Mathematics 3J04: Plan of Lectures (Fall, 2000)

Note: Numbers in brackets refer to sections of the main textbook. The filled discs denote the topics covered by the lectures (updated weekly).

1. Linear eigenvalue problems
• linear systems of equations, matrices and determinants, Cramer's rule (6.3,6.5,6.6)
• eigenvalues and eigenvectors (7.1-7.2,18.6)
• eigenvalues of symmetric and skew-symmetric matrices (7.3-7.4)
• basis of eigenvectors and diagonalization (6.4,7.5)
2. Numerical methods in linear algebra
• Gauss elimination (6.3,18.1)
• LU-factorization (18.2)
• iterative methods: Gauss-Seidel and power methods (18.3,18.8)
• QR-factorization (18.9)
3. Systems of differential equations
• basic results for systems of differential equations (3.1-3.2)
• linear systems of differential equations: homogeneous and inhomogeneous (3.3,3.6)
• nonlinear systems, phase plane (3.5)
• linearization of nonlinear systems, critical points and stability (3.3-3.5)
4. Numerical methods for differential equations
• Euler method and predictor-corrector methods (19.1)
• Runge-Kutta and Adams methods (17.3,19.1-19.2)
• numerical methods for systems of differential equations (19.3)
5. Fourier series and integrals
• Fourier series for periodic functions (10.1-10.4)
• Fourier integrals for non-periodic functions (10.8-10.9)
• basic Fourier transforms (10.10)
6. Partial differential equations
• PDE: elliptic, hyperbolic and parabolic (11.1,19.4)
• solutions of the wave equation (finite and infinite intervals) (11.2-11.4)
• solutions of the heat equation (finite and infinite intervals) (11.5-11.6)
• solutions of the boundary-value problems for the Laplace equation (11.5)
• double Fourier series for the two-dimensional wave equation (11.7-11.8)
7. Numerical methods for partial differential equations
• difference methods for the Laplace equation (19.4-19.5)
• explicit and implicit methods for the heat equation (19.6)
• explicit and implicit methods for the wave equation (19.7)
8. Probability theory
• events and probability (22.1-22.3)
• probability distributions, mean and variance (22.5-22.6)
• binomial and Poisson distributions (22.4,22.7)
• normal (Gauss) distribution (22.8)
9. Data analysis
• random data and point estimates (23.1-23.2)
• confidence intervals, t-distribution (23.3)
• testing of hypothesis (23.4)