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Order of independent math study, before starting college

  1. Apr 23, 2012 #1
    Hey everybody!

    I will be attending Notre Dame this following fall, and I am hoping to major in physics, as well as research (I haven't really decided what I will to focus on, esp. at the graduate level, but I hope I can figure that out later on down the road).

    Over the summer, I want to independently study math, just to be familiar with it and have a decent background in it (My physics teacher said that he had wished he had a stronger math background before he was taking certain classes, as opposed to studying math WHILE he was in the classes, because he thinks the two would have linked together and explained things a tad bit better.)

    As far as my current level of math, I seem to catch on and understand concepts, formulas, etc. very well. On the other hand though, my school has disappointing math offering, so I have Algebra II + Trig, but each had a very narrow focus, and I don't have much of a background in series or trig identities, and maybe other topics as well. For calculus, I had some level of independent study starting in Sophomore year ( I wanted to find, analytically, how to find turning points of functions, and my teacher wouldn't tell me because I "just wouldn't be able to understand it", and I found calculus very interesting, etc.) Also, I took a Brief Calculus I and II at a community college, which, as far as I can tell, was a survey of basic calculus topics without any trig at all. I only took the class to get out of my AP class, b/c the calc teacher was a jerk and wouldn't let me take it without Probability and Statistics, and he taught too slowly anyways.

    The calculus that I know (and trig/algebra, for that matter), though, I know VERY well, at the conceptual level. It all just makes sense to me.

    As for Physics, I took a standard AP B class with the Giancoli book, and a wonderful, fabulous teacher. I really haven't struggled at all with it (except maybe with visualizing E&M, right at first, but I could always use the formulas just fine).

    As far as I can tell, I need to study Calculus I through IV very thoroughly, as well as diff.eq, linear algebra, and complex analysis. I have books on the diff.eq, linear algebra, and complex analysis, but I am really wondering what order I should approach these (and which Calc book I should buy, given my background), as well as any other suggestions for books/topics I should look into. If it is worth anything, I am considering a focus in theoretical physics, and I am not (necessarily) expecting to get through everything this summer, but I am dedicating most of the summer to my studies.

    I apologize for the length, as this post definitely grew to be much longer than I had intended it to, and I thank everybody for all help in advance!
  2. jcsd
  3. Apr 23, 2012 #2
    First of all, trig will be quite important in physics, so you better understand it well. I'm not suggesting that you should memorize every single formula, but you should at least know which formulas exist and when to apply them.

    As for calculus. Since you already know some calculus, I think it would by quite worthwhile to step it up a notch and to study calculus the way it should be done. One of the best rigorous calculus books out there is "Calculus" by Spivak. This is usually a book only mathematicians study, but the people who do study it have a completely different and better grasp of calculus. I think it is very worthwhile for a theoretical physicist to study it.
    Warning: Spivak has very difficult exercises at times, so don't be discouraged if you can only solve half of them without peeking.

    After that, it's about time you learn about linear algebra and calc III. An excellent book for this effort is "Vector calculus, linear algebra and differential forms: A unified approach" by Hubbard and Hubbard. If you know some single-variable calculus, then this book shouldn't be too difficult.

    For further topics, I highly recommend the book "Mathematical methods in the physical sciences" by Boas. This contains about all the math you need to know before starting physics.

    If you finish these three books, then I'd say you have an excellent knowledge of the math.
  4. Apr 25, 2012 #3
    Thanks! I was looking at Spivak, but I wasn't sure if it was something I could delve into at this point, but I found a copy at a college library, checked it out, and it certainly seems do-able. I just didn't know if complex analysis/linear algebra ought to follow calculus, or precede it, but it looks like I'll tackle the calculus first!

    I appreciate the help and suggestions!
  5. Apr 25, 2012 #4
    Get the Boas book he recommended. I love that book so much. Not only does it explain most any concept that my feeble little mind can think of, it does so with brevity. It's a great book to read through, since it can be used as a sort of "sample platter" of mathematical things for physics. It has the "bulk" of most any subject in there, and if you find topic particularly interesting you can then seek out a book or advice already knowing a great deal about the topic.
  6. Apr 25, 2012 #5
    Complex analysis is a long way from now. You should focus on Calculus I through Calculus III. Once you have a good foundation of that you can jump to differential equations. And like Micromass, take a look at Apostol's Calculus.

    P.S. What is Calc 4? I think in most cases Calculus four is differential equations?
  7. Apr 25, 2012 #6
    Just to repeat some of what micromass said, a reasonable order for early undergrad (applied-ish) math is:

    Calc I (Differential)
    Calc II (Integral)
    Linear Algebra
    Calc III (Multivariable)
    Ordinary Differential Equations (ODEs)
    ...and so on.

    If you feel comfortable with the material, there is no reason you can't learn Linear earlier than that. If you hit something you really can't figure out, it usually means you need tools from somewhere else, that you haven't gotten to yet. Finding the right puzzle pieces can be fun in itself. Just start somewhere and see where you get.

    A book like Spivak will also increase your "mathematical maturity," which will help when/if you get to more proof-based courses like Analysis or Abstract Algebra. Don't be discouraged if you need to look at a more "mainstream" Calculus book some of the time.
  8. Apr 25, 2012 #7
    My vote is for Spivak for calculus if you want to take it up a notch.

    If you just want to do loads of calculus drills that will prep you for being able to easily handle University Physics: go with either an older version of Stewart Calculus (which would be cheap to buy online), or one of those Shaum's 3000 calculus problems books.

    I would next become familiar with basic vector stuff and polar + spherical coordinates.

    After that, look into learning some differential equations solution methods, just the basic 1st order and 2nd order solution techniques should last you well into when you take a formal course in ODEs.

    Lastly, if you manage to get through that still in the summer, look into MIT's OpenCourseWare and find the linear algebra lectures by Prof. Strang.

    Good luck at Notre Dame, it's a great school :-)
  9. Apr 25, 2012 #8
    I have a related question that I think will help the OP. I've taken Calc A & B and Physics 1 & 2 as a soon-to-be sophomore (meaning I'm a freshmen for another week). Well, this summer I plan to learn some new programming language (learned java this year) and maybe something else. Any suggestions? I was thinking python but saw a thread on here saying C is very important and beneficial to know. Next semester I'll have mathematical phyiscs and modern physics. Should I get started by reading through the Boas book this summer, or maybe I should learn some new math.. I know I should keep busy the next three months, any advice is welcome.

    Edit: Might want to be a geophysicist but haven't taken any courses and won't be able to until spring. Should I learn some optics or some geology?
  10. Apr 26, 2012 #9
    The programming language you learn itself doesn't matter at all. If you learn one language well, then picking up other languages will be almost self-evident. Python is very good for beginners, but it really doesn't matter that much. Just pick what is most convenient to you.

    As for the Boas book. I actually recommend that you do a bit of multivariable calc first before doing the Boas book. I think it's easier that way.
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