Outline:

This module aims to provide students with understanding of some advanced mathematical methods, and further experience and skills in mathematical manipulation and problem solving. Topics include: Partial Differential Equations, Series solution of Second-order Ordinary Differential Equations, Legendre Functions, Fourier Analysis, Lagrangian and Hamiltonian Mechanics and Special Relativity.

Aims:

This course aims to

Provide the remaining mathematical foundations for all the second- and third-year compulsory Physics and Astronomy courses.
	Prepare students for the second semester Mathematics option MATH6202.
	Give students practice in mathematical manipulation and problem solving at second-year level.
Teaching and Learning Methodology:

This course is delivered via weekly lectures supplemented by a series of problem solving tutorials and additional discussion.

In addition to timetabled lecture and PST hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

Partial Differential Equations [5 hrs]. Superposition principle for linear homogeneous partial differential equations; Separation of variables in Cartesian coordinates; Boundary conditions; One-dimensional wave equation; Derivation of Laplace's equation in spherical polar coordinates; Separation of variables in spherical polar coordinates; the Legendre differential equation; Solutions of degree zero.
	Series solution of Second-order Ordinary Differential Equations [5 hrs]. Series solutions: harmonic oscillator as an example; Ordinary and singular points; Radius of convergence; Frobenius method; Fuchs’ theorem; Applications to second-order differential equations.
	Legendre Functions [5 hrs]. Application of the Frobenius method to the Legendre equation; Range of convergence, Quantisation of the l index; Generating function for Legendre polynomials; Recurrence relations; Orthogonality of Legendre functions; Expansion in series of Legendre polynomials; Solution of Laplace's equation for a conducting sphere; Associated Legendre functions; Spherical harmonics
	Lagrangian and Hamiltonian Mechanics [4 hrs]. The Lagrangian and Lagrange's equation; Variation of Action; The Euler-Lagrange equations; Variational Principles; From the Lagrangian to the Hamiltonian; Derivation of Hamilton’s equations.
	Fourier Analysis [5 hrs]. Fourier series; Periodic functions; Derivation of basic formulae; Simple applications; Differentiation and integration of Fourier series; Parseval's identity; Complex Fourier series; Fourier transforms; Derivation of basic formulae and simple applications; Dirac delta function; Convolution theorem.
	Special Theory of Relativity [9 hrs]. Implications of Galilean transformation for the speed of light Michelson-Morley experiment; Einstein’s postulates; Derivation of the Lorentz transformation equations and the Lorentz transformation matrix; length contraction, time dilation, addition law of velocities, “paradoxes”; Four-vectors and invariants; Transformation of momentum and energy; Invariant mass; Conservation of four-momentum; Doppler effect for photons, threshold energy for pair production.