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.
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| Start |
End |
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| 04 Oct 2025 |
Jun 2026 |
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| Registration closes 11/09/25 (places subject to availability) Click to register
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This module is expected to start for the last time in October 2028.
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).
Entry
If you’re studying this module on its own or as part of a postgraduate qualification, you should have:
- an honours degree in mathematics (minimum 2:2) or
- an honours degree in a subject with a high mathematical content (minimum 2:1).
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).
Preparatory work
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.
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.
Qualifications
M820 is an option module in our:
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.
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 forums.
Computing requirements
- Primary device – A desktop or laptop computer. It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
- Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
- Our OU Study app operates on supported versions of Android and iOS.
- Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
- Internet access – Broadband or mobile connection.
- Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.
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.
Assessment
The assessment details can be found in the facts box.
Students also studied
Students who studied this module also studied at some time:
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 2025.
We expect it to start for the last time in October 2028.
How to register
To register a place on this module return to the top of the page and use the Click to register button.
Regulations
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.