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 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. Revision: Functions of more than one variable; partial differentiation; the chain rule; change of coordinates; total differential; directional derivative
2. Lagrange Multipliers: Maxima/minima of a function subject to a constraint
3. Multiple Integrals: Integration and choice of coordinate system: basis vectors; area and volume elements; Jacobians; path, area and volume integrals
4. Vector calculus: Scalar and vector fields; grad, div, curl operators and their physical interpretation
5. Vector Integration: Green’s Theorem in the plane; Divergence Theorem in 2D and physical interpretation
6. Stokes’s Theorem: The curl and its interpretation. Conservative fields, irrotational fields.
   Stokes’s theorem and its derivation
7. 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
8. 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
9. Interference and diffraction phenomena: the Huygens-Fresnel 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, problem-solving, self-study