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Note: I've edited this question on October 9th, after establishing a bounty on it.

What are the best introductory calculus textbooks that

  • explain why calculus is important in a broad intellectual and scientific context, justifying its inclusion in a liberal education. Necessarily, this means it puts great emphasis on applications to science and engineering, with specifics;

  • are non-dogmatic, i.e. they justify what they say, in most cases in a serious way, i.e. heuristic rather than rigorous (logical rigor in mathematics, a beautiful thing in some contexts, can approach sarcasm in other contexts);

  • Are at a level that can be understood by students adequately prepared in prerequisites but primarily interested in other things than mathematics and not inclined to develop their mathematical ability beyond what is needed to become liberally educated and to know that such a field as mathematics exists (in a way in which most educated people currently do not know there is such a field).

Such a book would not say "This differential equation arises in the study of fluid flow", but rather "This section will explain how to derive this differential equation from these physical principles, and some of the exercises are on understanding steps in this and related derivations, and some of them are on applying the techniques." That's a part of the "non-dogmatic" nature of the book. It would also explain why, or at least that, a particular equation is consequential beyond mathematics.

Such a book would necessarily be ruthless about refusing to include some topics that are part of the (far too) ritualistic standard course.

PS: Could answers describe the books' contents and say why they are judged to answer the points above?

Martin Sleziak
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Michael Hardy
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    Stewart's Early transcendentals? But you are probably aware of that already – Prahlad Vaidyanathan Oct 04 '13 at 19:05
  • A true story: I mentioned to a certain professor of mathematics that a section in Chapter 3 of Stewart that is putatively about applications is dishonest. I said an honest account would be about derivation of differential equations from physical ideas. She said that would be a good thing but there's not enough time for it. I said "So scrap ninety percent of it and then there's time." She indefinitely deferred further discussion of that topic. Are there more important things for mathematicians to do? Standard dogma says work at the forefront of research is more important. That's just dogma. – Michael Hardy Oct 04 '13 at 19:07
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    fundamentals of mathematical analysis by G.M. Fikhtengolts – ILoveMath Oct 04 '13 at 19:08
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    @PrahladVaidyanathan : Stewart's Early Transcendentals is the polar opposite of what I have in mind. At some time in the past someone invented a calculus course for people who want to understand calculus. Then people started pushing hordes of completely unqualified students to take calculus. Then people like Stewart watered down that course to the point of making it phony. I'm looking for honest books for honest students. – Michael Hardy Oct 04 '13 at 19:10
  • I can imagine comments provoking me into writing some detailed opinions of some specific aspects of Stewart's book...... But not now. – Michael Hardy Oct 04 '13 at 19:17
  • PS: Some people may feel tempted to read my emphasis on "honest" as meaning they're books for mathematically inclined students rather than those capable of learning mathtematics but who might not seek that subject out. I.e., I'm not looking for a high difficulty level, but I am looking for a book suitable for students who show up in order to learn the subject rather than to get a grade to put on their resumes. – Michael Hardy Oct 04 '13 at 19:31
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    I know thit is not what you want - but any good introductory physics course. From my "childhood" I have a preference for Feymnan's lectures, but there are several other choices (some of them might be better from this perspective). The only downside is too much physics :) – user8268 Oct 04 '13 at 19:37
  • Which standard parts would you drop, aside from most differential equations? – dfeuer Oct 04 '13 at 20:39
  • @user8268 : If you make your comment an answer and list two or three or twenty such books with a few comments on the pros and cons of each, I'll probably up-vote it. – Michael Hardy Oct 04 '13 at 20:57
  • @dfeuer Depending on abilities of the students and time constraints, I'd drop much of the great variety of different sorts of things that can happen when one thinks about the concept of limit of a function and emphasize that the main thing to know about that, as far as differential calculus is concerned, is that it deals with the indeterminate form $0/0$. I'd drop the mean value theorem except as maybe an optional section that's primarily about what it's used for. I'd stop pretending "related rates" is one topic within differential calculus and make it clear that it's the whole subject. – Michael Hardy Oct 04 '13 at 21:01
  • ....and a lot of other stuff. To be continued, maybe..... – Michael Hardy Oct 04 '13 at 21:02
  • I notice that I used the word "serious" in a potentially provocative way. I can't resist mentioning this: C. S. Lewis may be best known for this children's fantasy series Chronicles of Narnia and his many books as an amateur theologian, but as an academic he was a Lecturer and Tutor in English literature for some decades at Oxford and later a Professor of Medieval and Renaisance English at Cambridge, and in fields in which he actually had credentials, one of his works was a book on the ways in which the meanings of words change over centuries. And he noted that at the beginning of the.... – Michael Hardy Oct 05 '13 at 03:35
  • ....20th century, the word "serious" had religious connotations. – Michael Hardy Oct 05 '13 at 03:35
  • "justifying its inclusion in a liberal education for purposes other than contemptible ones like using it as a weeder for medical school or business school applicants." This kind of normative statement seems unnecessary in the question. I, for one, am glad these things help weed out such applicants, even though there are many better reasons to study the calculus. – ely Oct 09 '13 at 20:22
  • @EMS : Obviously the sequence of courses that you took will be of no use to most students who take calculus because they won't do what you did. Most students who take calculus never take another math course after that. – Michael Hardy Oct 09 '13 at 20:32
  • @EMS : Would you seriously propose that conventional calculus courses using a book like Stewart are useful to typical students who take them? Maybe you've never taught such a course. They learn algorithmic things and nothing else, and they don't come to even suspect anything else exists, no matter how explicitly you tell them it does. – Michael Hardy Oct 09 '13 at 20:34
  • I disagree with you about using calculus as a weeder, but I've deleted that phrase from the question. I think those who promote its use for that purpose are either gullible or criminal. – Michael Hardy Oct 09 '13 at 20:36

