University of Oxford Physics Year 2 - Advanced Mathematics (Mathematical Methods) Complete Lecture Notes & Revision Guide

Comprehensive handwritten lecture notes for the University of Oxford Year 2 Physics course "Mathematical Methods". These notes provide a structured and detailed treatment of the material covered throughout the course, including derivations, worked examples, key formulas, diagrams, and explanations designed to support both understanding and revision. Topics covered include: • Linear operators, linear maps, adjoint operators, self-adjoint operators, eigenvalues, eigenvectors, diagonalisation, spectral decompositions, and operator methods used throughout theoretical physics • Vector spaces, Banach spaces, Hilbert spaces, completeness, orthonormal bases, dual spaces, Riesz representation theorem, and the mathematical foundations of infinite-dimensional spaces • Functional analysis, inner-product spaces, norms, convergence, completeness, orthogonality, and applications to physical systems and differential operators • Fourier analysis including cosine series, sine series, real Fourier series, complex Fourier series, Parseval's theorem, convergence properties, and spectral decompositions of functions • Fourier transforms, inverse Fourier transforms, convolution theory, transform methods, translation and scaling properties, and applications to physical and mathematical problems • Orthogonal polynomial theory, Gram-Schmidt orthogonalisation, recurrence relations, Rodrigues formulae, generating functions, and expansions in orthogonal bases • Legendre polynomials, associated Legendre polynomials, spherical harmonics connections, multipole expansions, and applications to electrostatics and central-force problems • Laguerre polynomials and Hermite polynomials, generating functions, orthogonality relations, recurrence relations, and applications in mathematical physics and quantum mechanics • Linear ordinary differential equations, existence and uniqueness theorems, Wronskians, reduction of order, variation of constants, Green-function methods, and boundary-value problems • Sturm-Liouville theory, self-adjoint differential operators, orthogonal eigenfunction expansions, eigenvalue problems, completeness relations, and physical applications of Sturm-Liouville systems • Green functions for inhomogeneous differential equations, driven oscillators, boundary-value problems, and solution methods for linear operators • Laplace's equation, harmonic functions, Dirichlet and Neumann boundary conditions, coordinate transformations, curvilinear coordinates, and Laplacians in Cartesian, polar, cylindrical, and spherical coordinate systems The document consists of carefully organised handwritten notes taken during the course and is suitable for lecture review, tutorial preparation, revision, and exam preparation. Ideal for Oxford Physics students and anyone studying mathematical methods, differential equations, functional analysis, Fourier techniques, and the mathematical foundations of theoretical physics.