Math 210A, Fall quarter, 2013

Part 1. Sequences and Series

• Sequences of numbers. Concept and properties of convergence. Cauchy's criterion. Monotonic sequences.
• Infinite series of numbers. Convergence tests of positive series. Alternating series. Product of series. Rearrangements. Infinite product.
• Sequences of functions. Different kinds of convergence of functions: pointwise, uniform, L², and weak convergence. Weierstrass Theorem. Dirac delta-functions.
• Series of functions. Term-by-term differentiation and integration. Power series. Methods of calculating infinite sums.

Part 2. Matrix Techniques

• Determinants and ranks. Definition and basic properties. Applications: Liouville Theorem in statistical mechanics. Rank-one matrices.
• Eigenvalues and eigenvectors. Bounds of eigenvalues. Singular-value decomposition.
• Symmetric positive definite matrices. Spectral decomposition.
• Orthogonal and rotational matrices: O(n) and SO(n). Polar decomposition.
• Matrix exponentials.

Part 3. Hilbert Spaces

• Vector spaces and subspaces. Basis and dimension. Sum and product of subspaces.
• Inner product. Hilbert spaces. Examples. Convergence in Hilbert spaces.
• Least-squares approximation.
• Orthogonal systems. Complete basis.
• Orthogonal polynomials.
• Bounded linear operators. Self adjoint operators. Trace.

Part 4. Fourier Series

• Definition and basic facts.
• Convergence of Fourier series.
• Gibbs phenomenon.

• partial differential equations, basic concepts, examples (Schrodinger, Navier-Stokes, elasticity, Fokker-Planck, etc.), and solution techniques, separation of variables, spherical harmonics, Green's functions.
• qualitative properties of solutions to Laplace equations and Poisson equations,
• methods of the calculus of variations
• Applications: fluid mechanics equations, Poisson equation and Poisson-Boltzmann equation for electrostatic interactions in molecular solvation and electrolytes,
• If time permits: basics of stochastic processes.
• if time permits: differential geometry of surfaces and curves with application to biological molecules, membranes, and cells.