math: calculus

Early Transcendentals is the way to go. This is the book that made James Stewart a very wealthy guy.

Prior to Stewart writing that book, virtually every calculus book used a sequence that is now known as "late transcendentals". They didn’t have a term for it back then, because they didn’t need one. It was the only way to go. The most fundamental difference is the way certain topics are defined, which results in a slightly different order of topics.

Calculus can broadly be divided into three topics: limits, derivatives, and integrals. You can throw series in there as a fourth if you’d like.

Nearly every book starts with limits, then introduces derivatives. But here is where they split. The "late transcendentals" approach doesn’t teach you how to do derivatives for every kind of function right away. They do most of them, but then need to move on to integrals before they can finish derivatives. This is because "late transcendentals" books define the log function as the integral of another function. Now that it’s defined, they can circle back to showing you a derivative for that function.

Stewart came up with a clever way to deal with logs without defining it as an integral. He just uses the same definitions for functions that you learned in your algebra classes. This allowed him to give a complete section on derivatives for all the main parents functions before moving on to integration. It tidies up the table of contents quite a bit.

After this, virtually every new calculus book took that approach. There are some old fashioned folks that still turn their noses up at it, but I think it’s just as rigorous as the old way in the context of a first-year calculus book.

Chapters 1-5 are typically calc 1
Chapters 6-11 are typically calc 2
Chapters 12-16 are typically calc 3
Chapter 17 may be taught in calc 3 but covers topics that are more appropriate for a Diff EQ class.