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Other Small Textbooks

  1. Nov 15, 2016 #1
    Hello PF,
    i am a first year physics student and i am taking a minor in math also, i have now a solid background in the 3 levels of calculus.
    I am searching for small textbooks about some mathematical topics, like in the range from 100 to 300 pages at max, as i am trying to fill my little spare time with something useful, but not overwhelming. I would also like the topics to not be covered in my courses as i don't want to study the same thing twice.
    I am planing to take: Linear algebra, Differential equations, Introduction to analysis, Differential geometry, and maybe topology. So give me advice outside of those topics, or just in the same topic but other directions.
    Thank you.
     
  2. jcsd
  3. Nov 15, 2016 #2
    Introduction to Topology 2e by Gamelin and Greene is my favorite introduction to topology now. It's 256 pages, and covers what I consider to be the most important introductory material without feeling too brief.

    I can't think of any of my favorite textbooks in the other topics you mentioned that are under 300 pages.
     
  4. Nov 16, 2016 #3
    No i meant that i want books about different topics than the ones mentioned
     
  5. Nov 16, 2016 #4
    Ah, I was a bit tired when I answered earlier, sorry.

    Naive Lie Theory by John Stillwell is a lovely introduction to Lie Groups, Lie Algebras and the exponential map using matrix groups. 217pp.

    A Book of Set Theory by Pinter is a nice introduction to modern set theory including the Axiom of Choice. 256 pp.

    An Introduction to Lebesgue Integration and Fourier Series by Howard J. Wilcox is a nice introduction to Lebesgue integration. 192 pp.

    If you want a supplemental book on topology, Topology by Klaus Jänich covers a lot of the intuition of basic concepts in topology well. 193 pp.

    Pinter's A Book of Abstract Algebra is 400 pages, but it has the "feel" of a thin book.
     
  6. Nov 16, 2016 #5
    Inside the list: Farlow's Partial Differential Equations for Scientists and Engineers is structured in mostly independent chapters than can be read in very little spare time.

    Outside of the list: The Book of Numbers, by... well, you'll find out by who.
    And you might find intriguing many books by Nahin.

    And BTW: "studying the same thing twice"? There is not such a thing. You look at it by a different vantage point.
     
  7. Nov 16, 2016 #6

    jasonRF

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    Topics I think you are missing if you are a physics major / math minor:

    *complex analysis. Very useful for physics - if you will be taking a math-methods for physics course that covers this topic then skip this. I'm not sure what the best "short" book is, though. Churchill and Brown is longer than 300, but you can skip the last couple chapters and still get a good background. It also feels thin, and is relatively easy reading. Used copies of old editions are fine:
    https://www.amazon.com/Complex-Vari...=sr_1_2?s=books&ie=UTF8&qid=1479352473&sr=1-2

    * Probability theory. This is an important topic. Ross's book feels thin, but is more than 300 pages. I really like it, but not everyone does. Old editions are fine:
    https://www.amazon.com/First-Course...9353815&sr=1-1&keywords=ross+probability+1988

    For both of those topics there longer books that are better, and probably easier (and perhaps quicker) to read. I also own books on these topics that are <300 pages but will take much longer to read than the ones listed above.

    * Fourier analysis and distribution theory (generalized functions). I really like Strichartz's book. The first 5 chapters cover the basics in < 80 pages and are easy reading. It doesn't get any easier, I think.
    https://www.amazon.com/Distribution...1479353040&sr=1-1&keywords=fourier+strichartz
    Lots of physicists prefer Lighthill for this topic. It has a lot to offer as well, but I find it harder to read. It follows a different approach than Strichartz, and is <110 pages:
    https://www.amazon.com/Introduction...=1479353142&sr=1-1&keywords=fourier+Lighthill
    EDIT: forgot about Folland, which is >300 pages but takes a more broad view, and would teach you how to really use fourier analysis to solve the kinds of boundary value problems you will see in physics. Strichartz covers boundary value problems too, but to a much lesser degree. It really requires you know some complex analysis, though:
    https://www.amazon.com/Fourier-Anal...d=1479355582&sr=1-5&keywords=fourier+analysis

    Jason
     
    Last edited: Nov 16, 2016
  8. Nov 16, 2016 #7
    Never looked at it this way. Thank you.
     
  9. Nov 17, 2016 #8
    Ordinary Differential Equations by Carrier and Pearson. This one is different than most of the other ODE texts for undergraduates will probably give you a different perspective than what you gain in the classroom.

    Application of Green's functions in science and engineering by Greenberg. Short but enlightening and furthermore readable.

    A first course in wavelets with Fourier analysis by Boggess. Good place to start.

    Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Grinfeld. Friendly approach.

    Sequences and Series by J.A. Green. Builds up slowly rather than the approach used in Calculus texts.

    A radical approach to real analysis by Bressoud. Inspiring to say the least.

    COMPLEX ANALYSIS (read in order):

    Elements of the theory of functions (~150 pages) by Knopp. Will prepare you for the next two of the sequence.

    Theory of functions (2 volumes both ~200 pages each) by Knopp

    Functions of a Complex Variable: Theory and Technique by Carrier, Krook, and Pearson. This is similar to Churchill's book but much better in my opinion. Includes a chapter on asymptotics. Moves rather quickly though which is why I would wait until after at least the first Knopp book.
     
  10. Nov 17, 2016 #9

    mathwonk

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    Complex analysis is a curious subject in that almost all books on the topic are good, possibly because the original treatments were excellent, or that the subject is so beautiful. But having said that, my favorite by far for a small treatment is the book by Henri Cartan. For some reason, perfectly constructed as they are, the little books by Knopp have never helped me at all, just goes in one ear and out the other.

    https://www.amazon.com/Elementary-A...8&qid=1479406766&sr=1-1&keywords=henri+cartan
     
  11. Nov 17, 2016 #10

    mathwonk

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    the basic fact that complex analysis is so nice is the miracle that all complex differentiable functions are equal to their taylor series, and functions given by a power series have very nice properties. A lot of books begin with the definition of complex differentiable functions and then work hard to show they equal their taylor series.

    in hindsight one should kniow something about series, and why they behave so well. the book of cartan begins with a discussion of series, which i think is very useful. another good book along those lines is the book of serge lang. books that focus entirely on the integral aspect tend to be harder to understand in my opinion, like ahlfors and maybe knopp.

    also those books do a terrible job of explaining my favorite subject, riemann surfaces, which cartan does beautifully, although very minimally.

    the book i could actually understand, as a student, is the one by frederick greenleaf. it opened my eyes, but may not be as short as you asked for. still it is worth it, and very clear, easier in fact than the books of cartan and lang to read as a student.
     
  12. Nov 19, 2016 #11
    Is the book written by Copernicus ??
     
  13. Nov 19, 2016 #12
    I'm pretty sure SredniVashtar meant the one by John H. Conway and Richard Guy.
     
  14. Nov 19, 2016 #13
    @mathwonk just curious what you thought of the three (translated) volumes by Markushevich? I have a copy weighing down my shelf but haven't gotten a chance to read it yet.
     
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