Differential&Integral Calc with Courant

[johnnyies]

Is this a good text? Compared to Stewart or Larson? If it isn't, are there any other texts that are better than Courant?

And how low can this text run? I looked on Amazon and just for both the volumes, a new copy can run up to $130 and a used copy printed in the 1930s is almost $40 after shipping.

 

[n!kofeyn]

Courant's book is extremely good. There is a modernized version of his book written by Courant and Fritz that might be cheaper. Search this forum, as Courant's book has been discussed everywhere and multiple, multiple times.

 

[qspeechc]

Don't bother with Stewart or Larson, they are both dangerous to your mathematical health. I second Courant and John:

https://www.amazon.com/dp/354065058X/?tag=pfamazon01-20

 

[johnnyies]

What about Stewart and Larson that everyone here seems to hate so much? Why would the major universities use it then?

 

[naele]

I think there's a lot of hate around here for books like Stewart's but I'm not sure why. I would actually recommend Stewart's book especially as a pre-requisite for studies in analysis since it helps you develop a fundamental intuition about the functions you work with and study in a first course of real analysis.

And I don't see what's to hate about the book. It presents the epsilon-delta definition of limits of a function and proves certain theorems where appropriate, among other things.

 

[snipez90]

While I don't think people around here necessarily hate Stewart, naele certainly brings up a good point. Texts such as Courant, Apostol, or Spivak are for all intents and purposes basic analysis texts. If you don't believe me, look through some of the exercises (or I guess specifically the first chapter w/ appendix of Courant) and take an analysis course or reference Rudin's Principles of Mathematical Analysis. The point is, many people already know some calculus (perhaps the more mechanical aspects) before they encounter such a text for a more rigorous understanding. This is perhaps not the the original intention of these authors, who might have written it for bright students who hadn't seen calculus before, but this hardly changes the fact that many people already have gained some calculus intuition before learning the basic proof techniques of analysis from the aforementioned texts. Of course, this does vary from text to text. For instance, Spivak and all its exercises will look more like the first 6 or 7 chapters of Rudin than Courant.

 

[qspeechc]

Books like Stewart are fine for people who don't intend studying mathematics or physics. It is ridiculous to recommend them to mathematics students.

I don't buy the argument that books like Stewart build analysis intuition. I've read Stewart, unfortunately, from cover to cover. It helped me not one iota when I read Courant and John, or proper analysis books like baby Rudin or Pugh's Real Mathematical Analysis.
Stewart and similar books are meant to teach the mechanics of calculus: how do I compute this integral, how do I maximise the area etc. I never studied any of Stewart's so-called "proofs" in his calculus book, and I was fine, because it is a cookbook. His "proofs" never helped me understand how to do anything in his book, and never helped me in analysis. Courant and John's exposition is self-sufficient, it is enough to understand the rigorous proofs in the book and no prior exposure to calculus is assumed or needed.

Furthermore, I do not see why one should first read Stewart, which is at over 1000 pages, and then Courant and John, covering basically the same material but more rigorously, and then an analysis book like baby Rudin. Why waste your time with 1000 pages of drivvel when there is so much more stuff you can learn in mathematics in that time? (I should know).

Stewart does not help you grow mathematically. Compare Stewart to a pre-calc book. Stewart is no more difficult, it does not train you in mathematical thinking, it does not advance you mathematically, it merely teaches you to perform calculus, which frankly, once you get to a certain level as a math student, your knowledge from Stewart is irrelevent. And the sections on vector calculus are appaling imo, and yet vector calculus is crucial to physics.

If you're still convinced you want to drop over $100 on Stewart, go ahead. Get one of the earlier editions if you can, they will be cheaper, and a bit shorter, but basically the same as the newest edition.

 

[snipez90]

I like how you start off with "I studied [Stewart] from cover to cover" and then proceed to claim that you never actually studied any of his proofs. You do realize that most of his proofs in say, the 5th edition, are actually quite rigorous? He does include epsilon-delta proofs, and the geometric arguments he makes for the validity of the definition is similar to what you would see in an analysis text. His proofs of the fundamental theorem of calculus and the mean value theorem, arguably the most important results of calculus, are standard (cf. Wikipedia) but entirely correct and detailed. Sure he doesn't include the proof of l'hopital's rule, but that proof isn't particularly interesting. Now in between epsilon-delta arguments and the FTC and MVT, I skipped over the Intermediate Value Theorem and the Extreme Value Theorem. Stewart does not include the proofs of these, but one could certainly argue that instead of introducing least upper bounds and employing some rather ugly arguments to prove these (as Spivak does), it is better to leave these kinds of proofs out, since the techniques (or rather tricks) they demonstrate matter little in the long run. Stewart certainly doesn't have to include connectedness and compactness, which I think make for much more intuitive proofs of the two theorems I mentioned. I'm also fairly sure that Stewart's development of integration is pretty good, and he opts to use the standard riemann integral definition instead of the perhaps slightly easier Darboux definition involving sups and infs.

