Investigation of convergence, continuity, differentiability, compactness and connectedness, the Riemann-Stieltjes integral, and sequences of functions. Groups, including subgroups, quotient spaces, and homomorphisms. Ring theory, including ideals and quotients, homomorphisms, and polynomial rings. An introduction to modules and fields, including field extensions. Numerical Analysis Introduction to approximation theory, interpolation, numerical differentiation and integration, solutions of linear and non-linear equations, numerical solution of differential equations, optimization.
Emphasis is on error analysis and stability.
Several practical examples and computer programs will be covered. Lebesgue integration, Integration with respect to a measure, absolute continuity, Fundamental Theorem of Calculus for the Lebesgue integral. Introduction to finite difference and finite element methods for the solution of elliptic, parabolic and hyperbolic partial differential equations. Complex integration and the calculus of residues. Analytical continuation and expansions of analytic functions.