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Do I need Spivak/Apostol before analysis?

  1. Jul 3, 2015 #1
    Hello, I'm currently studying math and want to become a mathematician. So far I know up to computational single variable calculus, i.e. Calculus I and II, but not III. Since I found the school textbooks easy I decided to follow the MIT course for Calculus III, so I have started watching the multivariable calculus videos by Denis Auroux. I want to move onto analysis after multivariable calculus. However, this gets me wondering. Seeing as many people see this (MIT) course as being computational multivariable calculus, would I need to re-study all my calculus using say apostol before moving onto analysis, or could I jump straight from the MIT course to something like Pugh's analysis book? What would I learn from spivak/apostol that I can't learn by jumping straight into analysis after having learned computational single and multivarialbe calculus? I'm good at problem solving (judging by my ability to solve 3 problems from this year's USAMO)

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  3. Jul 3, 2015 #2
    Tough call. It really depends on the depth to which you master the material. Watching the videos is a much lower level than working all the assigned problems correctly.
  4. Jul 3, 2015 #3
    Hello, thanks for your reply Dr. Courtney. What about someone who completely masters the MIT course and have only previously done school math up to Calc I and II but are quite good at problem solving. Do you think they would be prepared for analysis?
  5. Jul 3, 2015 #4
    Students are seldom as good as Calc III as they think they are. I've taught Calc III, and my experience is that students almost always overestimate their abilities going into exams and are suprised when their mastery proves far below what they thought.

    Analysis is hard. Most students who passed by Calc III class with Bs or Cs would not have been ready for analysis. Maybe half the students with As would have been ready. There are a combination of issues, but usually a longer list of course success is needed to be ready for analysis. It's not just about pre-requisite skills, it is about mathematical maturity.
  6. Jul 5, 2015 #5


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    I think those books look good. I haven't read them, but they look like they cover a lot of the stuff that is important for analysis. You probably don't have to 'work through the entire book' since you already know calculus but i think spivak has sections on like mathematical foundations that you could work through
    Last edited: Jul 5, 2015
  7. Jul 5, 2015 #6


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    I think working through entire Spivak is very beneficial, even if you know calculus already. There are other nice alternatives to Spivak though, such as Nitecki.
  8. Jul 5, 2015 #7


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    I can't comment very specifically about spivak but i remember there being like a hundred pages of preliminaries which covered like induction, set theory etc... I think that these are the neccesary and often unstated prerequisites for analysis. i think that working through the entire book would ideally be helpful, but that seems like it would be more like 'relearning calculus', which may not be necessary if you already have a good understanding of those concepts. Imo what you would miss by going directly from calculus to analysis is experience with fundamental concepts in mathematics and experience solving more abstract problems.
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