We discuss basic mathematics indispensable for every student of physics.
The course is co-listed with mathematics 6802.
Synopsis: We discuss advanced mathematical methods absolutely indispensable for any research project in either experimental or theoretical physics. We start with complex variable theory and complex contour integration, with an emphasis on the indispensable practical knowledge regarding the application of these concepts “for serious physics research”. We continue with a discussion of coordinate transformations, based of course on matrix representations of the coordinate transformations, and basic vector analysis. Topics will include, among others things, Stokes’s theorem in both differential as well as integral form, and transformations into curvilinear coordinates, as well as Christoffel symbols and the different forms of gradient and divergence operators, in different coordinate systems (e.g., spherical and cylindrical). Tensors will be discussed. The separation ansatz for the solution of partial differential equations will be discussed and illustrated. A discussion of the most indispensable special functions necessary for physics research follows: orthogonal functions and solutions to ordinary differential equations, Gamma function, hypergeometric, confluent hypergeometric, Legendre, Laguerre, and Bessel functions, and Hermite polynomials. The course may end with a discussion of Green functions in one dimension, and possibly higher dimensions, or, interactively, with discussions on any topics where students feel the need for a refreshment of their mathematical background knowledge. The necessity of diligence, and the presence of pitfalls in the mathematical discussions, will be highlighted.
Here is the Syllabus for 6403.
Here is Exercise #1.
Here is Exercise #1X.
Here is an (evolving) scriptum (2022/01/26).