Mathematical Physics I

0. Introduction
1. Finite-dimensional vector spaces
• Vector spaces and linear operators
• Eigenvalues and eigenvectors
• Diagonalization and quadratic forms
2. Ordinary differential equations
• Linear equations with constant coefficients
• General first-order and second-order equations
• Power series solutions about a regular singular point
3. Sturm-Liouville theory
• Orthogonal polynomials
• Sturm-Liouville eigenvalue problems
• Fourier series expansions
4. Partial differential equations
• Wave and heat equations: one dimension
• Laplace equation: two dimensions
• Schrodinger equation: three dimensions
• Characteristics of second-order PDEs
5. Integral transforms
• Fourier transform and applications
• Laplace transform and applications