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Order To Learn Mathematics

  1. Jan 1, 2012 #1
    I'm currently doing some broad studying in math to be a physicist. I want to know what order to i need to learn certain subjects in. Somewhat like prerequisite for a class. Right now im finishing a calculus book that covers all the 3 calculus courses. i have a lot of math books to read beyond my current one. im just confused about the order i should read them in to get the most out of them.

    I have a book on linear algebra, differential equations, abstract algebra, discrete mathematics, Real analysis, logic and set theory, Combinatorics, algebraic geometry, complex analysis, differntial geometry, number theory, and reimann geomerty.

    I know some are grouped together like abstract, linear algebra, and logic and set theory. i just dont know where to start with any of them after calculus. i would also be pleased to know of any other subjects im missing to become a theoretical physicist.Any help would be greatly appreciated thank you!
  2. jcsd
  3. Jan 1, 2012 #2


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    I'm not an expert at all on theoretical physics and what math it uses, but as far as the math essentials go, you'll want to know:

    differential equations
    several variable calculus
    linear algebra
    abstract algebra
    real analysis

    As far as anything beyond that, there are a lot of branches of mathematics (complex analysis, numerical analysis, combinatorics, PDEs, etc.) You might get more pertinent answers if you ask this in one of the physics sections.
  4. Jan 1, 2012 #3
    I don't know if you need to learn all of that stuff. There is a lot of material out there to choose from, and you can't learn it all, so you might be better served by a more goal-oriented approach. What is the ultimate goal of studying all this?

    One answer is that it's not easy to predict what will be useful. If you study a subject that is not conventionally used in your own field, maybe you can find a connection. Also, it is nice to pursue subjects just for their own sake because you are interested in them. But, there's a trade-off. It takes a lot of time and adds to your maintenance problem if you don't want to forget what you learn.

    The other answer is that you should try to find out what it's like to do physics research. What skills will you actually need? What is most important?

    I know nothing about it, but I suspect relatively few physicists would need to know number theory. Of course, there are physicists out there who are making use of their knowledge of number theory, but probably not your average physicist. You can also try to learn these subjects on an as needed basis, instead of just aimlessly trying to learn a bunch of topics whose relevance to your future research is questionable.

    I think you should devote a certain amount of time to studying things that are not directly related to your work, but it needs to be a lower priority.

    Undoubtedly, right now linear algebra is the most important thing. I strongly recommend doing linear algebra before or at least concurrently with diff eq, even though it's not usually required. I hate the way undergraduate differential equations are normally presented, but the undergraduate level perspective is useful because you learn how to solve lots of examples explicitly. Personally, I think it would have been nice to do the more theoretical side of diff eq along side the basic stuff because then you would have a better understanding of what you are doing. However, that might require some real analysis background, which people don't usually get to until later.

    As far as the rest, there's no fixed order that you have to learn things in. Real analysis and then abstract algebra might come next, but it might help to read a books about naive set theory and/or proofs to warm-up for these subjects, since many people find them difficult.
  5. Jan 1, 2012 #4


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    As far as I am aware, a lot of physics courses include a mathematical methods course that covers specific things that physicists need to know.

    This is taken on top of your standard calculus series, linear algebra, and differential equations (and possibly partial differential equations) courses.

    As homeomorphic stated above, you will probably have to learn whats required when you need to learn it. One thing you should be aware of is that in these kinds of 'knowledge based' fields, your learning is only at the beginning stages. You will have to keep learning, even beyond your doctorate if you get one, if you decide to work in these knowledge-based areas.
  6. Jan 1, 2012 #5


    Staff: Mentor

    check out the order in the arfken weber book on mathematical physics. there they assume some knowledge of calculus and start with a chapter on vectors. see amazon for table of contents. The book is often used in courses on mathematical physics. Also it really depends on the physics you want to do but I guess that would depend on the math you can do kinda circular.
  7. Jan 2, 2012 #6
    In summary, as others have said, you must note that the mathematics required for different areas is best picked up as you go.

    Why? Because if you try to do too much guesswork, you'll not develop the physics background properly and just learn a bunch of math.

    Different areas of physics require different levels of sophistication with different areas of math.
  8. Jan 2, 2012 #7
    dang... what book is that???? that covers all of that o_O?
  9. Jan 2, 2012 #8
    Thank you, i guess i going to do differential equations ad linear algebra next. then read on set theory, logic, and proofs before getting to abstract algebra and analysis. o yeah lol its not one books, i have several books, sorry bout the confusion lol. im also very interested in computer science and computational science but im quite fond of mathematics . i currently persuing a double degree in physics and computational science but im trying to stay a step ahead of the game and go ahead and learn the matierial. Does anyone know any math subjects that will help me with computational science and computer science. i imagine, combinatrics, number theory, discrete math and computational thoery would be the big ones. Thank you for the help!
  10. Jan 2, 2012 #9
    Computer science usually has kind of light math at the undergraduate level. Discete math, algorithms, theory of computing. Mathematical logic is close to theory of computing. There's basic logic, but then there's the slightly more advanced stuff where you do things like Turing machines, which are basically computers. This is interesting for logic because what you can compute is related to what you can prove. So, that's relevant from a pure math perspective, but also for computer science.

    You could also look into graph theory because it's a beautiful subject, but probably just learning something very basic about it would be sufficient. Graphs are one type of data structure you can use in programming, and there are all sorts of algorithms that go along with them.

    You probably shouldn't try to learn too much too fast.
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