1  Review of Apostol’s Calculus

This is my review of

Executive summary: Uncommonly good. The gold standard.

This is not "Calculus for Dummies". This is calculus for students who are somewhat clever and somewhat sophisticated, and would like to become more so.

There are two volumes. Roughly speaking, Volume I is for the first year of the course, and Volume II is for the second year.¹

In some narrow sense, mathematics is all about equations and proofs. These books certainly have the equations and proofs ... but that’s not what makes them special. Apostol does a good job with the words that go between the equations, explaining where the equations come from, what they mean, and what you can do with them.

Whereas most calculus books are pretty much the same, Apostol’s books stand apart from the herd. They are full of surprises: sometimes more practical, sometimes more rigorously formal, and sometimes both at once.

Here’s an example of what I mean: Volume II contains a couple of chapters on the Calculus of Probabilities. That’s already a pleasant surprise, because most “calculus” books avoid that subject, and the math and science curriculum suffers accordingly. The next surprise is that Apostol formalizes probability in terms of set theory. This has vastly more formality and generality than the usual hand-wavy heuristic approach. On the other hand, it’s really rather simple, using not much more than the set theory you learned in 5th grade. It is vastly simpler than the measure-theory approach that mathematicians use when talking to other mathematicians.

This is not formality for the sake of formality, but rather generality for the sake of real practical power.

Specifically, Apostol makes the point that in any given application, there is more than one way to assign probabilities to the elements of a set. Mathematics won’t tell you the right assignment; only an understanding of the application will tell you that, and you might need to do some searching before you find it. This is serious business. I’ve seen people who were ranked among the world’s most expert statisticians who wasted many man-years because they failed to understand this point. I’ve discussed this with some of these guys, and some of them still don’t get it.

Apostol uses a similar trick when introducing Lebesgue integrals: He uses simple set theory, which turns out to be sufficient to obtain great generality, without having to drag out the heavy machinery of measure theory. Obviously this is a book written by somebody who has taught the course more than once, and who knows how to obtain the best result with the least effort.

We see a different conjunction of generality and practical power in the section on differential equations. For linear differential equations, Apostol sets forth the usual systematic methods for obtaining a solution in closed form. That’s not particularly surprising. Next we come to nonlinear differential equations, which is more of a surprise, because most introductory books avoid this topic like the plague. It turns out that most nonlinear differential equations do not have closed-form solutions, and there is not much in the way of fancy axiomatic methods that are of value. There are however powerful practical methods, such as drawing the flow-fields and trajectories numerically and looking at them. More generally, there are ways you can understand an equation and its solutions even when you did not or cannot write down a closed-form solution. This is excellent advice.

Bottom line: If you can only own one calculus text or calculus reference, this is the one you want.

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There is also Mathematical Analysis by the same author, which will keep you busy for a third year, but is not the subject of this review.