PX275 Mathematical Methods for Physicists

Weighting: 15 CATS

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)

This module is not available to Maths or Maths/Physics students.

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

Objectives:
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
• be able to define and calculate the gradient, divergence and curl of a vector field and understand Gauss’s and Stokes’ theorems
• be able to define a partial differential equation and to be able to solve the wave and diffusion equations using the method of separation of variables
• be able to represent simple, appropriate functions using Fourier transforms
• possess a good understanding of diffraction and interference phenomena and be able to solve problems involving Fraunhofer diffraction.

Syllabus:

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.
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.
Multiple Integrals: (some revision) Line, surface and volume integrals. Length of curves, surfaces of
revolution. Change of variable and change of order.
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.
Gauss's Divergence Theorem: Demonstration of its validity for rectangular and cylindrical volumes. Examples.
Stokes’s Theorem: The curl and its interpretation. Conservative fields, irrotational fields. Stokes’s theorem and its derivation. Examples.

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.
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.
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.
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.

Commitment: 40 Lectures + 20 All Class Sessions

Assessment: 2 hour examination (80%) + class tests/coursework (20%)