PX27515 Mathematical Methods for Physicists
Introductory description
The module reviews the techniques of ordinary and partial differentiation and ordinary and multiple integration. It develops vector calculus (Term 1). The theory of Fourier transforms and the Dirac delta function are also covered. Fourier transforms are used to represent functions on the real line using linear combinations of sines and cosines and are the basis for describing many interference and diffraction phenomena in optics (Term 2).
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.
 Revision: Functions of more than one variable; partial differentiation; the chain rule; change of coordinates; total differential; directional derivative
 Lagrange Multipliers: Maxima/minima of a function subject to a constraint
 Multiple Integrals: Integration and choice of coordinate system: basis vectors; area and volume elements; Jacobians; path, area and volume integrals
 Vector calculus: Scalar and vector fields; grad, div, curl operators and their physical interpretation
 Vector Integration: Green’s Theorem in the plane; Divergence Theorem in 2D and physical interpretation
 Stokes’s Theorem: The curl and its interpretation. Conservative fields, irrotational fields.
Stokes’s theorem and its derivation  PDEs: The wave equation, Poisson's equation, Schrödinger's equation. The diffusion equation and Fick's law. The role of boundary conditions. Separation of variables
 Fourier Transforms. Definition of Fourier transform, case of Gaussian and Lorentzian. Delta function and properties, Fourier's Theorem. Convolutions, example of instrument resolution, convolution theorem
 Interference and diffraction phenomena: the HuygensFresnel principle. Criteria for Fraunhofer and Fresnel diffraction. Fourier relationship between an object and its diffraction pattern. Convolution theorem demonstrated by diffraction patterns. Fraunhofer diffraction for single, double and multiple slits. Diffraction at a circular aperture; the Airy disc. Image resolution, the Rayleigh criterion and other resolution limits
Learning outcomes
By the end of the module, students should be able to:
 Deal with multiple integrals and know how to evaluate the length of a curve and the volume of a three dimensional object
 Define and calculate the gradient, divergence and curl of a vector field and understand Gauss’s and Stokes’ theorems
 Define a partial differential equation and solve the wave and diffusion equations using the method of separation of variables
 Represent simple functions using Fourier transforms
 Demonstrate a good understanding of diffraction and interference phenomena and solve problems involving Fraunhofer diffraction
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: Vector calculus, separation of variables, Fourier transforms and their application to describe diffraction
Transferable skills
Analytical, communication, problemsolving, selfstudy
Study time
Type  Required 

Lectures  40 sessions of 1 hour (27%) 
Other activity  20 hours (13%) 
Private study  90 hours (60%) 
Total  150 hours 
Private study description
Working though lecture notes, solving problems, wider reading, discussing with others taking the module, revising for exam, practising on past exam papers
Other activity description
20 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  

Assessed Coursework  20%  
Inperson Examination  80%  
Answer 4 questions

Assessment group R1
Weighting  Study time  

Inperson Examination  Resit  100%  
Answer 4 questions

Feedback on assessment
Personal tutors, group feedback
Courses
This module is Core for:
 Year 2 of UPXAF304 Undergraduate Physics (BSc MPhys)
 Year 2 of UPXAF300 Undergraduate Physics (BSc)
 Year 2 of UPXAF303 Undergraduate Physics (MPhys)
 Year 2 of UPXAF3N1 Undergraduate Physics and Business Studies
 Year 2 of UPXAF3F5 Undergraduate Physics with Astrophysics (BSc)
 Year 2 of UPXAF3FA Undergraduate Physics with Astrophysics (MPhys)
 Year 2 of UPXAF3N2 Undergraduate Physics with Business Studies