OUP published Visual Complex Analysis in 1999. MAA published Visual Group Theory in 2009. But no visualization counterparts have been published for other subjects like (Commutative) Algebra, Combinatorics, (Algebraic, Differential) Geometry, Harmonic Analysis, Matrix Theory, Measure or Ergodic Theory, (Advanced) (Real) Analysis, Rings Fields Modules, Topology. These subjects are abstruser than univariate calculus, but are the mathematicians listed below incapable of writing books on them?

Why don't these tyro authors publish, and wouldn't they profit more from, the aforementioned Visual equivalents to assist students visualizing other subjects, or problem books with full solutions to each problem? Why do these tyros persist in publishing univariate calculus textbooks that remain untrodden? These tyros must know that the market for univariate calculus textbooks is saturated and has little chance of being disrupted. In actuality, these tyros failed, because most instructors are unfamiliar with their untrodden textbooks probably because they don't say anything new or reform calculus education.

Instructors don't have time to read and compare the glut of univariable calculus textbooks. I read this review of Peter Mercer's More Calculus of a Single Variable (2014), but it doesn't distinguish it from other allegedly revolutionary books like Karl Menger's Calculus: A Modern Approach (2007), Daniel Velleman's Calculus: A Rigorous First Course (2017) or Dan Sloughter's Calculus From Approximation to Theory (2020).

Alex Himonas, Calculus: Ideas & Applications (2003).

Since there are so many calculus books and the coverage in them is so similar, I cannot say that this book is outstanding.

Dale Varberg, Edwin Purcell, and Steven Rigdon, Calculus (2006).

So: a mostly traditional book, with the real virtue of brevity but otherwise not too imaginative.

Robert Smith and Roland Minton, Calculus: Early Transcendental Functions (2007).

There's a numbing sameness to most calculus books. In part, that's inevitable: how many paths are there through this particular garden? Given that one must visit the usual tourist spots, there is little space for being really original. [...] Beyond that, this strikes me as plain-vanilla.

Jon Rogawski, Calculus (2008).

But now we come to the key question: Given that there are already thousands of calculus books in print, is it valuable to have a new calculus book that is just like nearly all of them, except that it is has clearer explanations and fewer errors? It's hard to make this sound exciting, and in fact I am not excited by it. Most calculus books are deadly dull, and even though this one has lots of pretty pictures and interesting sidebars I still had a hard time getting through it. I would have liked it much better if it had addressed some of the issues raised by the reform movement.

Arnold Ostebee, Calculus: From Graphical, Numerical, and Symbolic Points of View (2008).

When reviewing a general calculus text, the main question the reviewer needs to answer is “how is this book different from the existing mainstream choices?” The answer, in this case, is that the book is more ambitious and moves at a faster pace than most competing textbooks. It also makes a few unusual choices in the order in which the topics are covered.

Laura Taalman and Peter Kohn, Calculus (2013).

The Calculus textbook market is crowded, and the books on that market are very similar to one another. It is completely normal for two books in that category to overlap by ninety percent or more in their coverage of topics

Peter Lax, Maria Terrell, Calculus with Applications (2013).

My criticisms and suggestions aside, this is an altogether excellent text.

Michael P. Sullivan and Kathleen Miranda, Calculus: Early Transcendentals (2014).

Other than these aspects, there is nothing to differentiate this book from other books designed for the three-semesters of calculus market. In many ways I find myself wishing that there would be a moratorium on new calculus books because the field seems to have hit a wall regarding anything new in textbooks. [Embolding mine] I could have used this book to teach calculus 20 years ago and I could use my book of twenty years ago to teach it now and the outcomes would not be significantly different.

Alexander Hahn, Calculus in Context (2017).

Overall, this reviewer would not choose to use this text as the main text for a calculus course, but would (and has) utilized the myriad of applications as starting points for student projects.

Deborah Hughes-Hallett, Andrew Gleason, Calculus: Single and Multivariable, 8th ed. (2018). But the MAA hasn't reviewed it since the 2005 4th ed..

  • 7
    $\begingroup$ So what's your question? Is it about single variable calculus texts or more advanced topics? Perhaps you're a tyro greenhorn tenderfoot yourself? (seriously, what is this question? who upvotes this stuff?) $\endgroup$
    – Thierry
    Jul 3 '21 at 15:51
  • 7
    $\begingroup$ Can you tell I was cranky earlier before I ate? I do think this question is kind of weirdly passive aggressive toward textbook authors and could be cleaned up a bit, but still, I suppose it's a decent one. Needham-style texts for, say, graduate real analysis just seems like such a niche market that I don't know why anyone would expect them to exist. For what it's worth, Needham is releasing a new book in a few weeks and Weissman has written An Illustrated Theory of Numbers, so there may be more such books than you think. $\endgroup$
    – Thierry
    Jul 3 '21 at 17:00
  • 6
    $\begingroup$ I'm reminded of a story (from the 1960s) about the late J C Burkill at Cambridge University. After a rant about the lack of a decent undergraduate textbook on real analysis, ending with a claim that he could write a better one in a single weekend, another academic bet him a case of expensive wine (value say 30,000USD at today's prices) that he couldn't. So he wrote the book, won the wine, and the book is still in print 60 years later - amazon.co.uk/J.-C.-Burkill/e/… $\endgroup$
    – alephzero
    Jul 3 '21 at 18:33
  • 4
    $\begingroup$ My suggestion: those answering should state how many years teaching elementary calculus they have. Just as a point of reference for readers. $\endgroup$ Jul 4 '21 at 0:01
  • 3
    $\begingroup$ Problems with this question: (1) It took several readings for me to figure out what the question was about. (2) It assumes that the Visual $foo approach, as defined by two examples, is superior to all other approaches, that it is generalizable to all other topics, and that everyone agrees with this opinion. (3) Applying negative language to authors comes off as an ad hominem, nor is "tyro" accurate for people whose book is currently in its ninth edition. (4) It's an "ammiright?" question. $\endgroup$
    – user507
    Jul 5 '21 at 17:37

Single variable calculus is in a very different situation from the other topics you mention. The number of students in the US who take single variable calculus is at least two orders of magnitude up from the number who study any of those other subjects, because it's a requirement at many schools for a wide variety of majors.

