PHYS 516 Mathematical Methods has as its principal aim to develop a working knowledge of standard mathematical techniques that are useful to physicists and astronomers, especially for problems involving differential equations (ordinary and partial), integrals and their transforms, special functions, complex analysis, series, vectors, matrices and tensors. As such, it collects together many of the concepts and techniques that are discovered in specialized undergraduate mathematics courses, highlighting their application so as to facilitate learning throughout Rice's Physics and Astronomy graduate curriculum. It is therefore to a large extent a survey course, and complete mathematical rigor and depth is not possible in such a condensed format. Yet the material presented provides an excellent starting point for those interested in delving deeper into particular subfields.
While much of the course is analytic in character by necessity, to maintain contact with contemporaneous computational methods as research tools, there is also a modest numerical component to the course. This will address algorithm development and also provide practical opportunities at numerics for contained problems in applied mathematics. Topics covered include interpolation and curve fitting, iterative solution of ordinary differential equations, numerical integration and quadrature, and matrix decomposition methods. For a more extensive outline of the material covered, go to the Syllabus page.
PHYS 516 has no formal prerequisites and as such is fairly well self-contained. Yet it is implicitly assumed that students will have a working knowledge of general undergraduate mathematics including calculus, algebra, vectors, geometry and matrices, that which would naturally be acquired in a normal undergraduate degree majoring in physics or astrophysics.