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
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.