Lecturer: Rudolf Roemer

Weighting: 10 CATS

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.

Aims: 
To teach mathematical techniques needed by second, third and fourth year physics modules.

Objectives:
Students should:

• Be able to represent simple functions in terms of Fourier series and Fourier transforms
• Possess a good understanding of diffraction and interference phenomena
• Be able to minimise/maximise simple functions subject to constraints using Lagrange multipliers
• Be able to express vectors in different coordinate systems, recognise some physical examples of tensors
• Be familiar with the derivation, and with some applications in physical contexts, of Stokes’s theorem

Syllabus:
1. Fourier Series:
Representation for function f(x) defined -L to L; brief 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.

Worksheet
7. Stokes’ Theorem: 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

Commitment: 20 Lectures + 10 Examples Classes

Assessment: 1.5 hour written examination (80%) + in-class tests/assessed coursework (10%) + 1 worksheet on Stokes's Theorem (10%)

This module has a home page.

Recommended texts: KF Riley, MP Hobson and SJ Bence, Mathematical Methods for Physics and Engineering: a
Comprehensive Guide, CUP; H D Young and R A Freedman, University Physics (11th
Edition), Pearson.