- UCAS course code
- F305
- UCAS institution code
- M20
Master of Physics (MPhys)
MPhys Physics
Join a physics Department of international renown that offers great choice and flexibility, leading to master's qualification.
- Typical A-level offer: A*A*A including specific subjects
- Typical contextual A-level offer: A*AA including specific subjects
- Refugee/care-experienced offer: AAA including specific subjects
- Typical International Baccalaureate offer: 38 points overall with 7,7,6 at HL, including specific requirements
Fees and funding
Fees
Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £36,500 per annum. For general information please see the undergraduate finance pages.
Policy on additional costs
All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).
Scholarships/sponsorships
The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.
For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .
Course unit details:
Complex Variables and Vector Spaces
Unit code | PHYS20672 |
---|---|
Credit rating | 10 |
Unit level | Level 2 |
Teaching period(s) | Semester 2 |
Available as a free choice unit? | No |
Overview
Complex Variables and Vector Spaces (M)
Pre/co-requisites
Unit title | Unit code | Requirement type | Description |
---|---|---|---|
Dynamics | PHYS10101 | Pre-Requisite | Compulsory |
Vibrations & Waves | PHYS10302 | Pre-Requisite | Compulsory |
Mathematics 2 | PHYS10372 | Pre-Requisite | Compulsory |
Follow-up units:
PHYS30201
PHYS30672
Aims
To introduce students to complex variable theory and some of its many applications. To introduce the concept of vector space and some ideas in linear algebra.
Learning outcomes
On completion successful students will:
- determine whether or not a given function of a complex variable is differentiable;
- use conformal mappings of the complex plane to solve problems in 2D electrostatics, fluid flow and heat flow;
- construct the Taylor-Laurent series for functions that are analytic in an annular region of the complex plane;
- find the location and nature of the singularities of a function and determine the order of a pole and its residue;
- use the residue theorem to evaluate integrals of functions of a complex variable, and identify appropriate contours to assist in the summation of series and the evaluation of real integrals;
- find an orthonormal basis for a given vector space;
- define the adjoint of a linear operator and determine whether a given operator is Hermitian and/or unitary;
- use methods from this and prerequisite units to solve previously unseen problems in linear algebra, using Dirac’s notation where appropriate.
Syllabus
1. Complex numbers (8 lectures)
Functions of complex variable
Functions as mappings
Differentiation, analytic functions and the Cauchy-Riemann equations
Conformal mappings
Solutions of 2D Laplace equation in Physics
Integration in the complex plane
2. Contour integration (8 lectures)
Cauchy’s Theorem
Cauchy’s integral formulae
Taylor and Laurent Series
Cauchy’s Residue Theorem
Real integrals and series
3. Vector Spaces (7 lectures)
Abstract vector spaces
Linear independence, basis and dimensions, representations
Inner products
Linear operators
Hermitian and unitary operators
Eigenvalues and eigenvectors
Assessment methods
Method | Weight |
---|---|
Written exam | 100% |
Feedback methods
Feedback will be given orally at examples classes during the semester. Solutions to the problem sheets will be provided electronically.
Recommended reading
Spiegel, M.R. et al. Schaum’s Outline of complex variables, 2nd Ed. (Schaum’s Outlines, 2009)
Riley, K.F. Hobson M.P. & Bence S.J. Mathematical Methods for Physics and Engineers, (CUP, 2006)
Boas, M.L. Mathematical Methods for Physical Sciences, 3rd edn. (Wiley, 2006)
Arkfen, G.B. and Weber, H.J. Mathematical Methods for Physicists, 6th Ed. (Academic Press, 2005
Study hours
Scheduled activity hours | |
---|---|
Assessment written exam | 1.5 |
Lectures | 24 |
Seminars | 6 |
Independent study hours | |
---|---|
Independent study | 68.5 |
Teaching staff
Staff member | Role |
---|---|
Michael Godfrey | Unit coordinator |