1 Answers1

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Try looking at Agnew's book, which now seems to be freely available on the internet:

Ralph Palmer Agnew, Calculus. Analytic Geometry and Calculus, with Vectors, McGraw-Hill Book Company, 1962, xiv + 738 pages.

Agnew's book is probably not as "strongly oriented towards applications in science and engineering" as some books (still, it is probably above average in this regard, being written back during a time when such applications were more thorough than is the case today), but the writing is remarkably fresh and the exercises are among the best you can find in an elementary calculus text.

Another book worth looking at is below. This one is a bit more strongly oriented towards applications and has some novel approaches to certain topics.

James Callahan, et al, Calculus in Context. The Five College Calculus Project, W. H. Freeman, 1995/2008, xxi + 845 pages. [This also is online. See .pdf file for Chapters 1 through 6 and .pdf file for Chapter 7 through 12.]

Dave L. Renfro
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  • Callahan's title looks promising, but the degree to which it can live up to the title is to be seen. – Michael Hardy Oct 04 '13 at 20:40
  • The table of contents in Callahan looks better than I expected. – Michael Hardy Oct 04 '13 at 20:56
  • @Michael Hardy: Chapter 12 looks especially promising for what you're looking for. I got a copy of this book back in 1995 or 1996 and used parts of it in some of the calculus classes I taught at LSMSA during 1996-1999. I seem to recall that it got a lot of good press when it came out, but I haven't heard much about it in quite a while. I was pleasantly surprised to learn (today) that it's now freely available on the internet. – Dave L. Renfro Oct 04 '13 at 21:38
  • Possibly I'll "accept" this answer because of Callahan, but maybe first I'll try a bounty and see what other answers it can attract. – Michael Hardy Oct 05 '13 at 21:35
  • Agnew may actually be public domain now. Its copyright was in 1962, and it doesn't appear to have been renewed. –  Jul 10 '14 at 17:44