These notes were prepared for PHYCS-498MMA, a fairly traditional one-semester mathematical methods course for begining graduate students in physics. The emphasis is on linear operators and stresses the analogy between such operators acting on function spaces and matrices acting on finite dimensional spaces. The operator language then provides a unified framework for investigating ordinary differential equations, partial differential equations, and integral equations.
Although this mathematics is applicable to a wide range physical phenomena, the illustrative examples are mostly drawn from classical and quantum mechanics. Classical mechanics is a subject familiar to all physics students and the point being illustrated is immediately understandable without any further specialized knowledge. Similarly all physics students have studied quantum mechanics, and here the matrix/differential-operator analogy lies at the heart of the subject.
The mathematical prerequisites for the course are a sound grasp of undergraduate calculus (including the vector calculus needed for electricity and magnetism courses), linear algebra (the more the better), and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required.