# TOPICS IN THEORETICAL PHYSICS - 2022/3

Module code: PHYM039

## Module Overview

This 15-credit M-Level module introduces important topics and techniques in theoretical physics that have a wide range of applications in many areas physics and engineering and which the students will not have met before. Both the mathematical techniques and their applications are covered at a level appropriate for Masters level students coming to the end of their degree and who should be able to pull many different ideas in theoretical physics together.

### Module provider

Mathematics & Physics

ROCCO Andrea (Biosc & Med)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 114

Lecture Hours: 26

Tutorial Hours: 10

Semester 2

None

## Module content

I. Functions of complex variables (12 hours)

• Continuity and differentiability

• The Cauchy-Riemann conditions

• Analyticity, singularities, poles.

• Complex integration

• Cauchy's theorem

• Residues and the residue theorem

• Taylor and Laurent series

• Laplace's equation in 2D and conformal mapping

• Laplace and Fourier transforms

• Dispersion relations

II. Calculus of Variations (9 hours)

• Integral principles in physics

• Principle of least action and other minimisation problems

• Lagrangian mechanics

• Euler-Lagrange equations

• Applications in configuration space

• Variation subject to constraints

• Extension to functions of more than one variable

• Isoperimetric problems and Lagrange multipliers

III. Integral Transforms (5 hours)

• Fourier transforms

• The Dirac delta function

• Laplace transforms

• Solving differential equations with Laplace transforms

• Convolutions

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework Coursework 30
Examination End of Semester Examination (Final Unit of Assessment) - 2 hours 70

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their understanding of mathematical techniques, their derivation (bookwork), their applications in physical examples, both of a type they have encountered in lectures and in the process of solving the examples in problem sheets and more original problems not encountered.

Thus, the summative assessment for this module consists of:

• A written end of semester Examination (Final Unit of Assessment)), two hours long, in which the student must answer 3 from 4 questions covering all areas of the course.

• A Coursework submission to be completed during the semester. The Coursework will be made available during week 8, with a submission deadline in Week 12.

Formative assessment and feedback

Regular feedback on previously taught material at the beginning of a lecture and discussion of problems and issues encountered in working through the problem sheets. These sheets will be discussed during the tutorial sessions. Model solutions to all problem sheet questions are made available after the students have had sufficient time to tackle them themselves. A revision class is set at the end of the module to go through past examination papers.

## Module aims

• To provide a sound grounding two important topics mathematical physics: Complex Variable Theory and Calculus of Variations. In particular, the basic theorems, methods and applications of functions of a complex variable, a range of advanced integration techniques and theorems and their applications in a range of physical examples and variational principles in classical mechanics leading to both Lagrangian and Hamiltonian formulations.

## Learning outcomes

 Attributes Developed 001 On successful completion of this module, students will have a solid understanding of complex variable theory. KC 002 Students will have a solid grounding in both Lagrangian and Hamiltonian mechanics. KC 003 Students will have a solid understanding of the principles of calculus of variations and its application in a range of physics problems. KC 004 Students will have a solid understanding of the definition and application of integral transforms in theoretical physics problems. KC 005 On successful completion of this module, students will have developed strong problem solving skills, and will be able to apply them across a wide range of physics problems. PT

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 teach students the practical and problem solving skills required to tackle mathematical physics questions.  As such the teaching strategy places equal importance on lecture content and practical skills. For each topic covered in lectures there is an equivalent tutorial session, in which students will have further learning opportunities to develop their hands-on skills on the topics discussed in the lectures.

The learning and teaching methods include:

• 26 h lectures
• 10 h tutorial sessions
• 114 hours of self-study

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: PHYM039

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Nuclear Astrophysics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Astronomy MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics with Quantum Technologies MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Physics MPhys 2 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 2022/3 academic year.