Mathematics is the natural language in which physics is expressed, and it is therefore important that any working physicist should be fluent in it. This module is the last compulsory mathematics module in all except theory programmes, and contains the remaining core mathematics needed for all physics modules in future years. The module is roughly divided into two pieces: calculus (~15 credits) & matrices and linear algebra (~5 credits). These are further sub-divided as shown below:

1. Calculus: Vector Calculus
 Distributions
 Fourier Series
 Fourier Transforms
 Partial Differential Equations

2. Matrices and Linear Algebra: Basic Matrix Algebra
 Eigenvalues and Eigenvectors

Most of the equations encountered in physics are linear partial differential equations (p.d.e.’s), and the calculus section is focussed on the techniques needed to formulate and solve these equations, using examples from physics. Vector
calculus is the language in which both the Maxwell equations of electromagnetism, and the Navier-Stokes equation of fluid mechanics are written; the Laplacian derived in vector calculus is also at the heart of most p.d.e.’s found in physics.
Fourier series and Fourier transforms involve splitting periodic and non-periodic functions respectively into their frequency components. They are extensively used in many areas of physics, including optics and wave phenomena, classical mechanics, and quantum mechanics. Finally the section on p.d.e.’s introduces the method of separation of variables, which reduces a p.d.e. to a set of ordinary differential equations (o.d.e.’s); the solution of such odes is largely beyond the scope of this module, and is a main part of the Eigenphysics module.

Matrices arise in the analysis of many physical situations; examples include
moments of inertia in rigid bodies, normal modes in coupled oscillators, Lorentz
transformations in special relativity, and perturbation theory in quantum mechanics.
In this module the properties of matrices are developed from basic algebra to the
solution of eigenvalue problems, using appropriate examples from physics.

By the end of the module the student will be able to:

• Calculate the gradient, divergence, curl, and Laplacian in cartesian or otherorthogonal coordinate systems
• Prove the standard vector calculus identities
• Test a coordinate system to see if it is orthogonal
• Evaluate line, surface and volume integrals
• State the divergence and Stokes theorems, and verify them in a particular situation
• Define and use Heaviside and Dirac delta functions, including delta functions of non-trivial argument
• Derive the formulae for real and complex Fourier series, including Parseval’s theorem
• Represent a given periodic function as a Fourier series
• Write down the formulae for Fourier transform, inverse Fourier transform and Parseval’s theorem
• Write down the formulae for Fourier sine and cosine transforms, and higher dimensional Fourier transforms
• Evaluate the Fourier transform of simple functions
• Explain the role of the Fourier transform in Fraunhofer diffraction, and use it to evaluate diffraction patterns
• Separate variables in partial differential equations
• Solve partial differential equations using a variety of Fourier techniques
• Add, multiply and find the determinant and inverse of a matrix
• Use Gaussian elimination to solve simultaneous linear equations
• Diagonalise a small matrix, determining both the eigenvalues and eigenvectors
• Define and identify symmetric, anti-symmetric, Hermitian, anti-Hermitian, orthogonal, unitary, and normal matrices