An undergraduate course.

This three-block course will be of interest if you wish to explore and experiment with mathematics using computers. The first block introduces you to computer-assisted algebra techniques using Maple, a software package that allows your computer to symbolically manipulate, numerically evaluate, and graphically visualise mathematical expressions. The second block is about dynamical systems with an emphasis on chaos. Much of this part of the course is taught using traditional pencil-and-paper methods but Maple still frequently comes into play. The final block mainly concerns computer simulations of random processes, again using Maple.

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What you will study

In recent decades, mathematicians have increasingly employed computer-assisted algebra packages in their calculations. Maple is one of the more popular packages, and can be used easily to expand functions as series, evaluate sums and integrals, solve differential equations and plot the results of calculations. It can also be used to write computer programs. The three equally weighted blocks in this course all use Maple to varying extents.

Block A introduces you to Maple and shows how to apply it to a range of mathematical problems. It starts by acquainting students with basic Maple concepts and commands. Examples include plotting functions and solving short exercises involving differentiation, integration, series, vectors, matrices and ordinary differential equations. Then, by working through a number of relatively longer mathematical problems, you are taught the rudiments of programming with Maple, eventually leading up to Maple procedures (often called subroutines in other programming languages). The block finishes by taking you through a series of case studies, thus showing how Maple can be used to treat more substantial mathematical problems.

Block B is about chaos and dynamical systems. Roughly speaking, a dynamical system is a rule that determines how a quantity evolves in time, and chaos is when extreme sensitivity to initial conditions occurs so that, for all practical purposes, it eventually becomes impossible to predict future values of the quantity. The modern approach to dynamics, including those aspects relating to chaos, has developed rapidly over the last four decades mainly through the availability of powerful computers with numerical and graphical capabilities. This block reflects this history by using Maple to provide students with experience of the behaviour of simple dynamical systems. However, a large part of this block also uses conventional pencil-and-paper methods necessary to consolidate this intuitive understanding.

Dynamical systems come in two types: (i) differential equations, where time is continuous; and (ii) iterative maps, where time is discrete. Most of this block is concerned with maps which are usually simpler and yet embody many of the essential features common to both types. This study is motivated by an initial discussion of some real-world systems (e.g. parts of the solar system) that behave chaotically and it is shown how these can often be treated as maps. One-dimensional maps are studied in some detail, leading to a precise definition of chaos. Dynamical systems in higher dimensions are then considered, again mainly (though by no means entirely) as maps. Topics studied at this point include area-preserving and area-contracting maps. The latter leads to a discussion of fractals, which are self-similar objects that also characterise the geometry of the attracting sets of chaotic orbits.

The final Block C is concerned with computer simulations of random processes such as, for example, gambling games and models of the growth of biological populations. Such a subject relies heavily on computing, which is achieved using Maple, although pencil-and-paper analysis also enters into this block. The block starts by introducing some basic notions of randomness and probability, illustrated by examples taken from coin-tossing and die-rolling experiments, followed by some simple models of population dynamics. Then follows a discussion of random walks and the central limit theorem, which relates to properties of random systems with many random variables. This theory is applied to the integration of functions, the so-called Monte Carlo method of integration. The rest of the block confines itself to exploratory simulations of a range of systems starting with the standard map (a type of dynamical system showing chaotic behaviour) and models for infectious disease spreading (modelled either by sets of differential equations or by random processes). Random models of traffic flow are simulated, showing the emergence of traffic jams. Finally the block closes with a detailed investigation of the dynamics of the trebuchet, which is a type of mediaeval siege engine.

You will learn

Successful study of this course should enhance your skills in understanding complex mathematical texts, constructing solutions to problems logically, using professional mathematical software and communicating mathematical ideas clearly.

Entry

This is a Level 3 course. Level 3 courses build on study skills and subject knowledge gained from studying at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with the OU. 

You should have a sound knowledge of differential and integral calculus, vectors and matrices (including eigenvalues), partial differentiation and ordinary differential equations. This course is designed to follow Mathematical methods and models (MST209). Please note that you may have difficulties completing this course if you have not acquired the prerequisite knowledge and skills through achieving a grade 2 or higher pass in the recommended course.

If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.

Preparatory work

Familiarising yourself with those units of MST209 covering mathematical methods would be good preparation. Knowledge of mechanics and modelling is not required for this course, though it may be advantageous in certain places.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

If you have a disability or additional requirement

If you use special hardware or software you must, well before the course begins, find out whether it will work with the course software. At the point of writing this it was not yet clear whether or not Maple would be fully accessible using a screen reader. The printed study materials are available in Adobe Portable Document Format (PDF). Some Adobe PDF components may not be available or fully accessible using a screen reader and mathematical notation may be particularly difficult to read in this way. Other alternative formats of the study materials may be available in the future. Our Services for disabled students website has the latest information about availability.

If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.

You can also find information about accessible study materials, the Disabled Students' Allowance, equipment and other services on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.

Study materials

What's included

Included in the study materials are the three main course texts, one for each of the three blocks in the course, and some shorter supplementary texts to guide you through the course. You will also receive the software package Maple on CD, and a CD-ROM containing some Maple worksheets and some visual material illustrating aspects of the course.

You will need

You require internet access at least once a week during the course to download course resources, including your assignments, and to keep up to date with course news.

A calculator is not required during the course, as you will be using Maple, but for the examination you will need a scientific calculator.

Computing requirements

You will need a computer with internet access to study this course. It includes online activities – you can only access using a web browser with Flash and Java – and some course software provided on disk.

• If you have purchased a new desktop or laptop computer running Windows since 2006 you should have no problems completing the computer-based activities.
• If you’ve got a netbook, tablet or other mobile computing device check our Technical requirements section.
• If you have an Apple Mac or Linux computer – please note that you can only use it for this course by running Windows on it using Boot Camp or a similar dual-boot system.

You can also visit the Technical requirements section for further computing information including the details of the support we provide.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work including computer work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the course.

Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

The assessment details for this course can be found in the facts box above.

Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Professional recognition

This course may help you to gain recognition from a professional body. You can view or download our Recognition leaflets 3.3 Professional Engineering Institutions and 3.6 Institute of Mathematics and its Applications for further information.

Future availability

The details given here are for the course that starts in February 2012 when it will be available for the last time. A replacement course is planned.

Students also studied

Students who studied this course also studied at some time:

• Mathematical methods and models (MST209)
• Pure mathematics (M208)
• Graphs, networks and design (MT365)
• Mathematical modelling (MSXR209)
• Exploring mathematics (MS221)

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

Student Reviews

Distance learning

The Open University is the world’s leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you’re at home, at work or on the move. As an OU student you’ll be supported throughout your studies – your tutor will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

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