Advanced calculus book recommendations.

Question

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.

Answer 1

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

The 2nd edition (xvii + 492 pages) was published by Addison-Wesley in 1974. The subtitle "A Modern Approach to Advanced Calculus" is omitted in the 2nd edition, which also omits the material on vector analysis and line integrals and surface integrals, while adding some basic material on metric spaces and a couple of chapters on Lebesgue integration. Thus, the 2nd edition is closer to what is generally known as real analysis than advanced calculus.

[2] Robert Creighton Buck, Advanced Calculus, International Series in Pure and Applied Mathematics, McGraw-Hill Book Company, 1956, viii + 423 pages.

The 2nd edition, with the collaboration of Ellen Fedder Buck, was published by McGraw-Hill Book Company in 1965. The 3rd edition, with the collaboration of Ellen Fedder Buck, was published by McGraw-Hill Book Company in 1978 (xii + 622 pages).

[3] Philip Franklin, A Treatise on Advanced Calculus, John Wiley and Sons, 1940, xiv + 595 pages. Internet Archive copy.

Reprinted (unabridged and corrected republication) by Dover Publications in 1964 (xii + 595 pages).

[4] Wilfred Kaplan, Advanced Calculus, Addison-Wesley Mathematics Series, Addison-Wesley Publishing Company, 1952, xiv + 679 pages.

The 2nd edition was published by Addison-Wesley Publishing Company in 1973 (xv + 709 pages). The 3rd edition was published by Addison-Wesley Publishing Company in 1984 (xiv + 721 pages). The 4th edition was published by Addison-Wesley Publishing Company in 1991 (xvi + 746 pages). The 5th edition was published by Pearson in 2002 (exact paging not known).

[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.

The 2nd edition, co-authored with William Robert Mann, was published by Xerox Corporation in 1972 (xx + 774 pages). The 3rd edition, co-authored with William Robert Mann, was published by John Wiley and Sons in 1983 (xviii + 732 pages).

[7] Frederick Shenstone Woods, Advanced Calculus, Ginn and Company, 1926, x + 397 pages.

The 2nd edition [= New Edition] was published by Ginn and Company in 1934 (x + 397 pages). The chapters and chapter paging for the (so-called) 1934 2nd edition is identical to the chapters and chapter paging for the 1926 1st edition. Note: This book has become rather famous because Richard Phillips Feynman worked through it during his last year of high school (Fall 1934 − Spring 1935) — he mentions this book in his 1985 book Surely You're Joking, Mr. Feynman!. In fact, since mid 2000s Woods’ book has been nearly impossible to purchase at a reasonable price due to the internet popularity of things Feynman has said about Woods’ book (e.g. among other things, the method of evaluating definite integrals by differentiating under the integral sign — a method, incidentally, that one can find in many advanced calculus texts, and not just Woods’ text).

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.

Reprinted by Springer-Verlag in 1989.

[9] Charles Henry Edwards, Advanced Calculus of Several Variables, Academic Press, 1973, xii + 457.

Reprinted (unabridged and corrected) by Dover Publications in 1994 (xii + 457 pages).

[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.

The 2nd edition was published by Springer-Verlag in 1977 (xii + 411 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.

The 2nd edition was published by Jones and Bartlett Publishers in 1990 (xii + 580 pages).

[13] Helen Kelsall Nickerson, Donald Clayton Spencer, and Norman Earl Steenrod, Advanced Calculus, D. Van Nostrand Company, 1959, ix + 540 pages.

Reprinted (unabridged) by Dover Publications in 2011 (x + 540 pages).

Answer 2

I recommend:

• Introduction to Electrodynamics by Griffiths. Although it’s a physics textbook, it has a very illuminating presentation of certain key topics in vector calculus. The presentation is non-rigorous but highly intuitive.
• Div, Grad, Curl and All That.
• Vector Calculus, Linear Algebra, and Differential Forms by Hubbard and Hubbard. It is rigorous but also very readable and insightful.
• Advanced Calculus by Folland.
• Multivariable Mathematics by Shifrin.

Answer 3

I am reading "Analysis on Manifolds" by James R. Munkres and I was reading "Calculus on Manifolds" by Michael Spivak.

First I tried to finish "Calculus on Manifolds" by Michael Spivak.
I thought this book has few pages and the proofs are short and clear and often easy to understand.

However, I gave up halfway through.

"Analysis on Manifolds" by James R. Munkres is extremely kind to the readers.
I believe I can finish this book.

I recommend both books.

Answer 4

The subjects that extend calculus tend to be rigorous courses like real analysis, or courses like vector calculus (calculus with multiple variables). The latter can be taught in a rigorous or less rigorous manner the first time around. For the less rigorous treatment, I found Paul's Online Notes rather useful.

Ultimately, however, advanced calculus is proof-based and rigorous – this is not a reflection of calculus per se, but of how modern mathematics is done in general. To ease the transition, one book I would recommend is A Hitchiker's Guide to Calculus, which bridges the gap between calculus and real analysis rather well. It is written by Michael Spivak, who also wrote one of the finest books on introductory real analysis there is. Ironically, that book is titled Calculus.

Answer 5

"What’s Next After Stewart?" If you want to know why the subject is sometimes called "infinitesimal calculus", you may want to check out Keisler's textbook:

H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach. Dover Publications, 2012.

The book is also available for free from the author's webpage: https://www.math.wisc.edu/~keisler/calc.html

Answer 6

An "anti-recommendation": The accepted answer recommends Loomis & Sternberg's Advanced Calculus. I strongly disagree with this and would advise people to avoid this book. The table of contents suggests a good idea that could have turned into a really good book. But the execution of the idea (the book they actually wrote) is just full of problems. Some examples (in roughly increasing order of severity):

1. Key definitions are not visually highlighted in any way. A particularly bizarre case is the definition of "proper" Riemann integrals (bounded domain and range) via Darboux sums, which is buried in the middle of a proof (of Theorem 10.1, Chap. 8). Cleverly hidden where no one would think to look for it!
2. The index is notably deficient, which together with point #1 makes it particularly painful to find a detail of earlier-read material about which you might need to refresh your memory.
3. Weirdly (dis)organized presentation:
   • The Heine-Borel theorem is never presented as a unified whole; instead, various pieces of it are scattered about chapter 4: page 206 has sequential compactness $\implies$ closed and bounded, page 208 has closed and bounded $\implies$ sequential compactness, and page 214 has sequentially compact $\iff$ compact. This despite the Heine-Borel theorem in its entirety being essential to understanding the later presentation of Riemann integration in $\mathbb{R}^{n}$.
   • The notion of compactness (of sets in a topological space) is used before it's defined. The definition comes on page 214, yet compactness is invoked multiple times on pages 206-213.
4. The definition of "improper" Riemann integral (chap. 8, sect. 13), as it's written, is obviously circular (the definition effectively assumes that integration over unbounded domains has already been defined, but it hasn't). It's possible that this is a typing mistake: perhaps "all $B$ with $B \cap {A}_{\epsilon} = \emptyset$" was supposed to be "all bounded $B$ with $B \cap {A}_{\epsilon} = \emptyset$". Hard to be sure.

I could go on, but I think this list gives the flavor of it. I believe this book was at one time (the 1970's maybe?) the textbook for the famous Math 55 course at Harvard, but has since been replaced in favor of other textbooks for the course. Not surprising to me it was discontinued: it's arguably the most notable example of just plain bad writing I've ever seen in a math book.