Introductory description

The module reviews the techniques of ordinary and partial differentiation and ordinary and multiple integration. It develops vector calculus and discusses the partial differential equations of physics (Term 1). The theory of Fourier transforms and the Dirac delta function are also covered. 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. The examples used to illustrate the module are drawn mainly from 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, chain rule, Taylor
   series. Change of coordinates for functions of more than one variable. Revision of vectors,
   the cross product and relationship of its modulus to parallelogram area. Vector areas for the
   context of surface integrals.
2. Multiple Integrals: Line, surface and volume integrals. Length of curves, surfaces of
   revolution. Change of variable and change of order.
3. Vector differentiation: Scalar and vector fields. Mathematical definition of grad and its
   physical meaning - examples. The divergence defined mathematically and a physical
   interpretation. Relationship with flux. The Laplacian, Solenoidal fields. Physics examples.
4. Gauss's Divergence Theorem: Demonstration of its validity for rectangular and
   cylindrical volumes. Examples.
5. Stokes’s Theorem: The curl and its interpretation. Conservative fields, irrotational fields.
   Stokes’s theorem and its derivation. Examples.
6. 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.
   Examples.
7. Fourier Series (revision): Representation for function f(x) defined -L to L; brief mention
   of convergence issues; real and complex forms; differentiation, integration; periodic
   extension
8. 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.
9. 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.

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