# Calculus 3: Transforms & Models

## Module description

This module develops further the ideas covered in Calculus 2. The important ideas of the Laplace transform and Fourier series will be introduced, and their applications in solving other problems in Calculus will be considered. Systems of differential equations will be studied, and used to model and investigate problems in the natural and social sciences.

Teaching for this module will take place throughout the year, with eight evenings of lectures in each of the autumn and spring terms, and two evenings of revision and consolidation in the summer term.

### indicative module 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: the methods introduced in the module will be to model and interpret problems in the natural and social sciences covering some  the following: 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 have:

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