[Mowicz]

I've decided to put up a few reviews for any wandering Math majors (or enthusiasts!) entering third year. I remember I had no clue which Math courses were interesting, since the names suddenly shift to stuff you've never heard of and your course options suddenly become more broad.

Math Stream: Analysis (Pure Math)
Taken with: Dr. Alama

First off, it is worth noting that this course, as well as Math 3X03, are manditory for any Math degree...regardless of your course interests. So get used to this course, you've got to take it. So this review is not aimed to convince you to take or avoid this course...instead it is intended to show you what to expect.

Course Description:

Welcome to what Math really is! This crash course will take a very deep look at the principles of calculus, and you'll gain a very deep understanding of things such as limits, continuity, and derivatives...A rich study of the real numbers awaits.

This course revolves around the concept of mathematical proof, and drastically increases one's mathematical maturity. Many people find that they are not suited for Masters/PhD work after this course, so brace yourself! But conversely, those who succeed and enjoy the material can rest assured that they have a place in Mathematical Academia.

There will be NO computation in this course. Infact, you will not need a calculator on your tests. A typical Real Analysis Question will be:


Let

U be an open subset of R^n and let f be a function from U ->R^m: Tell what it means for f to have a derivative at a point x in U,and prove or disprove the assertion that f is continuous at x whenever f is differentiable at x.

This may seem scary and hard, but you will be gradually built up to this level...don't worry.

Something interesting you will learn (and prove) that you may not have thought of, is this:

Suppose you are at McMaster, and have a map of Hamilton in your hand. Somewhere on that map, is exactly one point which lies at its true location in the world...but there is only one such point. (The math version of this is called the Banach Fixed Point Theorem)

Professor: Dr. Alama is hilarious. He paces his material perfectly, and makes a lot of very nice analogies to help you understand.

Good luck!

The first real math course for most (if you can, take math 2S03 before this course). In this course, you pretty much learn real analysis.

When I took this course, it was taught by Dr Bronsard; she explains things via many examples. We start with the basic properties of the real numbers (unfortunately, the reals aren't themselves constructed -- you will do this in math 3EE3 however) and move on to formalizing limits and the entire limit-based calculus in one real-valued real variable.

I strongly recommend taking math 4A03, be it after or instead of this course (waivers are easy to get); everything is explained much more thoroughly (you learn "why" and begin to generalize).

Originally Posted by Mahratta

I strongly recommend taking math 4A03, be it after or instead of this course (waivers are easy to get); everything is explained much more thoroughly (you learn "why" and begin to generalize).

I find your suggestion of taking 4A03 instead of 3A03 very intriguing. What topics from 3A03 do you think one should study rigourously before asking for a waiver for 4A03? Obviously taking 3A03 would be the best preparation for 4A03, but it would be nice to have a general idea of what I should be focusing on the most during my self-studying (especially if I were to go as far as replacing 3A03 with 4A03).

Originally Posted by oranges

I find your suggestion of taking 4A03 instead of 3A03 very intriguing. What topics from 3A03 do you think one should study rigourously before asking for a waiver for 4A03? Obviously taking 3A03 would be the best preparation for 4A03, but it would be nice to have a general idea of what I should be focusing on the most during my self-studying (especially if I were to go as far as replacing 3A03 with 4A03).

First, it may be hard to actually replace 3A with 4A (I was forced to take them concurrently for bureaucratic reasons, 3A is required to graduate, but I was also too lazy to argue my case with the registrar). But even taking them concurrently is a good idea; a lot of the ideas in 3A may seem theoretically unmotivated, and 4A provides the motivation.

Next year, first semester of 4th year math is really cluttered with courses (4A, 4X, 4B, 4E all in the same semester), so taking 4A early gives me much more flexibility in 4th year.

We covered ch. 1-11 and 16-20 of Carothers, with some omissions and some extras.
We started by reviewing metric space theory (i.e. metric topology stuff), and then from there generalized the results we got from 3A, which was all in 1 variable. In other words, we were simply talking about open n-balls instead of open intervals.

After this generalization, we had a bit of time to do some new stuff.

We covered Lebesgue theory, which is the model of integration that mathematicians today actually use, more-or-less; to be precise, they use a more general notion of measure, Lebesgue measure being a particular case.

A last-ditch argument for 4A; it lets you look at mathematics differently. If you are interested enough to do mathematics on your own, outside of coursework, then consider this problem:

"Suppose (X, m) is a measure space and (M(x)) is a family of metric structures indexed by X. What does it mean for a sequence (i.e. function) over M(x) [i.e. an element of the product over X of the M(x)] to be integrable? What restrictions do we have to put on X, m, or on each M(x)?"

As you can see, this massively generalizes the notion of integration, you look at it in a different way -- and I don't think that I would be able to make sense of the question at all without 4A.

Ah, interesting. I'll definitely consider taking 3A and 4A concurrently. Thanks again for your advice, Mahratta, I really do appreciate it!

Has anyone had Dr. Sawyer for this course before? I'm curious about his teaching style and the types of questions he may put on tests and exams.

Currently taking the course. Sawyer's tests are relatively straightforward. Midterms were "answer 2 of 3" style, all proofs and definitions. If the proof was short, there was an accompanying definition, otherwise just the proof. Each question worth 10 points no matter how much work was involved or how difficult it was. Exam "answer 6 of 9". I like his testing style, but his teach style is horrid. Over-complicates the simple topics and doesn't explain difficult concepts thoroughly enough. Seems unwilling to answer questions, you'll have your hand up for upwards of 5 minutes on average. You'll be doing a lot of independent learning from his online lecture notes (or text book if you choose to buy it, but it's not necessary).