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I would like to ask for some intermediate level textbook for calculus (single variable), or, at least, some supplement to Spivak's Calculus for better understanding on how to approach and solve his problems.

I read and done a lot of Stewarts and Leithold Calculus and was REALLY easy and boring. They arent so rigorous and sophisticated mathematically; the exercises are extremely easy. So I went straight to Spivak's Calculus, because everyone say it's rigorous. Indeed, it was TOO rigorous, haha. No problems until the limits section, nor in the theory of limits. I was enjoying it. However, I have huge difficult in solving the problems in this chapter, specially because there aren't many examples (at least, not in the same level of difficult of the proposed problems).

I have a good background in high-school math (Theory of Sets, Functions, Trigonometry, Polynomials, Complex Numbers, Analytic Geometry, Combinatorics, Basic Linear Algebra, Number Theory etc.).

Thanks! :D

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    $\begingroup$ Why don't you try Apostol's book? $\endgroup$ – Daniel Apr 6 '15 at 15:43
  • $\begingroup$ @DanielEscudero Maybe because it's shamefully expensive ;-) $\endgroup$ – Jean-Claude Arbaut Apr 6 '15 at 15:56
  • $\begingroup$ @DanielEscudero Escudero, thanks for the reply! I've heard that Apostol's book is even harder than Spivak; this is why I tried Spivak's Calculus first. Specially because integration starts Apostol's book. Do you have another opinion about it? Could you explain what you think? Thanks again! :) $\endgroup$ – Matthew Apr 7 '15 at 1:40
  • $\begingroup$ @Matthew I've explained it in an answer. $\endgroup$ – Daniel Apr 7 '15 at 2:35
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I personally like Apostol's book, as I mentioned in a comment. Here are the reasons:

  1. It has a very good an rigorous exposition, without letting go away the intuitive ideas behind calculus (but it's not impossible to read).
  2. It presents integral before derivatives, which is historically right and shows the importance of the fundamental theorem of calculus and how an integral is really some kind of "sum" instead of being the backwards process of taking derivatives.
  3. It has a good amount of challenging (and doable) exercises and very ingenious proofs (as I've pointed before), and is a perfect bridge between "engineering" mathematics and "pure" mathematics.

That's my opinion, I hope you don't downvote me (as use to happen with subjective answers).

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  • $\begingroup$ Would you say Apostol's calculus I and II books almost cover the calculus part of his analysis book? $\endgroup$ – StubbornAtom Nov 5 '16 at 11:54
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One text that I personally found to be a very good reference is Leon Simon's An Introduction to Multivariable Mathematics. It was compiled from the lecture notes for the Math 51H course at Stanford (their equivalent of Harvard's Math 55 series), which he taught for many years, and aims to cover linear algebra, analysis in $\mathbb{R}^n$, and an introduction to real analysis. The physical version is quite inexpensive relative to other texts and there is an online version!

Simon's book is a serious introductory text (however self-contradictory that may sound) designed for prospective mathematics majors at Stanford and will certainly prove to be a decent challenge. It is quite short in length but is fairly dense and does not contain many exercises. For more exercises in this style, you can search for previous Math 51H problem sets.

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I find that getting it "just right" for lower level calculus books is quite hard. Books like Stewart tend to somehow miss most students. For students that are very good and have a lot of background, this text is wholly boring, and does not bring anything to the table. For students with little background, the applications seem trite and it doesn't exactly go out and excite them to study more. The book struggles to answer questions like "why is that theorem true?" without appealing to heuristic arguments at best.

A personal favorite which is definitely harder than Stewart, but easier than Spivak is George Simmons, Calculus with Analytic Geometry. It has a great many problems in it, some are quite easy, and others require more creativity. I would not go as far as to call the book "hard," but some of the tougher exercises kept me for a little while when I took calculus - it is simply harder than Stewart. It is not 100% rigorous, but many of the important proofs that are not in the sections can be found in appendices.

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Stewart Calculus: Concepts and Contexts 4e

You really can't go wrong with it. It has thousands of exercises ranging in difficulty from mundane to nearly impossible. The examples are great and it explains even the most nuanced of ideas very clearly with beautiful graphics.

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  • $\begingroup$ Firstly, the OP mentions that he bores of Stewart's book. Secondly, I must admit I heartily disagree with your estimation of Stewart. It is good for mass market, particularly for schools which may lack professors or mathematicians sufficiently capable or willing to design and maintain a calculus curriculum. But the level of sophistication is very low, so low as to not adequately prepare aspiring mathematicians. It certainly would barely help with Spivak, as the OP requests, since Spivak is pitched so drastically higher than Stewart. Further, Spivak demands that his readers think. $\endgroup$ – davidlowryduda Apr 6 '15 at 18:42
  • $\begingroup$ @mixedmath: There are a lot of Stewart textbooks and I have found this particular one to be leagues ahead of the others. It's my personal suggestion and he does not need to follow it if he doesn't want to. Furthermore, I think it is presumptive of you to state what you did about the quality of the professors and departments who would prescribe this textbook considering its popularity with universities globally. Indeed, it would be a perfect SUPPLEMENT to Spivak (rather than a replacement) as OP requested. $\endgroup$ – aidandeno Apr 6 '15 at 18:46
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    $\begingroup$ Aidandeno, thanks for the reply, but Stewart really doesn't help me. I dont see much difference between his books in level of difficult, actually, but thanks, anyway. And yes, I would like a replacement OR a supplement for Spivak. :) $\endgroup$ – Matthew Apr 6 '15 at 20:39
  • $\begingroup$ Mixed Math, thanks for the reply. I really appreciate that Spivak demands that his readers think, but I feel that he wants that I discover analysis - something like to demand that a child discover the arithmetic series at 8. I know that a more independent approach is necessary for constructing a mathematical reasoning, but, again, I'm feeling that i'm not really learning because i dont see how to approach and solve his kind of problems. I'm more used to something like the method "learn by problems" in Andreescu books.Of course it demands a lot of hours of reasoning, but nothing like Spivak. :( $\endgroup$ – Matthew Apr 6 '15 at 20:42
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    $\begingroup$ In my experience as a tutor, this book, like many other elementary calculus texts, pretends to appease the reader with "application" which is either overtly trivial, or much too complicated for most readers. $\endgroup$ – Alfred Yerger Apr 7 '15 at 4:19

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