Course outline

The sections we'll cover are 15.1-9, 14.5-6, and 16.1-10, plus possibly a little extra material if there's time; we'll go more or less in that order, but a few things may get juggled around compared to the book. A more detailed outline follows.

Chapter 15

We'll start out with Chapter 15; the first few sections on double integrals are likely familiar from Math 126. Next come triple integrals; these are exactly like double integrals except with 1.5 times as many integrals, but actually setting up a triple integral over an arbitrary region of 3-dimensional space can be challenging, because of the difficulty in visualization compared to 2 dimensions.

Rounding out Chapter 15 is the change of variables formula for double/triple integrals. This is the analogue of substitution for single-variable integrals, but there's now a more geometric flavor. With one-variable integrals, you only integrate over intervals, which are not very complicated geometrically, and so in that setting, substitution is usually just a way to turn difficult integrands into easier ones. With multiple integrals, the domain of integration can become complicated, and our goal with change of variables will generally be to transform it to a simpler one, rather than to transform the integrand.

Chapter 14

Next come two sections about derivatives. The chain rule for single-variable functions says how to take the derivative of a composition of two functions, and we'll go over how to do the same thing for multivariable functions. The expressions that result can look confusing, but Stewart gives a nice way to organize them.

Given a function of two variables, its two partial derivatives tell you its rate of change in the x and y directions. You might reasonably want to know the rate of change in a different direction, which we'll see how to do. The notion of gradient introduced here will turn out to be an important one, and we'll encounter it throughout the rest of the class.

Chapter 16

This is where the really new and most interesting material is. We start by introducing vector fields, and the rest of the chapter is about doing calculus with them. Recall the fundamental theorem of calculus: the integral of f' over the interval [a,b] is f(b) - f(a). That is, you can compute an integral over the one-dimensional object [a,b] by looking at something on its boundary, just the two points a, b. Likewise, we'll see that Green's theorem and Stokes' theorem relate a double integral over a two-dimensional object to another integral over its boundary, which will be some one-dimensional curve. Then, the divergence theorem will relate a triple integral over a three-dimensional object to another integral over its boundary, which will be some two-dimensional surface. There are quite a few new ideas in this chapter, but they will all turn out to be interrelated, and often just different pieces of a single larger picture (although fully seeing this big picture will have to wait for future courses).