Course Summary

Math 302, Elements of Analysis I, is a course which explores the formality behind much of the content students learn through multivariable calculus. In this way the class is the opposite of many proof based math courses, where typically a new class of objects or unfamiliar quality is introduced, and the goal is to explore its ramifications (often by showing how the new idea generalizes, synthesizes or otherwise sheds light on previously understood content). In 302, by contrast, students typically have an established intuition for the topics we consider, and the challenge is to replace this intuitive understanding with an axiomatic one. In short, we will mostly be focused on resolving the “why” questions of calculus. As a consequence of trying to understand the “why” questions, we'll often be interested in examples of extreme behavior, in contrast to most introductory calculus classes which instead focus almost exclusively on “nice” functions or sets.

The class begins by detailing the axioms we need to describe the real numbers, then exploring their ramifications. We'll ultimately list a handful of axioms which uniquely characterize the real numbers, and we'll spend a considerable amount of time thinking about how these axioms distinguish the real numbers from other familiar sets of numbers (like the rational numbers). With this information in hand, we'll be able to start “doing analysis” in $\mathbb{R}^n$. This will require us to formalize notions like “limit” and “continuity,” but will also require us to think carefully about “open” and “closed” sets. Our objective will be to establish the right abstract formulation for some concepts so that it is easy to prove more powerful theorems. As an example, instead of showing that all polynomials are continuous directly, we'll instead argue that the functions $f(x) = 1$ and $g(x) = x$ are continuous, and then we'll prove that the sum, difference and product of continuous functions is again functions. Taken together, these results will tell us that all polynomials are continuous, without requiring a direct “$\varepsilon-\delta$ proof” of this fact.

The conceptual highlights of the course include (but are not necessarily limited to): the least upper bound property, the uncountability of the real numbers, basic point set topology, limits (of functions and sequences), continuity, connectedness, open covers, compactness, and the canonical results from single-variable (real) differentiation and integration. In addition to this content, students will exercise a number of skills that extend far beyond the mathematical classroom: recognizing patterns; connecting ideas across disparate contexts; extending intuition from a familiar setting to solve problems in an unfamiliar setting; and communicating in a variety of ways and to a variety of audiences (ranging from informal discussions to very formal write-ups).

Course Instructor

The professor for this class is Andy Schultz. His office is on the fourth floor of the Science Center, room W408. His office hours are listed on the syllabus.

You are highly encouraged to attend office hours, and you never need an appointment to do so. If scheduled office hours don't fit with your schedule, contact the instructor to set up an appointment to help you outside of office hours.

You can contact the instructor at . Though he is always happy to receive emails from you with questions or concerns about the course, he can't guarantee that he'll be able to promptly reply to emails late at night or over the weekend. If you do contact the professor by email, please be sure to follow standard email etiquette. In particular, please make sure you include a greeting and signature and avoid abbreviations. If you're contacting him to ask about a problem, please be sure to specify what the problem asks (as opposed to asking something like “I can't get problem 2 and need your help”).