# Unit information: Calculus of Variations in 2022/23

Please note: It is possible that the information shown for future academic years may change due to developments in the relevant academic field. Optional unit availability varies depending on both staffing, student choice and timetabling constraints.

Unit name Calculus of Variations MATH30005 10 H/6 Teaching Block 1B (weeks 7 - 12) Dr. Slastikov Not open MATH20015 Multivariable Calculus and Complex Functions and MATH20101 Ordinary Differential Equations 2 None School of Mathematics Faculty of Science

## Unit Information

Unit Aims

To introduce students to the calculus of variation, and to illustrate its use in the solution of some elementary problems arising in mathematics and in physics.

Unit Description

Some simple problems typical of the subject are as follows: Amongst all the curves joining two given points on a manifold (such as the plane, or the sphere), find the one of shortest length; for a given perimeter, find the planar shape of largest area. Calculus of Variations develops the tools necessary to answer such questions; it is an important branch of optimisation in which the quantity (the functional) to be minimised depends on infinite-dimensional vectors that may for instance represent curves or surfaces. The subject has deep connections with various fields in the natural sciences, including differential geometry, ordinary and partial differential equations, materials science, mathematical biology, etc. It is one of the oldest and yet one of the most used tools for the investigation of problems involving the concept of "free energy". The aims of this course are (1) to cover the basics of the calculus of variations, including the one-variable case, and (2) to illustrate the theory by considering various applications arising in the natural sciences.

Relation to Other Units

The unit builds on Calculus 1, Linear Algebra and Geometry 1, and Multivariable Calculus. In terms of the mathematics involved, it is also closely connected with other units that pertain to differential and partial differential equations.

## Your learning on this unit

After taking this unit, students will:

1. know the basic techniques and results of the calculus of variations
2. be able to apply these techniques to solve some problems arising in other areas of science that can be formulated in terms of the minimisation of some functional.

## How you will learn

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## How you will be assessed

100% Timed, open-book examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH30005).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.