Methods of Mathematical Physics

This subject is not offered in 2014.

120 hours

One of:

Plus:

MAST30021 Complex Analysis may be taken concurrently with MAST30031 Methods of Mathematical Physics

For the purposes of considering request for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Student Support and Engagement Policy, academic requirements for this subject are articulated in the Subject Overview, Learning Outcomes, Assessment and Generic Skills sections of this entry.

It is University policy to take all reasonable steps to minimise the impact of disability upon academic study, and reasonable adjustments will be made to enhance a student's participation in the University's programs. Students who feel their disability may impact on meeting the requirements of this subject are encouraged to discuss this matter with a Faculty Student Adviser and Student Equity and Disability Support: http://services.unimelb.edu.au/disability

This subject builds on, and extends earlier, related undergraduate subjects with topics that are useful to applied mathematics, mathematical physics and physics students, as well as pure mathematics students interested in applied mathematics and mathematical physics. These topics include:

• Special functions: Spherical harmonics including Legendre polynomials and Bessel functions, including cylindrical, modified and spherical Bessel functions;
• Integral equations: Classification, Fourier and Laplace transform solutions, separable kernels, singular integral equations, Wiener-Hopf equations, and series solutions;
• Further vector analysis: Differential forms, covariant derivatives, and integrating p-forms;
• Further complex analysis: The Schwarz reflection principle, and Riemann-Hilbert problem.

On completion of this subject, students should:

• Be familiar with the most important special functions of mathematical physics, namely the spherical harmonics including Legendre polynomials and Bessel functions, and how they arise in solving the Laplace equation in different coordinate systems using separation of variables.
• Learn how a physical problem formulated as a differential equation and a set of boundary conditions can be recast as an integral equation, and how that may offer a way to solve the problem that is not available in the original formulation.
• Be familiar with the calculus of differential forms as a powerful set of tools that allows one to solve intricate physical problems that involve differentiation and integration of vector fields over curved manifolds efficiently and with maximal notational simplicity.
• Learn new, fundamental concepts, including the Schwarz reflection principle and the solution of the Riemann-Hilbert problem, that extend the basic concepts of a first subject in complex analysis to allow for the solution of more sophisticated physical problems.

Three written assignments of up to 60 pages due at regular intervals during the semester (30%); 3-hour written exam in the examination period (70%)

This subject is not available as a breadth subject.

In addition to skills that are useful in careers in science, engineering, commerce and education, students will develop useful generic skills that include:

• The problem-solving skills of identifying strategies to solve unfamiliar problems;
• The analytic skills of constructing and expressing logical arguments, and of working in abstract, general terms to clarify and improve available solutions;
• The time-management skills of meeting regular deadlines while balancing competing commitments.