Now obviously Stewart is hardly the most rigorous and comprehensive text out there. But do realize that when you argue against a text that not only builds the basic feel of calculus but also does include detailed proofs of the important results that you may have failed to look at, you are liable for spreading misinformation.

*EDIT* Your point about going through multiple texts is good, but this is largely an issue of "mathematical maturity". I could certainly see people not being able to make the leap from say Stewart, to Rudin just as much as I can see people not being able to tackle Courant or Spivak as a first encounter with calculus. I'm not saying that it can't be done, but I think that many people who have only studied say precalculus would find say the first chapter of Courant hard to digest. We could disagree on this, but the fact is neither of us took this route.

 

[qspeechc]

I don't have Stewart at hand to check your claims that his proofs are rigorous (I sold it), but as far as I remember, it is not as rigorous as, say, Courant and John, and I don't recall Stewart's "rigour" helping me any when I read Courant and John.

Indeed I read Stewart from cover to cover. Yes I read his proofs, but I never paid much attention to them. The truth of the matter is that Stewart's "proofs" are not an integral part of the text in that one could leave them out entirely and still be comfortably able to work through the text. Could you say the same about Courant and John? No, the focus of Courant and John is rigour, whereas it is not in Stewart's book. Just compare the questions in the books, for example. Stewart is mostly about "compute this integral", whereas Courant and John's questions are more theoretical and proof-oriented.

I think we agree Stewart is not enough preparation for baby Rudin. Like I said, Stewart's Calc book is not more sophisticated than a pre-calc book, so how it prepares you for Rudin, let alone Courant and John, I have no idea. We both agree that a book like Courant and John or Spivak is a necessary prerequisite for Rudin. Yes Courant and John will be a little harder for the noob, but it is necessary to read. And I still do not see why one should read 1000 pages of Stewart before Courant and John, since it never helped me at all to read Stewart first. And reading the chapters on vector calculus in Stewart's books are a complete waste of time. So you're saying it's necessary to read Stewart AND a rigorous calc book before Rudin? I did this and I can vouch Stewart was a 1000 page waste of my time. If a book lke Spivak is necessary, why waste the timewith Stewart first, when it doesn't help at all? It only teaches the mechanics of calculus.
Btw, I read Stewart because I was forced to, and this was before I knew any better, i.e. before I knew there was such a thing as a rigorous calc book.
And many students have had Courant and John or Spivak or similar books as there first calculus books. Indeed, those books were written for students with only pre-calc and no prior exposure to calculus.

But like I said, if your heart is set on reading Stewart, go ahead, but get an earlier edition, which will save you $$.

 

[naele]

Stewart will not prepare you for Rudin, or any other analysis book. I think we can all agree to that. I do think it is foolish to try and learn analysis without calculus though. After going through Stewart you will know how the standard real valued functions behave, especially when it comes to limits and convergence. The other thing, in Stewart you develop a lot of the little algebra tricks which are necessary for a first course in analysis. I certainly don't regret taking calculus before analysis. I'm quite glad I did.

That said, there are probably better books than Stewart's out there, as far as calculus goes.

 

[jbunniii]

I echo the recommendation to get the more modern "Introduction to Calculus and Analysis" [Courant and John] instead of the somewhat old-fashioned "Differential and Integral Calculus" [Courant].

Courant and John is unlike the typical freshman calculus book because it gives proofs for everything and is careful to state under what hypotheses the various theorems hold.

I will say that Courant's proofs sometimes feel a little "informal" or "sketched" compared with Spivak, and I'm not sure I would want it to be my first encounter with delta-epsilon proofs, but it's still more rigorous than the typical introductory calculus book, and it covers a lot of interesting applications that are more or less ignored by Spivak.

I'm not all that familiar with Stewart. But looking at the preview at Amazon I can see that it's aimed at a less mathematically mature audience than Courant and John: there is a much greater emphasis on worked-out examples, boxes around important equations, various sidebar "projects," and simple-minded exercises like "Find the numerical value of sinh(0)." It looks like my high-school calculus text from 1985, except with more "graphing calculator" material. Courant and John isn't anything like that. My advice to the original poster is to take a look at both and decide for himself, because I'm betting that if one of these books appeals to him, the other one won't.

 

[johnnyies]

Should I finish the second part of single-variable calculus (calc2) before moving into courant and analysis? Or will reading through Courant give me a better understanding and make the calc2 class easier?

 

[johnnyies]

I got the second volume, hardcover 1936 printing. I still can't find the first volume for a good price. Any suggestions?

 

[etoipi]

Go to http://kr.cs.ait.ac.th/~radok/math/mat/startall.htm and it's about halfway down. Both volumes online.