 Course Details in 2018/19 Session

If you find any data displayed on this website that should be amended, please contact the Curriculum Management Team.

Module Title Mathematics for Physicists 2 Physics and Astronomy Physics & Astronomy 03 12497 Dr Smith Intermediate Level 20 Full Term Mathematics for Physicists 1A - (03 19751) None Lecture-66 hours Practical Classes and workshops-22 hours Guided independent study-112 hours Total: 200 hours 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 12497-01 : Exam : Exam (Centrally Timetabled) - Written Unseen (80%) 12497-02 : Examples sheets : Coursework (20%) 1 x 3 hour exam (80%) and continuous assessment (problem sheets) (20%) none