MAT 303 Real Analysis

The real number system. The least upper bound axiom. Limit superior and limit inferior.

Sequences in a metric space. Continuity of functions. Cartesian product of metric spaces. Completion of a metric space. Everywhere and nowhere dense sets. Baire’s theorem. Compact sets. Compactness and convergence of sequences. Heine-Borel Theorem. Continuity and compactness. Connected sets.  Contraction mapping. Contraction mapping theorem.

The Arzela-Ascoli theorem. Peano Theorem.

MAT 341 Differential Geometry

Calculus on Euclidean Space, Tangent vectors, Tangent Space, Directional Derivatives, Curves, 1- forms, Differential forms, Mappings, Covariant Derivative, Frenet-Serret Formulas, Tensor Product, Osculating plane.