This leads to several effects

  • The market for selling such books is huge, so an unsuccessful calculus book may sell substantially more copies than a highly successful measure theory book. (Even if we assume the people writing these books aren't particularly motivated by the royalties, those sales are also one plausible measure of the impact of the book.)
  • Professors have to teach the subject a lot, which motivates them to want to teach it in a way they like.
  • Because the topic is taught to younger, less mathematically mature students, issues with the textbook are harder to smooth over.
  • Because the course has obligations to many departments outside math, it's relatively hard to innovate. Someone who wants to significantly change what specific topics, say, a combinatorics course covers can often just do that unilaterally, or at least negotiate it with just a couple other people. But even curricular changes to the calculus course may have to be discussed, not only at the departmental level, but possibly negotiated with other departments as well.

Put together, I think it's clear why that would motivate some people to write books which, from their perspective, improve the way calculus is taught while still being compatible with the basic strictures of the standard American calculus course.

  • 1
    $\begingroup$ This is a good answer, but it's worth noting that the market logic described by this question doesn't apply at all to OER books, and there are many OER freshman calc books out there that are not at all like the standard non-free offerings. $\endgroup$
    – user507
    Jul 5 '21 at 17:40

I agree with Henry Towsner's answer, but I also believe the premise of the question is false.

For example, my college adopted Sullivan & Miranda 2e (one of the "glut" of calculus textbooks listed in the question) in our most recent calculus textbook search, in part because it seemed more readable than the usual options. Every example in the exposition directly indicates to the student which exercises they should try next, before they read further in the text. This supports the instructors in my department who focus on teaching students how to read textbooks (a skill many students have not learned before reaching us).

If a reviewer is not focused on this aspect of teaching mathematics, perhaps looking at the Sullivan & Miranda text would cause the reviewer to despair that "the field seems to have hit a wall regarding anything new in textbooks." To my committee, it looked great. Perhaps many of the textbooks listed have a non-trivial following like this one.

  • $\begingroup$ This reminds me of something I especially like in more advanced monographs, at least when it makes sense to do, which is to give for each of the bibliographic entries a list of the page(s) where that bibliographic item is cited. I find this extremely useful in getting information about little known papers that I don't have copies of (or maybe I do, but the paper is not in English), or simply looking up additional information about a very narrow topic that I happen to know is discussed in a certain paper (e.g. other relevant papers and/or aspects not in the paper I know). (continued) $\endgroup$ Jul 4 '21 at 18:55
  • $\begingroup$ Three examples I found just now on my bookshelves (I didn't look very far) are: The Higher Infinite by Akihiro Kanamori (1994) and Symmetric Properties of Real Functions by Brian S. Thomson (1994) and Continuous Nowhere Differentiable Functions by Marek Jarnicki and Peter Pflug (2015). $\endgroup$ Jul 4 '21 at 18:55

I think the question as asked here is just misinformed. There are numerous books on pretty much all advanced mathematical topics which contain both answers and solutions. If you wish to read on intuition there is so much available on Mathoverflow etc. that this question and it's presuppositions must surely be an affront to those who labored to gain insight in previous eras where the internet did not allow us to see such vistas.

Perhaps this is in part an answer, why write a book when most new students are too busy searching the net for information. If you seek an audience, is a book really the format of choice now ?

Of course, I value books and I'd like to finish a few of my own, but I will admit this is more about quality control and finishing some of my own thoughts. If I wanted to influence students and push the edge of understanding, I think You Tube video tutorials are probably more where it's at.

  • $\begingroup$ " If you wish to read on intuition there is so much available on Mathoverflow etc. that this question and it's presuppositions must surely be an affront to those who labored to gain insight in previous eras where the internet did not allow us to see such vistas." Did you mean Math Stack Exchange rather than Math Overflow? But Stack Exchange continues to get new legitimate, excellent questions asking for intuition, which suggests that intuition isn't taught or discussed enough, which commends more books on intuition! $\endgroup$
    – NNOX Apps
    Jul 9 '21 at 2:32
  • $\begingroup$ @ugro not a typo, but certainly I would include MSE and MESE in the etc. Also, reddit and Wikipedia and mathworld and... the internet. But, my choice of MO is because we're talking about "advanced math topics", when I think advanced I think MO. I can read MSE or blogs about advanced math, but often the source is someone who doesn't really know much more than I, so they're not giving that high-level big picture that you might get from a more high-powered mathematician who is interested in making more disciples for their cause... just my opinion, I love the MSE too. $\endgroup$ Jul 12 '21 at 19:25
  • $\begingroup$ Isn't it obvious that a book would be more productive and concentrated than sifting through "reddit and Wikipedia and mathworld and... the internet"? $\endgroup$
    – NNOX Apps
    Jul 13 '21 at 5:48

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