Math 517

Instructor: Dr. Clayton Shonkwiler
Time: Monday, Wednesday, Friday 12:00–12:50
Location: Engineering E106
Office: Weber 216
Office Hours: Monday 3:00–4:00, Wednesday 11:00–12:00 and 3:00–4:00, Thursday 2:00–3:00
Text: Principles of Mathematical Analysis, by Walter Rudin
Email Address: clay@shonkwiler.org
Syllabus

Overview

The goal of this course is to develop the theory of limits, continuity, differentiation, and linearization in a relatively general setting. These are the same concepts that are taught in a typical undergraduate analysis course (like MATH 317), so you should already have some familiarity with them. However, in this course we want to get a little more serious and develop this theory in general metric spaces (to the extent possible) and in higher-dimensional Euclidean spaces.

Here are the topics for the course as listed in the Qualifying Exam syllabus:

Metric spaces, compactness, completeness.
Sequences, convergence, Cauchy sequences.
Series, power series, nonnegative and absolutely convergent series.
Continuity, uniform continuity, intermediate value theorem.
Sequences and series of functions, pointwise and uniform convergence.
Weierstrass approximation theorem, equicontinuity, the Arzela-Ascoli theorem.
Differentiation in several variables, partial derivatives, the chain rule.
Linearization, mean value theorems, sequences of differentiable functions.
Higher order derivatives, power series, Taylor’s theorem.
Contraction mapping principle, implicit and inverse function theorems.