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Studying Struggling with end chapter problems (Spivak)

  1. Sep 2, 2016 #1
    I'm a physics student trying to get a more in-depth understanding of math. A few weeks ago, I started studying from two textbooks, Spivak's Calculus on Manifolds and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. So far, the stuff from Hubbard's text is pretty straight forward. The problems are, for the most part, fairly easy, and even if I come across a difficult one, it's at least something I can take a stab at. On the other hand, I'm having a much different experience with Spivak. I read the text and understand the proofs, but when it comes to actually solving the problems, I struggle. I'm happy if I can work 30% of them. It makes me feel pretty dumb, I won't lie. At this point, I may very well give up on the end of chapter problems and just use it as a supplement to Hubbard.

    I've never really sat down and studied analysis from a text like Rudin; my only experience with the subject comes from Spivak's Calculus, which I went through maybe 50% of. Is it unreasonable to tackle a text like Calculus on Manifolds without being well practiced at analysis problems? Maybe if I just try and hammer out problems, it'll eventually start making sense? It doesn't help that Spivak doesn't include many problems to begin with.
  2. jcsd
  3. Sep 2, 2016 #2


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    There may be two things going on. First, if you're having problem with chapter 1 problems, then you're not really prepared for the book. Most of those problems are rather standard. Secondly, you cannot read a book like Calculus on Manifolds on your own and expect to make quick progress. You need to dissect every single proof and appreciate the subtleness of the arguments. This is true for most analysis books. The thing is, it won't start to make sense as you read on. You need to get a good grasp of each section because each subsequent section will depend completely on the previous one. If the finer points of compactness alludes you, then it'll only get worse.

    My advice: Get a better understanding of Analysis with a simpler book before you attempt to study a book like this. There do exist books with more intuitive explanation. Gain that intuition first, then focus on the details.
  4. Sep 2, 2016 #3


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    the tell tale remark to me is that you only read 50% of the more elementary calculus/analysis book by spivak. why would you expect to jump successfully into modern advanced calc when you apparently don't know beginning rigorous calc thoroughly? and there are a lot of problems in that more elementary book to work on. and you don't say whether you also have the linear algebra prerequisite spivak mentions in his preface.
    Last edited: Sep 2, 2016
  5. Sep 2, 2016 #4
    Well, when I say 50%, I mean I went over the parts that I thought were important. I don't feel like finding the book again, but I did the chapters on limits, continuity, differentiability, integration etc. I've studied linear algebra, so that's not really a problem. I could definitely use some brushing up, but I'm fairly comfortable with it.

    So what do you recommend doing? What sort of analysis text should I be looking for?
    Last edited: Sep 2, 2016
  6. Sep 5, 2016 #5
    Work through SPivak doing all the exercises.
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