Advanced Calculus

by Angus E. Taylor, W. Robert Mann

Outlines theory and techniques of calculus, emphasizing strong understanding of concepts, and the basic principles of analysis. Reviews elementary and intermediate calculus and features discussions of elementary-point set theory, and properties of continuous functions.

Genres: Mathematics, Reference, Textbooks

732 pages, Paperback

First published January 1, 1955

About the author

Angus Ellis Taylor (October 13, 1911 – April 6, 1999) was a mathematician and professor at various universities in the University of California system. He earned his undergraduate degree at Harvard summa cum laude in 1933 and his PhD at Caltech in 1936 under Aristotle Michal with a dissertation on analytic functions. By 1944 he had risen to full professor at UCLA, whose mathematics department he later chaired (1958–1964). Taylor was also an astute administrator and eventually rose through the UC system to become provost and then chancellor of UC Santa Cruz. He authored a number of mathematical texts, one of which, Advanced Calculus (1955, Ginn and Co.), became a standard for a generation of mathematics students.

Reviews

[Adam · Rating 2 out of 5]

I find this book pretty un-readable. The proofs often are given in a way that seems "out of nowhere" so that, unless you already know the subject and understand why the author began the proof in the way that he did, you'll be unclear on the logic itself and how you might reproduce similar reasoning in other contexts. There's also a lot in it that seems unnecessarily complex, especially at the point when the material is exposed. I'm certainly not opposed to a level of mathematical maturity in a text with "advanced" in the title, but I am against gratuitous distraction in a complex topic when the important central subject can be presented much more simply. For instance Kennath Ross's text in Elementary Analysis, covering much of the same topics, is not my favorite but it at least has the virtue of being efficient. That one might be a little excessively simple to be used for Math majors but I expect students to still get more out of it than this one by Taylor because they will at least see the flow of ideas and the patterns of reasoning--even if they might not learn enough to quite qualify as "having taken an advanced undergraduate course in Calculus / Analysis".

I still haven't found an Analysis book that I really like. As I said, the Ross book feels like it's a little too simple, which is not bad as a primer but is bad for students to be really qualified in the subject. I also just occasionally dislike the presentation in Ross's book, as there are typos, definitions are not always clearly presented, and proofs are not always as clear as possible. Apostol gives a somewhat refreshing geometric entrance into the subject, especially good for diversity of perspectives given that the analytic approach is so much more universal (of course for good reason)--but I would say Apostol is best used as second perspective, and a student should instead rely primarily on some other text. For that role, so far, my favorites have been Rudin or Spivak, possibly complemented with occasionally checking in **Counterexamples in Analysis** by Gelbaum et. al.

[Ian Paredes · Rating 3 out of 5]

it's a REALLY old book (i believe the bulk of the book is at least 40 or 50 years old), but it's still a great book to learn out of. just be sure to scrap a lot of the things that you learn in this book once you move on to "baby rudin" and other texts that are meant to generalize the concepts in this book; while this does give a great overview of advanced calculus, it's not the greatest resource to refer to.