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Mathematics classes that will help with physics (list included)

  1. Feb 22, 2013 #1
    Hello folks,

    I was wondering if anybody could give me some suggestions on which mathematics courses will be of the most use for theoretical physics. I am a sophomore at Wayne State university and am taking intro to quantum mechanics and a first course in optics this semester.

    I am taking linear algebra to finish the basic math sequence over the summer here, (calculus, differential equations, and linear algebra)

    And I was just wondering if somebody could help me with finding out which classes would be most helpful to pursue studies in theoretical physics. I am dual majoring in mathematics, but I am mainly concerned not with getting a degree, as with getting knowledge

    Here is a listing of the classes offered at my school

    http://www.bulletins.wayne.edu/ubk-output/lib_ucl.08.73.html#47937 [Broken]

    Thanks
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Feb 22, 2013 #2
    Numerical methods.
     
  4. Feb 22, 2013 #3
    For theoretical physics mate?
     
  5. Feb 22, 2013 #4
    Remember that theoretical physics is not a field of physics. All fields have theorists and they all have their own techniques and methods. Numerical methods can be a part of that for sure.

    I did bio-physics theory as an undergrad and we used numerical methods.
     
  6. Feb 22, 2013 #5
    So I have to confine myself to an area and theorize there? The undergrad stuff at my school is this.. I have left (in semester order)

    Thermodynamics/stat mechanics, mechanics 1

    quantum physics 1, mechanics 2

    Quantum physics 2, electromagnetism 1

    Electromagnetism 2, modern physics lab

    4 semesters. But over the summers they do not offer these classes, so I want to take a lot of math classes over the summers to be the best I can be, I really like quantum mechanics a lot, I want to take a lot of quantum mechanics classes as a grad student, if that helps isolate classes.

    I was told elementary analysis (which is the class required to get into all those upper level classes on that list) is good, as well as partial differential equations and complex analysis, but then I heard algebra was good, probability theory, basically every teacher I ask tells me something different so I don't know what to do.
     
  7. Feb 23, 2013 #6
    It is quite hard to say since little of math (at least at the level you are considering taking) is useless for physics... following your own interests towards math can help too!

    But anyway definitely take an analysis class, and definitely an abstract algebra class (for QM). Taking classes like complex analysis, probability theory, more abstract algebra, more analysis etc can all definitely be useful, but understand that whenever math is necessary in a physics class, you usually learns that math within the physics class, just much more quick and dirty and bare-boned than in a full math class, but it's not like you'll ever get stuck if you don't take them. That being said two very important mathematics topics that usually get taught quite shabbily within a physics context (although it is definitely very useful to know them properly) are: representation theory (very important for QM) and differential geometry (very important for GR). (The problem is that they might be grad courses in your math department.)

    If you are really planning on going to the mathematical physics sides of things, i.e. you know you will be studying a lot of math in the future, then take as much analysis, algebra and topology classes to ensure a firm foundation for self-study down the road!
     
  8. Feb 23, 2013 #7
    Thanks for the advice. They teach a class next semester called 'Methods of theoretical physics 1', where you learn differential equations with boundary conditions, Fourier transforms, PDEs, vector analysis, and probably some other stuff, matrix methods I believe if I recall correctly.

    There is a class called elementary analysis which is the prereq for all the higher math courses. I'm not great at math, I mean, I get A's, but I don't really feel like I understand it, so I want to focus on the things that I can apply towards physics. Unfortunately the elementary analysis is not offered this summer so I will have to take it next year so next summer I can take some algebra and perhaps something else. They do offer a lot of topology and things like that. I should just become a monk and go to school for the rest of my life.
     
  9. Feb 23, 2013 #8

    Fredrik

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    Abstract algebra and then maybe more abstract algebra? For QM? I would say linear algebra, and then more linear algebra. (If the first course is about "how to calculate" aspects of linear algebra, and the second is about theorems about vector spaces and linear operators, you definitely need the second one).

