Calculus of variations and advanced calculus

The Calculus of Variations is a crucial mathematical tool in optimisation concerned with integrals (functionals) taken over admissible paths. The paths are varied, leading to the Euler–Lagrange differential equation for a stationary path. Euler, Lagrange, Jacobi, and Noether, amongst others, developed the theory, which has important applications in modern physics, engineering, biology, and economics. You’ll develop your knowledge of the fundamental theory and the advanced calculus tools required to find and classify stationary paths. Topics include functionals, Gâteaux differential, Euler–Lagrange equation, First-integral, Noether’s Theorem, Second variation/Jacobi equation, and Sturm–Liouville systems.


Module code


  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
Study level
Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU postgraduate modules correspond to these frameworks.
OU Postgraduate
Study method
Distance learning
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

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.

This module 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 module 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 module should enhance your skills in understanding complex mathematical texts, communicating solutions to problems clearly and interpreting mathematical results in real-world terms.

This module and Analytic number theory 1 (M823) are entry-level modules for the MSc in Mathematics (F04), and normally you should have studied one of them before progressing to the intermediate and advanced intermediate modules in the degree.

Note you must complete this module if you wish to take the ‘Variational methods applied to eigenvalue problems’ topic for your Dissertation in mathematics (M840).

Teaching and assessment

Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

  • Marking your assignments (TMAs) and providing detailed feedback for you to improve.
  • Guiding you to additional learning resources.
  • Providing individual guidance, whether that’s for general study skills or specific module content.

The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.


The assessment details can be found in the facts box.

Course work includes

4 Tutor-marked assignments (TMAs)

Future availability

Calculus of variations and advanced calculus (M820) starts once a year – in October.

This page describes the module that will start in October 2024.

We expect it to start for the last time in October 2028.


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.

Entry requirements

You must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention.

You should normally have a minimum of a 2:2 honours degree in mathematics or a 2:1 honours degree in a subject with a high mathematical content. If you do not have such a qualification, your application will be considered and you may be asked to complete an entry test. Non-graduates will not normally be admitted to M820 unless as part of another Open University qualification. Students already registered for a qualification of which M820 is a constituent part will normally be admitted to M820.

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 the discontinued module MST209), and/or Pure mathematics (M208) (or the discontinued module M203) and some study of mathematics at third year honours level should provide adequate preparation.

Whatever your background, you should assess your suitability for this MSc in Mathematics module by trying our diagnostic quiz.

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 module, please speak to an adviser.


Start End England fee Register
05 Oct 2024 Jun 2025 £1360.00

Registration closes 05/09/24 (places subject to availability)

This module is expected to start for the last time in October 2028.

Future availability

Calculus of variations and advanced calculus (M820) starts once a year – in October.

This page describes the module that will start in October 2024.

We expect it to start for the last time in October 2028.

Additional costs

Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

Study events

This module may have an optional in-person study event. We’ll let you know if this event will take place and any associated costs as soon as we can.

Ways to pay for this module

We know there’s a lot to think about when choosing to study, not least how much it’s going to cost and how you can pay.

That’s why we keep our fees as low as possible and offer a range of flexible payment and funding options, including a postgraduate loan, if you study this module as part of an eligible qualification. To find out more, see Fees and funding.

Study materials

What's included

You’ll be provided with printed module notes covering the content of the module, 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:

  • a week-by-week study planner
  • course-specific module materials
  • audio and video content
  • assessment details and submission section
  • online tutorial access
  • access to student and tutor group forums.

Computing requirements

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

If you have a disability

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.

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