Problems such as determining 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.
This module provides an introduction to the central ideas of variational problems and some of the mathematical background necessary for the subject. It describes many of the simple applications of calculus of variations and discusses, where possible, the historical context of these problems.
The module also contains more advanced material, such as an analysis of the second variation and 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.
Successful study of this module should enhance your skills in understanding complex mathematical texts, communicating solutions to problems clearly and interpreting mathematical results in real-world terms.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
You’ll receive printed module notes covering the module's content, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques. In addition, you will have a printed handbook.
You’ll also have access to a module website, which includes:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M820 is an option module in our:
Sometimes you cannot count a module towards a qualification if you have already counted another module with similar content. To verify any excluded combinations with this module, check with an adviser before registering.
Calculus of variations and advanced calculus (M820) starts once a year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2032.
As a student of The Open University, you should be aware of the content of the academic regulations, which are available on our Student Policies and Regulations website.
If you’re studying this module on its own or as part of a postgraduate qualification, you should have:
We’ll consider all applications but may ask you to complete an entry test.
If you’re studying towards our undergraduate Master of Engineering (M04), you must have passed one of the following (minimum Grade 3 pass recommended):
If you’re studying towards our undergraduate Master of Physics (M06), you must have passed all your Stage 3 modules (minimum Grade 3 passes recommended).
You should have a sound working knowledge of undergraduate calculus and have studied the elements of vector spaces. Mathematical methods, models and modelling (MST210) (or equivalent) or Mathematical methods (MST224), and some study of mathematics at third-year honours level should provide adequate preparation.
Whatever your background, you should assess your suitability with our diagnostic quiz.
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.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
| Start | End | Register by | England fee |
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| 03 Oct 2026 | 30 Jun 2027 | 10 Sep 2026 | Not yet available* |
| *This start date is open for pre-booking, which means you can reserve your place ahead of the fees being confirmed. We’ll publish updated 2026/27 fees and funding information on the 25th of March. |
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
This module may have an optional study weekend. There will be a charge to attend in person or online. If attending in person, you’ll also need to pay for your travel, and accommodation if you need it. More details will be available after the module has started.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in England.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate tuition fee loans in Northern Ireland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate loans in Scotland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in Wales.
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As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average, the processing time for a joint loan application is five working days from receipt of the required documentation.
Read more about Open University Student Budget Accounts (OUSBA).
Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.
Over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.
You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.
We accept American Express, Mastercard, Visa and Visa Electron.
We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).
We know that sometimes you may want to combine payment options. For example, you may get support from your employer to pay part of your tuition fee and pay the remainder by credit or debit card.
For more information about combining payment options, contact an adviser.
Please note: your permanent address/domicile will affect your fee status and, therefore, the fees you are charged and any financial support available to you. The fee information provided here is valid for modules starting before 31 July 2026. Fees typically increase annually. For further information about the University's fee policy, visit our Fee Rules.
