Section 1 What is this course about?
The title of this course is Advanced Calculus, which is a little misleading: it makes it sound like the honors version of basic calculus. But that’s not what we’re about here.
A better title might have been Introductory Real Analysis. So, what is real analysis and what is its relationship to calculus? The word analysis has a specific technical meaning, the details of which I’m sure other mathematicians might quibble over. For us, let’s just say that analysis is short for analysis of functions. That is, we’re going to try and understand the behavior -- qualitatitve and quantitative -- of functions. The real part is that, for the most part, our functions will take in and spit out real numbers.
So you can see, from one perspective, that calculus isn’t too bad a name -- we’ll be talking about limits, derivatives, and integrals of real-valued functions. The difference between this coruse and your calculus course is: we’re going to prove everything!
But real analysis is much more than proof-based calculus. In the process of developing tools to make sense of what limits, derivatives, and integrals are, we’ll discover that functions are a good deal more complicated than even a good calculus student might give them credit for. Even worse, we’ll find that the real numbers have a surprising amount of depth. And the ways that we talk about the real numbers to understand them properly wind up applying much more broadly.
And just like any other language you might learn, this one will allow us to pose questions that wouldn’t otherwise have occurred to us.
So let’s get started. Our first order of business will be to understand what the real numbers are.
