Postgraduate

Calculus of variations and advanced calculus

On this page

• What you will study
• Entry
• Qualifications
• If you have a disability or additional requirement
• Study materials
• Teaching and assessment
• Students also studied
• Future availability
• How to register

The calculus of variations, which plays an important role in both pure and applied mathematics, dates from the time of Newton. Development of the subject started mainly with the work of Euler and Lagrange in the eighteenth century and still continues. This course, which for most students begins the MSc in Mathematics programme, develops the theory of the calculus of variations. It also introduces other topics including the calculus of functions of several real variables; and ideas of convergence, particularly of sequences of functions and normed vector spaces – a working knowledge of which is required both in this and other modules in the MSc in Mathematics programme.

Course facts

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What you will study

Problems such as the determination of the shortest curve between two points on a given smooth surface and the shapes of soap films, are most easily formulated using ideas from the calculus of variations. The calculus of variations also provides useful methods of approximating solutions of linear differential equations; furthermore, variational principles also provide the theoretical underpinning for the coordinate-free formulations of many laws of nature.

M820 provides an introduction to the central ideas of variational problems, as well as some of the mathematical background necessary for the subject. Many of the simple applications of calculus of variations are described and, where possible, the historical context of these problems is discussed.

The course also contains some more advanced material, such as an analysis of the second variation and of discontinuous solutions; it ends with a discussion of the general properties of the solutions of an important class of linear differential equations, namely Sturm-Liouville systems. Throughout, the emphasis is on the mathematical ideas and one aim is to illustrate the need for mathematical rigour. Applications will be discussed but you are not expected to have a detailed understanding of the underlying physical ideas.

You will learn

Successful study of this course should enhance your skills in understanding complex mathematical texts, communicating solutions to problems clearly and interpreting mathematical results in real-world terms.

Entry

You must be registered for the MSc in Mathematics or for another qualification towards which the course can count. You should have at least second-class honours in mathematics or in another degree with a high mathematical content, such as engineering or theoretical physics. In exceptional circumstances applicants without such a qualification will be considered, although non-graduates will not normally be admitted to the MSc programme. You should have a sound working knowledge of undergraduate calculus and have studied the elements of vector spaces. Mathematical methods and models (MST209) (or the discontinued MST207), and/or Pure mathematics (M208) (or the discontinued M203) and some study of mathematics at third year honours level should provide adequate preparation.

All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

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

Qualifications

M820 is an optional module in our:

• Postgraduate Certificate in Mathematics (C90)
• Postgraduate Diploma in Mathematics (E23)
• MSc in Mathematics (F04)

Some postgraduate qualifications allow study to be chosen from other subject areas. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

If you have a disability or additional requirement

The material contains small print and diagrams, which may cause problems if you find reading text difficult. Written transcripts of any audio components and Adobe Portable Document Format (PDF) versions of printed material are available. Some Adobe PDF components may not be available or fully accessible using a screen reader. 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 study 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, financial support and 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

Study notes and other printed materials.

You will need

We recommend that you have access to the internet at least once a week during the course and would like to point out that vital material, such as your assignments, will be delivered online.

Computing requirements

This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.

You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.

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, and whom you can ask for advice and guidance. 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 can be found in the facts box above.

You will be expected to submit your tutor-marked assignments (TMAs) online using a special maths eTMA processor, which is used in place of the main eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper.

You will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of your answers to produce an electronic TMA would take you an unacceptably long time.

Assessment is an essential part of the teaching, so you are expected to complete it all. But if you unavoidably miss or do badly in an assignment, some courses allow you a ‘substitution score’, calculated as a weighted average of all your scores for the course. In M820 this rule can apply to one assignment only. You will be given more detailed information when you begin the course.

Students also studied

Students who studied this course also studied at some time:

• Analytic number theory I (M823)
• Nonlinear ordinary differential equations (M821)
• Applied complex variables (M828)
• Fractal geometry (M835)
• Advanced mathematical methods (M833)

Future availability

The details given here are for the course that starts in February 2012. We expect it to then be available in February 2013, October 2013 and thereafter once a year in October.

How to register

To register a place on this course return to the top of the page and use the Click to register button. For more information and advice about registration see OU Study Explained.

Study explained

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Study explained

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Student Reviews

This is the maths course that most postgrad students end up doing first. As such it is well presented. I ...
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Good course and great Tutor. Exam was fairly written but marked a bit harshly. Recommended. ...
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