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Accessibility statement
Qualification dates
StartEnd
03 Oct 2026Jun 2027
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.
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.
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.
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:
We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or a remotely invigilated (proctored) exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.
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:
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.
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.
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