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Rigorous Intro to analysis/calculus

  1. Dec 6, 2008 #1

    tgt

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    One is Spivak's book. What are other good ones which compare to it?
     
  2. jcsd
  3. Dec 6, 2008 #2
    Calculus (books 1 and 2) by Tom M. Apostol
     
  4. Dec 6, 2008 #3

    mathwonk

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    rudin's principles of analysis (baby rudin), goursat's course of mathematical analysis, and several old books from the early days of rigorous analysis, like g.h. hardy's pure mathematics, and some others you can find in the library.

    joseph kitchen also has a very rigorous calculus book as spoivak's is. another excellent calculus book with more rigor than average is courant or courant and john.

    all these questions are answered in detail in the first few pages of who wants to be..?

    one of the most rigorous is dieudonne's foundations of modern analysis.

    If like me, you ever find yourself entering a second year rigorous calculus class without having prepared sufficiently in a first year rigorous calculus class, you may as I did, find useful the 20 page or so summary of first year rigorous calculus in the appendix of the book by wendell fleming, calculus of several variables.
     
    Last edited: Dec 6, 2008
  5. Dec 8, 2008 #4
    hah I was just going to recommend Spivak. I took honors calc as a freshman for 3 quarters and we used it- very interesting, *tough* material. Thankfully none of the tests were as hard as the problem sets.

    Why are you looking for an alternative, out of curiousity?
     
  6. Dec 8, 2008 #5

    mathwonk

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    well there is no physics ion there and no applications of any kind. so courant might be a good supplement.
     
  7. Dec 9, 2008 #6
    I'd rather poke my eyes out than read that crap compiled by Apostol. Obstruse elementary mathematics, it was pushed down my throat in grade 11. Atleast Spivak had substance, Apostol just talked in circles in his own little world. The only thing I got from Apostol was history. Save rigor for analysis, and learn calculus from application (ie. the reason it was invented). See Real Analysis by Pugh, the book I'm currently reading.
     
  8. Dec 10, 2008 #7

    tgt

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    Anyone tried "Fundamentals of Mathematical Analysis" by Rod Haggarty?
     
  9. Dec 11, 2008 #8

    mathwonk

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    khemix, maybe apostol is not as much fun as spivak, but it is very scholarly and any strong student will greatly benefit from it.

    well actually your post does put the blame where it belongs if you read it twice, namely on your teacher and your preparation, not on apostol. unfortunately you attributed your bad experience to the book, not to your lack of readiness for it.

    but thats all right. you are well served by spivak. but apostol is a great book for the right student and teacher. it was the first book used by one of my best colleagues at MIT, and he liked it fine. i also used it for returning high school teachers one summer and was very impressed by it. but it is an odd choice for eleventh graders.

    as to rigor, here's a tiny example, compare the proofs of the intermediate value theorem in apostol to that in spivak. see which is clearer. to me apostol was a bit simpler and better, although both are rigorous.

    one fine feature of apostol is that he denies the student the crutch of starting out using the fundamental theorem to do integral calculus, and thus insures ones learning the meaning of integration. i.e. he does integration first, as it should be done. when the derivative is done first, hundreds of years out of order and context, the average student never again makes any effort to understand what the integral means, but always uses antidifferentiation to do integrals.

    The student never learns that this approach does not always work, because it is the only one he ever uses. Apostol gives a thorough study of the integral first, so the student learns that the integral really is something quite distinct from an antiderivative, and can be studied very thoroughly without derivatives. Then when the derivative is introduced and the FTC, it really is a connection between two different ideas, both of which the student has learned.

    In the usual approach the student never learns what an integral is, and no amount of explanation can make it clear after the fact. Once the average student has the FTC, he will never again listen to an explanation of what an integral is. Then when he meets an integral that cannot be done by the FTC he is lost.

    Indeed many students even think that a step function is not integrable because it is not continuous and so the FTC does not apply. How foolish is this? That's like saying a figure made of two rectangles does not have area!

    apostol gives the proof, due to newton, that any monotone function is integrable. this very easy proof makes the idea of integrability quite clear, and shows it is not dependent on continuity. most books omit this theorem, and rely instead on a less easy one that all continuous functions are integrable, which they then leave to the appendix, or leave out altogether.

    spivak's approach is to do the hard technical theorems first, allowing him to prove that continuity does imply integrability. this is now a standard advanced approach, but i think apostol's historical one is even better pedagogically.

    by the way notice that newton, the man who invented calculus for application, also supplied the rigorous proof mentioned above. rigor is a tool for making sure the applications we make are actually correct.

    I suggest you try to realize that just because you yourself had trouble learning something at one point, it may not be because the book was a bad one. maybe you just tried to read it too early in your career. otherwise you run the risk of closing yourself off from some great sources that will be accessible to you later.
     
    Last edited: Dec 11, 2008
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