I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John.I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the problems provided.After reading that book, should I go for one of the two above or should I study something else like Rudin?My focus is on being rigorous and also adept at problem solving. Thank you.


I think Courant and John's book is the richest of the three textbooks you mention: it essentially contains the other two.
Spivak is the most rigorous (and is very, very aesthetic) but I think that if you want rigour, it would be boring to apply it to material you already know: better start learning more advanced analysis.

1) Rudin is of course good but a bit dry.

2) John and Barbara Hubbard wrote a very geometric book,Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach whose center of gravity is the subject traditionally called "calculus of several variables".
It is probably the best book in that category, because of the rigour and depth of the presentation, its modernity (everything done in the language of manifolds and differential forms), the wealth of material presented (including linear algebra and integration) and, last not least, the interesting (mostly historical) vignettes in the margins.
There are many, many friendly easy little calculations made in the text; moreover the exercises are carefully thought out, starting from trivial computations and aiming for more difficult results, provided with hints.

3) Lang wrote a very impressive book, Real and Functional Analysis, which contains an amazing lot of mathematics packed in less than 600 pages.
As always with Lang his forte is not in explicit, down-to-earth calculations, but the book is an amazing synthesis of basic analysis: you will be introduced to topology, functional analysis, integration theory ( in $\mathbb R^n$ but also on locally compact spaces), manifolds, differential forms,...
And the book profits from Lang's vast culture: it must be the only textbook which explains the role that Hironaka's resolution of singularities (one of the most difficult theorems in algebraic geometry) might play in the statement of Stokes' theorem!

3 1/2) As a stepping stone to 3), you might use Lang's more elementary Undergraduate Analysis, which you could consider a review of Courant-John in more modern language, and an ideal preparation for the more ambitious 3).

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    $\begingroup$ I'd also like to point out that Lang's Undergraduate Analysis has a companion volume, Problems and Solutions for Undergraduate Analysis, which provides solutions to the exercises in the former. A similar statement is true for Lang's Complex Analysis. This makes these texts very useful for the self-learner. $\endgroup$ – ItsNotObvious Nov 7 '11 at 23:34
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    $\begingroup$ Thank you very much, Mr Elencwajg.That was really helpful.I now see the picture quite clearly as to where I want to be in the next 1 and half years. $\endgroup$ – Eisen Nov 8 '11 at 2:25

I'm familiar with all three books, although I've never fully read through or taught from any of them. For me, Spivak's book is more fun to dip into from time to time, at least if you already know the basic material. Indeed, the exercises in Spivak are an excellent place to look if you want a pure math supplementary topic or student project when teaching calculus. Spivak's writing is more energetic and the page layouts (pictures, side bars, etc.) are much more engaging. A possible negative aspect of Spivak's book is that there are virtually no physics or engineering applications.

Courant's book is by far the best for physics applications and plane curve topics, and I've made use of small selected parts of it many times when teaching calculus (e.g. Lorentz transformations as imaginary rotations, big "Oh" and little "Oh" notation, the big picture of how to integrate rational functions, the $u = \tan \frac{\theta}{2}$ substitution, etc.). In general, the problems in Courant's book are much harder than the problems in the other two books, and this could cause some difficulty in using Courant's book for self-study. On the other hand, the problems in Courant's book are less "intertwined" with the book's exposition, and thus many of them can be skipped without being much of a hindrance to reading the text.

Apostol's book is probably the best of the three books for a strong mathematics student who is beginning to study calculus. Apostol is dryer than Spivak or Courant, but I think it would work better as a class text than Spivak or Courant. I would recommend Apostol even more strongly over the other two books in the case of someone studying calculus on their own. [I'm sure there will be different opinions on this, but for what it's worth, I say this as someone who never took a calculus course (high school or college), but rather learned the material by self-study from a text (Edwin Joseph Purcell's 1968 Calculus With Analytic Geometry, which was the only calculus book in our county public library).]

