What calculus books should I use to learn more advanced calculus after Stewart's, Larson's book and Thomas' book?
I just finished a calculus $3$ course and I want to learn more about calculus. Also, it seems that famous calculus book like Thomas', Larson's and Stewart's books are considered basic and elementary books and they don't cover many topics in calculus like special functions, proofs of many theorems and rigorous arguments, etc.I figured that rigour and proofs is in a separate course called Real analysis and I found some good sources to learn it, but I don't want to learn rigorous mathematics yet,I want a book that has more theorems of calculus that elementary books like Stewart didn't cover like (the proof of $\pi$ is irrational, more techniques of integrals and special functions, etc …) or cover them in more detail and depth and I am not sure what books to use to learn more about these topics or to learn more advanced calculus.
Best Answer
It’s important to distinguish “advanced calculus” from “real analysis”. Advanced calculus is more inclusive on topics such as: vector analysis, multivariable methods including surface and volume integrals, often some complex analysis, often some Fourier series and the gamma function. Real Analysis is more focused on topics such as: metric and topological concepts (even if metric spaces and/or topological spaces are not introduced), pathological and nuanced counterexamples, differentiation and integration considered more as topics to be studied in-of-themselves rather than as tools for other mathematical investigations.
In what follows I’ve restricted myself to those books I’ve become fairly familiar with over the past 45+ years. Of these books and based on what you’ve said, Buck [2] and Kaplan [4] and Taylor/Mann [6] are what I’d most strongly recommend you consider. I’ve grouped the books into 3 categories.
Traditional/Standard Level Advanced Calculus Books
Some traditionally used books for advanced calculus courses (2 semesters length, U.S. upper undergraduate level) are the following. FYI, the 1972 2nd edition of Taylor/Mann [6] was the text used (for many years) where I was an undergraduate.
[1] Tom Mike Apostol, Mathematical Analysis. A Modern Approach to Advanced Calculus, Addison-Wesley Mathematics Series, Addison-Wesley Publishing Company, 1957, xii + 553 pages. Internet Archive copy
[2] Robert Creighton Buck, Advanced Calculus, International Series in Pure and Applied Mathematics, McGraw-Hill Book Company, 1956, viii + 423 pages.
[3] Philip Franklin, A Treatise on Advanced Calculus, John Wiley and Sons, 1940, xiv + 595 pages. Internet Archive copy.
[4] Wilfred Kaplan, Advanced Calculus, Addison-Wesley Mathematics Series, Addison-Wesley Publishing Company, 1952, xiv + 679 pages.
[5] Murray Ralph Spiegel, Schaum’s Outline of Theory and Problems of Advanced Calculus, Schaum Publishing Company, 1963, viii + 384 pages.
[6] Angus Ellis Taylor, Advanced Calculus, Ginn and Company, 1955, xiii + 786 pages.
[7] Frederick Shenstone Woods, Advanced Calculus, Ginn and Company, 1926, x + 397 pages.
Slightly Higher Level Advanced Calculus Books
What follows next are texts pitched at a slightly more advanced level. FYI, the 1977 2nd edition of Fleming [11] was the text used (for many years) for the honors version of the upper undergraduate advanced calculus course where I was an undergraduate. Typically only 2 or 3 students each year took the honors version, which was conducted as an independent reading course (i.e. no lectures).
[8] Richard Courant and Fritz John, Introduction to Calculus and Analysis, Volume II, with the assistance of Albert Abraham Blank and Alan David Solomon, John Wiley and Sons (Wiley-Interscience), 1974, xxvi + 954 pages.
[9] Charles Henry Edwards, Advanced Calculus of Several Variables, Academic Press, 1973, xii + 457.
[10] Harold Mortimer Edwards, Advanced Calculus, Houghton Mifflin Company, 1969, xv + 508 pages.
Reprinted (and retitled Advanced Calculus. A Differential Forms Approach) by Birkhäuser in 1994 (xvi + 508 pages).
[11] Wendell Helms Fleming, Functions of Several Variables, Addison-Wesley Publishing Company, 1965, x + 337 pages.
Very High Level Advanced Calculus Books
Finally, the last two books are well known as very high level honors texts, pretty much only suitable for the strongest undergraduates at universities such as Harvard, Princeton, etc.
[12] Lynn Harold Loomis and Shlomo Zvi Sternberg, Advanced Calculus, Addison-Wesley Publishing Company, 1968, xii + 580 pages.
[13] Helen Kelsall Nickerson, Donald Clayton Spencer, and Norman Earl Steenrod, Advanced Calculus, D. Van Nostrand Company, 1959, ix + 540 pages.