Introductory description

The module covers the theory of Fourier transforms and the Dirac delta function. The module also introduces Lagrange multipliers, co-ordinate transformations and cartesian tensors illustrating them with examples of their use in physics.

Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. 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 far-field limit are the Fourier transforms of the "diffracting" object. 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 the product of the Fourier transform of repeated delta functions and the Fourier transform for a single copy of the motif.

Module web page

Module aims

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.

1. Fourier Series:
   Representation for function f(x) defined -L to L; mention of convergence issues; real and complex forms; differentiation, integration; periodic extensions
2. 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
3. Interference and diffraction phenomena:
   Interference and diffraction, the Huygens-Fresnel 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
4. 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
5. Vectors and Coordinate Transformations:
   Summation convention, Kronecker delta, permutation symbol and use for representing vector
   products. Revision of cartesian coordinate transformations. Diagonalizing quadratic forms
6. Tensors:
   Physical examples of tensors: mass, current, conductivity, electric field
7. 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, problem-solving, self-study