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Course of Action beyond Calculus

  1. Dec 4, 2007 #1
    I completed a study of Michael Spivak's Calculus recently. If I want to broaden my mathematics knowledge to the point where I can take a rigorous differential geometry course at a nearby university (I think they are using a differential geometry book also by Spivak), what should I do over the next year?

    What (preferably rigorous) texts should I study and do a few problems from?

    I heard I should probably know some advanced calculus, what texts would you recommend to me for that?
  2. jcsd
  3. Dec 4, 2007 #2
    calculus by tom apostol
  4. Dec 4, 2007 #3


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    If you completed Spivak, I would guess you know enough already!

    If you want to broaden your knowledge, do some linear algebra or calculus with several variables. No need to go through calculus all over again.
  5. Dec 4, 2007 #4


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    Here's just a little warning comment: Going through Calculus again could be a good thing because you can improve when you study a course a second time; you can also relearn what you forgot after finishing studying the first time. Still, going into multivariable Calculus is a very good idea
  6. Dec 4, 2007 #5
    Isn't Apostol mostly just one-variable? I am not so sure about Apostol's Calculus, but I think Spivak very rigorously covered a lot about one-variable calculus.
  7. Dec 5, 2007 #6
    you could try calculus on manifolds by spivak, it's a small book, and should be easy to read at this point
  8. Dec 5, 2007 #7
    Do I still need linear algebra?
  9. Dec 5, 2007 #8
    There are two volumes of Apostol, the first is mainly single variable, but the 2nd is linear algebra mixed with multi variable.
  10. Dec 5, 2007 #9


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    A little bit of set theory might be useful. It might be covered in linear algebra books as an aside however.
  11. Dec 5, 2007 #10
    Linear algebra is generally the next logical step in a math progression. It makes differential equations a bit easier. I can't really suggest a text book for linear algebra, but if you look in Mathwonk's thread in the academic section of this forum, I think you'll be able to find a linear algebra book in there.
  12. Dec 5, 2007 #11


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    Why waste your time going through it all over again. Just reference it whenever you need it. Going through Calculus won't broaden your knowledge much beyond "getting" what you didn't "get" last time. Doing Linear Algebra is an entire subject you already don't "get", so it will broaden your knowledge MUCH MORE, which makes complete sense to do something else.

    Why else is good to change subjects? It's good because it will also increase your mathematical maturity.
  13. Dec 5, 2007 #12
    You have to do linear algebra before you can do multivariate calculus anyway.
  14. Dec 5, 2007 #13
    I guess I'm a little confused! :confused:

    So do I need to study:

    1. Linear Algebra,
    2. Multivariable,
    3. And then I would be ready for Spivak's Differential Geometry

    Someone earlier mentioned something in diff. eqs... do I need that too?

    What books do you recommend for whatever sequence of study you would pursue if you were in my place?
  15. Dec 5, 2007 #14


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    From JasonRox:
    The point is sometimes students don't understand some topics well enough the first (or second) time through; restudying the whole (yes all of it) course again helps very much.

    Certainly studying something like linear algebra can broaden the mind since it is different than Calculus - new skills, new ideas.

    homeomorphism said:
    Check the prerequisites for courses in college catalogs. Usually you need one year of Calculus before doing multivariable Calculus. some also say you need three semesters calculus before linear Algebra; some may say you need just one year Calculus before linear Algebra. (Many schools still offer a combination linear algebra + differential equations course)
    Last edited: Dec 5, 2007
  16. Dec 5, 2007 #15
    I actually have pretty good experience with one-var Calc (through AP classes and Michael Spivak's Calculus).

    I am only familiar with multivariable application, but I don't know the deep aspects: such as how and why Fubini's theorem works and where Riemann fails and Lebesgue works, etc. if you see what I mean. So I want to study multivariable in a deep/rigorous way. What books would you recommend?
  17. Dec 5, 2007 #16


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    You could pick just about any standard old undergraduate Calculus book, since many are (were) designed for three-semester Calculus sequences. You could do well if you find the old Sallas & Hill book, or an old Larson & Hostetler book. They have the theoretical development, but I'm not exactly certain about the detailed rigor.
  18. Dec 5, 2007 #17


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    Yes, but he studied from Spivak! Not just some basic Calculus textbook. I can't see someone going through that text barely knowing what's going on.

    I don't even have that thorough of knowledge of Calculus and I would consider myself ready for Differential Geometry. (Of course, I have many other subjects in my background. The idea is that I touched many areas and am familiar with different aspects of mathematics. As well as I gathered some mathematical maturity in the mean time.)


    Try that. It would requires you to know basic knowledge of linear algebra and multivarible calculus.

    Honestly, you're really not that far from reading this. If you're going to wait for school to teach you linear algebra and multivariable calculus, then yeah you're far from it.
    Last edited by a moderator: May 3, 2017
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