PX2767.5 Methods of Mathematical Physics
Introductory description
The module starts with the theory of Fourier transforms and the Dirac delta function. Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. Fourier transforms are a powerful tool in physics and applied mathematics. A Fourier transform will turn a linear differential equation with constant coefficients into a nice algebraic equation which is in general easier to solve.
The module explains why diffraction patterns in the farfield limit are the Fourier transforms of the "diffracting" object. It then looks at diffraction generally. The case of a repeated pattern of motifs illustrates beautifully one of the most important theorems in the business  the convolution theorem. The diffraction pattern is simply the product of the Fourier transform of repeated delta functions and the Fourier transform for a single copy of the motif. The module also introduces Lagrange multipliers, coordinate transformations and cartesian tensors illustrating them with examples of their use in physics.
Module aims
To teach mathematical techniques needed by second, third and fourth year physics modules
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Fourier Series:
Representation for function f(x) defined L to L; mention of convergence issues; real and complex forms; differentiation, integration; periodic extensions  Fourier Transforms:
Fourier series when L tends to infinity. Definition of Fourier transform and standard examples  Gaussian, exponential and Lorentzian.
Domains of application: (Time t  frequency w), (Space x  wave vector k).
Delta function and properties, Fourier's Theorem.
Convolutions, example of instrument resolution, convolution theorem  Interference and diffraction phenomena:
Interference and diffraction, the HuygensFresnel principle. Criteria for Fraunhofer and Fresnel
diffraction. Fraunhofer diffraction for parallel light. Fourier relationship between an object and
its diffraction pattern. Convolution theorem demonstrated by diffraction patterns. Fraunhofer
diffraction for single, double and multiple slits. Fraunhofer diffraction at a circular aperture; the
Airy disc. Image resolution, the Rayleigh criterion and other resolution limits. Fresnel diffraction,
shadow edges and diffraction at a straight edge  Lagrange Multipliers:
Variation of f(x,y) subject to g(x,y) = constant implies grad f parallel to grad g. Lagrange
multipliers. Example of quadratic form  Vectors and Coordinate Transformations:
Summation convention, Kronecker delta, permutation symbol and use for representing vector
products. Revision of cartesian coordinate transformations. Diagonalizing quadratic forms  Tensors:
Physical examples of tensors: mass, current, conductivity, electric field  Stokes’ Theorem (Worksheet):
Line integrals, circulation; curl in Cartesians; statement and proof of Stokes’ theorem for triangulations; dependence on region of integration and vector field; gradient, irrotational, solenoidal and incompressible vector fields; applications drawn from electromagnetism, fluid dynamics, condensed matter physics, differential geometry
Learning outcomes
By the end of the module, students should be able to:
 Represent functions in terms of Fourier series and Fourier transforms
 Demonstrate understanding of diffraction and interference phenomena
 Minimise/maximise functions subject to constraints using Lagrange multipliers
 Express vectors in different coordinate systems, recognise some physical examples of tensors
 Derive, and apply in physical contexts, Stokes’s theorem
Indicative reading list
KF Riley,MP Hobson and SJ Bence, Mathematical Methods for Physics and Engineering: a
Comprehensive Guide, Wadsworth, H D Young and R A Freedman, University Physics 11th
Edition, Pearson.
View reading list on Talis Aspire
Subject specific skills
Mathematical methods including: Fourier transforms and their application to describe diffraction, Lagrange multipliers.
Skills in modelling, reasoning, thinking
Transferable skills
Analytical, communication, problemsolving, selfstudy
Study time
Type  Required 

Lectures  20 sessions of 1 hour (27%) 
Other activity  10 hours (13%) 
Private study  45 hours (60%) 
Total  75 hours 
Private study description
Working through lecture notes, solving problems, wider reading, discussing with others taking the module, revising for exam, practising on past exam papers
Other activity description
10 example classes
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D2
Weighting  Study time  

Class Tests and Assessed Coursework  20%  
Class Tests/Assessed Coursework 

2 hour online examination (April)  80%  
Answer 2 questions

Feedback on assessment
Personal tutors, group feedback
Courses
This module is Core for:
 Year 2 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 2 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
This module is Option list B for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year