Jane in NC Posted February 28, 2008 Share Posted February 28, 2008 With apologies for hijacking the Kimber's thread. You can take a look at the 3rd edition of Spivak's Calculus on Google books: http://books.google.com/books?hl=en&id=7JKVu_9InRUC&dq=michael+spivak+calculus&printsec=frontcover&source=web&ots=wiHIDhH0Vd&sig=RzR9aaub_1NpDjGsBXHkhjIAGKw#PPA185,M1 In the meantime, I'm pulling my Spivak off the shelf. Jane Quote Link to comment Share on other sites More sharing options...

Jane in NC Posted February 28, 2008 Author Share Posted February 28, 2008 I compared Spivak to Apostol on limits. Frankly I am a bit surprised by the handwaving Apostol does on this subject. Spivak begins his chapter on limits (which I now see is unavailable through Google books) with a provisional definition. He offers numerous graphic examples, classic limits involving sin 1/x, xsin 1/x and x^2 sin 1/x. He then considers an "amusing" function f(x) = x for x rational, 0 for x irrational before confessing that his demonstrations lack proof. It is then that he launches into epsilon/delta. His initial comments remain intuitive, but then he gets serious. This material covers fifteen pages of text followed by five pages of problems. The next chapter covers continuous functions, followed by a chapter entitled "Three Hard Theorems" all dealing with continuity on intervals. This is real analysis at a conversational level, leaving students exercises to prove some meaty theorems. The normal Calculus stuff is there but so is real mathematics. A book to consider for multivariable calculus is Modern Mathematical Analysis by Protter and Morrey. The text assumes a one year study of Calculus and presents multivariable material in a superior manner than the traditional Calculus text. It would be a great follow up to Spivak. Jane Quote Link to comment Share on other sites More sharing options...

Charon Posted February 28, 2008 Share Posted February 28, 2008 His section on limits is awesome. I really was expecting another Apostol like outcome. But, at least for that section, he really does go into great detail about the idea of a limit and how it is formalized and why. And, he backs it up in the exercises with a bunch of good problems. Problem 3 on page 106 is essentially 6 epsilon-delta proofs. By contrast, despite all the wonderful development of the material within Apostol, when you get to his section on limits you get a very sexy looking discussion of epsilon neighborhoods. You get exotic theorems for such a section, like that the indefinite integral is continuous which is something that only with Apostol's integration first approach could you do. (Otherwise, you would just have to do it later when you cover integrals.) There is nothing at all lacking in the presentation, and if anything, it beats Spivak hands down on neat-o-ness. But, you get to the problems and it is just a bunch of "use the limit theorems to calculate the following limits". I don't want to be too harsh -- the last eight or so problems are pretty good. They are kind of hard, though. You shouldn't have to skip, for instance, to working through Apostol's outline of an alternate proof of the fact that sine and cosine are continuous straight out of doing just a bunch of computational problems. Nevertheless, this section is not altogether that bad if you make sure and do all the problems. (It would be really easy, though, for a student to do the first 25, get stuck on the last eight and decide he had more less done enough for that section.) But, while Spivak lacks a certain amount of sizzle, I have to say he really is taking a no nonsense direct approach to really getting the student to prove something with epsilons and deltas. And, you have to start doing it on problem 3 not on problem 26. I have to say I am pleasantly surprised. I'll have to get the book and check it out more thoroughly. Thanks, Jane! Quote Link to comment Share on other sites More sharing options...

Charon Posted February 28, 2008 Share Posted February 28, 2008 I compared Spivak to Apostol on limits. Frankly I am a bit surprised by the handwaving Apostol does on this subject. Spivak begins his chapter on limits (which I now see is unavailable through Google books) with a provisional definition. He offers numerous graphic examples, classic limits involving sin 1/x, xsin 1/x and x^2 sin 1/x. He then considers an "amusing" function f(x) = x for x rational, 0 for x irrational before confessing that his demonstrations lack proof. It is then that he launches into epsilon/delta. His initial comments remain intuitive, but then he gets serious. This material covers fifteen pages of text followed by five pages of problems. The next chapter covers continuous functions, followed by a chapter entitled "Three Hard Theorems" all dealing with continuity on intervals. This is real analysis at a conversational level, leaving students exercises to prove some meaty theorems. The normal Calculus stuff is there but so is real mathematics. A book to consider for multivariable calculus is Modern Mathematical Analysis by Protter and Morrey. The text assumes a one year study of Calculus and presents multivariable material in a superior manner than the traditional Calculus text. It would be a great follow up to Spivak. Jane No -- I was able to get to it on google. I think I wasn't able to get to it in Amazon. It's not Baby Rudin -- definition-theorem-proof... now prove the Baire Category Theorem for an "exercise". But, I really do think he is trying to get his students to do epsilon-delta proofs, and at any rate, they are there in the exercises. It looks like maybe even every other exercise or something. It's pretty solid. I don't think you would walk away from spivak unable to do an epsilon-delta proof, for instance. I think you really could do that with Apostol, thinking to yourself "Well I did the vast majority of the exercises, so I'm pretty solid". We'll have to see what the rest of the book looks like. So far, I definitely like it better than Apostol, hands down. I'm not altogether sure Courant isn't as good in terms of what you can do after getting through the exercises. Quote Link to comment Share on other sites More sharing options...

Charon Posted February 28, 2008 Share Posted February 28, 2008 It doesn't seem to have any multivariable calculus or differential equations or anything like that. That must be why everybody wants to do something else like Apostol or Courant. Quote Link to comment Share on other sites More sharing options...

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