An introduction to proofs in advanced mathematics, intended as a transition to upper division courses including MATH 4107, 4150 and 4317. Fundamentals of mathematical abstraction including sets, logic, equivalence relations, and functions. Thorough development of the basic proof techniques: direct, contrapositive, existence, contradiction, and induction. Introduction to proofs in analysis and algebra.
Method of characteristics for first and second order partial differential equations, conservation laws and shocks, classification of second order systems and applications.
Real numbers, topology of Euclidean spaces, Cauchy sequences, completeness, continuity and compactness, uniform continuity, series of functions, Fourier series
Linear algebra in R^n, standard Euclidean inner product in R^n, general linear spaces, general inner product spaces, least squares, determinants, eigenvalues and eigenvectors, symmetric matrices.