[loose-cannon]

If you just pick one of those subjects, you'll probably find a textbook just as long as his entire PDF trying to cover 13+ subjects.

Sorry to be negative Nancy over here, but you're going to need more than 54 pages to cover calculus. There is value in organizing the major theorems in the different disciplines. But, to be honest, this doesn't really serve the beginner.

[morcus]

Two thoughts here:

1. I don't think it is at all intended to serve the beginner.

It's geared towards readers wait a reasonable amount of mathematical maturity already (it explicitly says that in the landing page).

2. Many, many of the pages of most introductory calculus textbooks are spent on exercises and on the specifics of computing integrals and derivatives of particular functions - none of this is necessary to understand the concepts themselves.

For example, Baby Rudin (the standard textbook for Analysis for math majors) covers Sequences, Series, Continuity, Differentiation, and the Riemann integral in less than 100 pages (including exercises).

[loose-cannon]

So this is aimed at somebody who has mathematical maturity but prefers... less content and detail? The point is that you are losing something in a shortened presentation. You're not just losing "unnecessary exercises" as you put it.

[bonoboTP]

From the book

> Philosophy behind the Napkin approach

> As far as I can tell, higher math for high-school students comes in two flavors:

> • Someone tells you about the hairy ball theorem in the form “you can’t comb the hair on a spherical cat” then doesn’t tell you anything about why it should be true, what it means to actually “comb the hair”, or any of the underlying theory, leaving you with just some vague notion in your head.

> • You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.

> Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here. Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs, or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof.

[morcus]

As I said, intro calculus books will spend a large amount of time teaching you the mechanics of finding closed form solutions for integrals and derivatives of various kinds of functions. Look at https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/pa... for an example. Most of that content is not that important to understand the concepts.

And yes, with more mathematical maturity you definitely don't need as much detail. The proofs get terser as you're expected to be able to fill out the more straightforward details yourself.

[schoen]

My first calculus class in high school was about 10% "conceptual explanation of limits, derivatives, and integrals", 30% "techniques for evaluating derivatives", 50% "techniques for evaluating integrals", and maybe another 10% (or less) "justifications of the correctness of those techniques". (I guess I'm putting the Fundamental Theorem of Calculus in the the last 10% here.)

The style of this textbook does seem to primarily skip the "techniques for evaluating" stuff, on the basis that you just wanted to understand what each branch of mathematics is about and what kinds of theorems it has that might relate to the larger edifice of mathematics.

[zozbot234]

I don't quite get how it's supposed to introduce calculus/analysis - the introductory chapters just start talking about metric spaces without even bothering to properly introduce the real numbers or their peoperties. I don't think that's quite sensible. For comparison, mathlib4 of course does it right by starting from topological spaces - and it manages to nicely simplify things throughout, by defining a basic "tends to" notion using set-theoretic filters.