# CALCULUS OF VARIATIONS - 2024/5

Module code: MATM059

## Module Overview

This module is an introduction to classical and modern methods in the Calculus of Variations. Variational methods have many applications in Mathematics, Physics and Engineering. Examples will be included in the module to illustrate the theory of the Calculus of Variations in practice.

This module complements module material in MAT3004 Introduction to Function Spaces, and MAT3008 Lagrangian & Hamiltonian Dynamics, but these modules are not pre-requisite.

### Module provider

Mathematics & Physics

BEVAN Jonathan (Maths & Phys)

## Overall student workload

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 1

None.

## Module content

Indicative content includes:

• Variational symmetries and conservation laws.

• Weak derivatives, the fundamental lemma of the Calculus of Variations, and the spaces L∞(a,b) and W1,∞(a,b)=Lip(a,b).

• The weak Euler-Lagrange equation.

• Convexity and the Legendre condition.

• Solving the Euler-Lagrange equation. Examples including the brachistochrone, minimal surfaces of revolution and the catenary.

• Examples of problems with no minimizer.

• Weak local minimizers: sufficient conditions for a solution of the Euler-Lagrange equation to be a minimizer.

• Constrained variational problems.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 mins) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions, theorems and propositions in the theory of Calculus of Variations.

• Experience of the classical and modern methods used to interpret, understand and solve variational problems.

Thus, the summative assessment for this module consists of:

• One in-semester test corresponding to all Learning Outcomes 1 to 4.

• A synoptic examination corresponding to all Learning Outcomes 1 to 4.

Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in office hours.

## Module aims

• Equip students with basic tools in the Calculus of Variations.
• Develop students' understanding of the applications of Calculus of Variations in Mathematics and other scientific disciplines.

## Learning outcomes

 Attributes Developed 001 Students will be able to identify accurately conditions under which the Euler-Lagrange equation can be derived, and to derive this equation from first principles under these conditions. KC 002 Students will recognize and determine appropriate function spaces in which to set given variational problems, and apply prior knowledge of ordinary differential equations to solve them. KC 003 Students will understand and apply theorems concerning weak local minimizers to given variational problems. KC 004 Students will be able to construct simple proofs similar to those encountered in the module. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Introduce students to the theory of the Calculus of Variations, including proofs of standard results.

• Provide students with experience of methods used to understand and solve variational problems.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM059

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM059 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MATM059 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to evaluate complex variational problems, break them into manageable components, and apply rigorous mathematical techniques and logical reasoning to arrive at solutions. These are highly sought after skills in any profession.

Global and Cultural Capabilities: Students enrolled in MATM059 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MATM059 is a module which demands a rigorous approach to the Calculus of Variations to which students will learn to adapt. Students will gain skills in analysing a variety of variational problems using lateral thinking, and will complete assessments which challenge them and build resilience.

Sustainability: Variational analysis can often be used to find the most efficient solution to a given problem, including those relating to sustainability. For instance, the design of the Wirtz pump, which is a `water-powered water pump`, can be tuned to maximize its output pressure, and thereby its efficiency. The module equips students with skills which will enable them to identify and solve the problems of least cost/great efficiency of the future.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 1 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.