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Calculus of variations and advanced calculus

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

Qualifications

M820 is an optional module in our:

This module can also count towards M03, which is no longer available to new students.

Module

Module code
M820
Credits

Credits

  • 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.
30
Study level
To enable you to make international comparisons, the information provided shows how OU postgraduate modules correspond to the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ).
OU Postgraduate
FHEQ 7
Study method
Distance learning
Find out more in Why the OU?
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

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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.

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

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 us 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 the 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.

Course work includes

4 Tutor-marked assignments (TMAs)
Examination
No residential school

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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 2019. We expect it to start for the last time in October 2024.

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.

    Entry requirements

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

    You should have at least a 2.2 honours degree in mathematics or a 2.1 honours degree in a subject 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, 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.

    Register

    Start End Fee Register
    03 Oct 2020 Jun 2021 Not yet available

    Registration closes 10/09/20 (places subject to availability)

    Register
    This module is expected to start for the last time in October 2024.

    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 2019. We expect it to start for the last time in October 2024.

    Additional costs

    Study costs

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

    Study weekend

    This module normally includes an optional study weekend. For each day you choose to attend, you must pay an additional charge of around £60 to cover tuition and refreshments during the day. You’ll pay this charge when you book, after you’ve registered on the module. You must also pay for your travel to and from the venue and your accommodation if you need it.

    Study materials

    What's included

    Study notes and other printed materials.

    Computing requirements

    A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

    Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

    A desktop or laptop computer with either:

    • Windows 7 or higher
    • Mac OS X 10.7 or higher

    The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

    To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

    Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

    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. 

    If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Disability support website to find more about what we offer.