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How much math should i have before i start analysis/advanced calculus?

  1. Oct 23, 2009 #1
    I want to this as a self study and so far I've only had a calc 1 & 2 class. should I go up to differential sq/linear alg before starting advanced calc?
     
  2. jcsd
  3. Oct 23, 2009 #2
    It's not really that you need specific prerequisites, it's more "math maturity" (a term that gets thrown around a lot but is never properly defined). It's a hard jump from algorithmic, computational math to abstract, proof-based math. I suggest you get really comfortable with proof techniques and logic before reading an analysis book.

    All that being said, usually in an intro analysis course all that is specifically required is knowledge of limits, derivatives, integrals, etc.

    WHICH analysis book you're reading makes a big difference too.
     
  4. Oct 23, 2009 #3

    lurflurf

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    Advanced calculus is an ambiguous term. One could start with high school math. A year of elementary calculus can be helpful. It could have to have seen some things proven. Linear algebra is good to know or be learning at the same time. Some vector calculus and differential equations would not hurt. Some complex numbers. With out a more specific list of topics the best thing to do would be (especially since self study should grant one some freedom), start is and if any difficulties are encountered shore them up then.
     
  5. Oct 23, 2009 #4

    lurflurf

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    Many do not develop those things from scratch. Some in enough detail that prior exposure is only "helpful", others with less detail so that prior exposure is "very helpful".
     
  6. Oct 23, 2009 #5
    what's a good book then?
     
  7. Oct 23, 2009 #6

    lurflurf

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    Among in print books I recommend
    Introduction to Analysis, by Maxwell Rosenlicht
    Elementary Real and Complex Analysis, by Georgi E. Shilov
    Undergraduate Analysis (Undergraduate Texts in Mathematics) by Serge Lang

    There are also many out of print books, particularly good and free online is
    Advanced Calculus by Shlomo Sternberg and Lynn H. Loomis

    Though not good Principles of Mathaematical Analysis (baby Rudin) by Walter Rudin is "standard".
     
  8. Oct 23, 2009 #7

    jasonRF

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    johnnyies,

    regarding books, for self-study with little background in proofs you will likely want a book that teaches about logic, set theory, quantifiers, and methods of proof. I recently finished working throug the book "analysis: with an introduction to proof" by Lay. The first two chapters worked through these topics efficiently but at a level I could understand. It includes answers to some of hte problems and in the early sections has some "fill in the blanks" proofs to get you warmed up. There are likely other, better, books to teach the basics of proofs but I thought Lay was adequate.

    Additionally, the remaining chapters of Lay present elementary analysis, including elementary topology in R and a couple of optional sections giving a brief intro to metric spaces. These were all at a readable level, but be prepared to work though the chapters with pencil and paper, and don't expect to zoom through pages quickly. Overall, I thought that a book that was any more complicated than Lay's would not have been within my reach given the time I had available. I should say that I am not what I would call mathematically gifted, although I do have a PhD in electrical engineering and have taken a reasonable amount of math (calc, multivariable calc, linear algebra, ODEs, PDEs, complex analysis, probability, stochastic processes, numerical methods). There are other books at a similar level as Lay, but I have not worked (or even looked) through them so cannot recommend. If you are gifted then more advanced book might be okay - and yes, there are substantially more advanced undergraduate texts that would be required to prepare you for math grad school but I think it unrealistic to start there.

    I purchased a used copy of the second edition of Lay that was cheap and just fine. If after working through the first two chapters you find the remaining sections of Lay too advanced, or if you just want a bunch of elementary anlaysis problems (meaning Proofs) to work, I like the book "mathematical analysis" by Binmore. It is basically calculus with theory, but about 1/3 of the book is complete solutions to all the exercises, and most of those are proofs. It does not include uniform anything (continuity or convergence) or work with general sets in R (only intervals), but otherwise is a great stepping stone that would significantly increase your "mathematical maturity" and is definitely not a waste of time.

    good luck - I am sure others will have other recommendations that would work as well.

    jason
     
  9. Oct 23, 2009 #8

    jasonRF

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    One great thing about Lang's book is that a solutions manual is available for purchase by anyone. So could be good for self-study.
     
  10. Oct 23, 2009 #9
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  11. Oct 24, 2009 #10
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