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    $\begingroup$ I think it is a virtue of Spivak that he doesn't write about physics. Calculus has many applications in many fields, physics only one of them. Biologists, economists, computer scientists, and surely a lot of other fields that don't come to mind right now use calculus. $\endgroup$ – Michael Greinecker Apr 3 '12 at 23:27

I have Courant, Apostol and Spivak, and the difference in difficulty is negligible between them. You could go right into Rudin, the difficulty might differ somewhat but not enough to dissuade you from approaching Rudin. I personally went through Apostol, and when comparing it with Courant I see many of the same questions in the exercise sections.

I plan to, in the future, go through Courant and Spivak as well. I want a very deep and thorough understanding of Calculus in particular, and more experienced mathematicians than I have suggested that the best way to get this is to go through the material in many different ways.

Personally, I feel that just going through one book is not enough to really embed the knowledge in me to the level that I want. I will, however, continue moving forward, as I would suggest you do as well; a certain amount of forward momentum is needed to keep the interest fresh and the brain actively involved. I think you would be bored if, immediately after going through Courant, you attempted to go through Spivak or Apostol. Too much of it is the same, and your brain would "check out" while reading and doing the exercises. It's better to let some time pass, go through some other books (such as Rudin), and then come back to a different approach in the future.

  • $\begingroup$ Thank you.But I would better wait for some more comments from the experts. $\endgroup$ – Eisen Nov 7 '11 at 15:21
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    $\begingroup$ No problem, I agree. I'll be looking to see what they say as well. $\endgroup$ – process91 Nov 7 '11 at 15:58
  • $\begingroup$ I hope to see some expert advice here!Thanks.:) $\endgroup$ – Eisen Nov 7 '11 at 16:28

For sheer beauty of writing and depth of understanding of the mathematical structure of basic calculus, nothing beats Spivak. Sadly,the lack of applications-not only to physics and engineering,but any other subject for that matter-is really its' Achilles' heel for using it as the text for a high-level learning of calculus. Otherwise,it would be the canonical choice against all comers. Courant/John is nearly as beautiful and it has a legion of wonderful applications and examples. Unfortunately,as Dave commented above,it's exercises are very bizarre, difficult and there are nowhere near enough of them. Apostol is written by one of the modern masters of classical analysis and it is probably the most complete of all 3 textbooks. Not only does it contain a careful and very rigorous presentation of both the theory and many the applications of calculus,it also has many wonderful historical notes that illustrate the sources of the material as well as a self-contained presentation of linear algebra. In short, all three have justifiably earned their classic status and they all have their advantages and disadvantages as textbooks for the talented student. If I had a gun to my head and had to pick one only-I'd probably go with Apostol simply because it's the most complete of the 3 and as a result,it's the most flexible. Even better would be Spivak supplemented by Apostol. Of course,this is all for calculus of a single variable. For functions of several variables,that's a horse of another garage and you can find many fine suggestions for textbooks on that at Good introductory book on Calculus on Manifolds.

  • $\begingroup$ I guess they don't approve of my daring to criticize the Holy Book Of Calculus by Spivak. And I'm a FAN of Spivak. $\endgroup$ – Mathemagician1234 Apr 4 '12 at 20:11
  • $\begingroup$ I liked it; I appreciate an honest and balanced review. There's no place for dogmatism in math, is there? ;) $\endgroup$ – user27634 Jul 10 '13 at 5:43
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    $\begingroup$ No,but that doesn't stop people,believe me. A friend of mine told someone in the Harvard math department he didn't like Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS. A year later he came up for a summer study program there for graduate students and the faculty passed him over. The faculty member of his department who recommended him told him later it was because they heard he didn't like Rudin-and to them,that meant they couldn't take him seriously as a mathematician. Amazing,but true. $\endgroup$ – Mathemagician1234 Jul 10 '13 at 6:50
  • $\begingroup$ I used Spikvak in AP Calc in 1977, and have never found another math text quite like it. $\endgroup$ – rbp Oct 3 '14 at 16:17

The best book is Courant-John because it contains all the topics associated with traditional calculus courses as it was understood in Europe and Russia. Second place is for Apostol's calculus, which is a typical calculus course based on linear algebra, as is understood in the US. Third place is for Spivak's calculus, this course is restricted to onedimensional calculus. If you want to buy one get Apostol's.

  • $\begingroup$ I disagree with the implication that certain books are somehow not understood in some geographical location. $\endgroup$ – smokeypeat Jul 17 '17 at 0:09

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