A-level

## Availability

This module is taught every year in both Semester 1 and Semester 2 at 9.00.

## Lecturers

First Semester: Prof K J Falconer (Module coordinator), Dr M Todd, Dr A Haynes, Dr A P Naughton

Second Semester: Prof K J Falconer (Module coordinator), Prof C E Parnell, Dr A P Naughton, Dr M Todd