# Calculus 3: Transforms & Models

## Overview

• Credit value: 30 credits at Level 6
• Convenor: Robert Russell
• Assessment: three problem sets (10% each) and a three-hour examination (70%)

## Module description

In this module we develop further the ideas covered in Calculus 2. We introduce the important ideas of the Laplace transform and Fourier series, and consider their applications in solving other problems in calculus. You will have the opportunity to study systems of differential equations, and use them to model and investigate problems in the natural and social sciences.

### Indicative syllabus

• Laplace transforms: the Laplace transform of elementary functions, properties of the Laplace transform, inversion of the Laplace transform, solving differential equations using Laplace transforms, the convolution of two functions, some families of integral equations, the z-transform and discrete systems
• Fourier series: periodic functions, the Fourier series of a function, finding the coefficients of a Fourier series, the range of validity of a Fourier series, even and odd functions and their Fourier series, half range Fourier series and the relationship between them and the corresponding Fourier series, application of Fourier series in the solution of partial differential equations, the Fourier transform
• Dynamical systems: systems of linear differential equations, coupled systems, autonomous systems, stability, critical points, phase plane analysis, trajectories, fixed solutions, periodic solutions, chaotic systems
• Models: mixing problems, tests for diabetes, nutrient exchange in the placenta, competing species, predator-prey models, the Lotka-Volretta equations, the transfer of heat, the arms race and combat, finance and economics

## Learning objectives

By the end of this module, you will be able to:

• use mathematical and/or statistical techniques
• understand a range of results in mathematics
• appreciate the need for proof in mathematics, and follow and construct mathematical arguments
• show awareness of the use of mathematics and/or statistics to model problems in the natural and social sciences, and formulate such problems using appropriate notation
• understand the importance of assumptions and show awareness of where they are used and the possible consequences of their violation
• understand a range of modelling techniques, their conditions and limitations, and the need to validate and revise models
• show a deeper knowledge of particular areas of mathematics
• use a modern mathematical and/or statistical computer package with a programming facility, and show knowledge of other suitable packages.