# 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**

**Exam 1 Practice Problems** (sketches of solutions)

**Final Exam Practice Problems** (sketches of solutions)

## 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, Taylorâ€™s theorem.
- Contraction mapping principle, implicit and inverse function theorems.