    I don't see a need for knowledge of more than the most basic definitions (groups, rings, fields, homomorphism, isomorphism) and maybe a few simple theorems about homomorphisms from abstract algebra. This is something like 1/4 of a typical abstract algebra course. So I'm not sure I'd even include abstract algebra on the list.
     
    Last edited: Feb 23, 2013
  10. Feb 23, 2013 #9

    Fredrik

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    Not at the undergrad level I think. I think for someone who sees himself as a future theorist, it makes sense to study (at least) linear algebra, real analysis, differential geometry, and maybe differential equations, representation theory, complex analysis, linear/harmonic analysis (i.e. Fourier series and stuff), and abstract algebra. Representation theory is super important, but some of it is taught in QM courses. So I can't say that it's essential to take a course on it, but I would definitely recommend it. A similar comment can be made about several of the other topics, in particular differential equations and stuff about Fourier series and integrals. You need some abstract algebra, but I'm not sure you need to take a course. It may be enough to read the early chapters in some book.

    If you want to go into mathematical physics, you also need topology, measure and integration theory, and functional analysis.

    There was a course like that at my university. I thought it was pretty useless to be honest. In my humble opinion, it's better to take "real" math courses.

    You will probably need this just to be able to read books on more advanced topics.
     
  11. Feb 23, 2013 #10
    Thanks a lot. I suppose I will just reference this thread and reopen the conversation as I get a bit closer. There is so much, it is a bit boggling. I suppose for right now I will content myself to focusing on understanding my classes now.

    And thank you everybody else as well. This gave me a lot to think about and I plan to revisit this question as I, and you, progress.
     
  12. Feb 23, 2013 #11
    As for linear algebra: I assumed one class treated both of those aspects, but if not yes I agree.

    As for abstract algebra: actually I agree that the material itself in an abstract algebra is not that important for QM (as in, all the theorems) but what seems immensely valuable to me from such a class is the reasoning skills you obtain when thinking about algebra (and it is a kind of mathematical maturity that is the distinct from the maturity you get from an analysis class, at least in my own experience). My opinion is that once you get the basics down ice-cold, it is much more possible to add to that the specific relevant physics-related pieces that you can self-study (e.g. representation theory), whereas to get the basics down can easily take the length of a proper math course on analysis and abstract algebra respectively.
     
  13. Feb 23, 2013 #12
    Last edited by a moderator: May 6, 2017
  14. Feb 23, 2013 #13

    micromass

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    The weird thing about your classes is that the Advanced Linear Algebra class requires two previous Abstract Algebra Courses. I don't really understand that. I'm not saying that Abstract Algebra isn't useful to understand before Linear Algebra, but I wouldn't put it as prereq.
     
  15. Feb 23, 2013 #14
    If I could redo my math education for theoretical physics I would:
    Take all the mathematical methods courses your schools physics and math departments offer.

    Take some numerical courses.

    Take real analysis, probability theory, combinatorics, complex analysis (Diff geo if I were more interested in GR), UD ODE's, PDEs and LA.
     
  16. Feb 23, 2013 #15
    Abstract algebra is a requirement at my University as well. Having taken the advanced course in Linear Algebra, I find this perfectly reasonable. Vector spaces are algebraic structures, and we used plenty of facts about the underlying group structure to prove theorems. The isomorphism theorems are invoked to construct quotient spaces, and modules are very hard to understand without a background in ring theory.
     
  17. Feb 23, 2013 #16

    micromass

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    His course doesn't cover modules. And you really don't need group theory to be able to understand quotient spaces and the isomorphism theorems. In fact, I might even say that it's better to first see quotient spaces in the setting of linear algebra than in group theory.
     
  18. Feb 23, 2013 #17
    Well this post got quite interesting. This is all obviously beyond me, but hopefully these responses and any continued will serve as a guide to me in the future, as well as others too.
     
  19. Feb 24, 2013 #18
  20. Feb 25, 2013 #19
    I'm wondering if I should take complex analysis over the Summer...........it's not required, but I want to go into theory, like GR.

    Of course, I say that now, with virtually no formal experience with GR.........
     
  21. Feb 25, 2013 #20

    Fredrik

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    Complex analysis is irrelevant to GR. What you need is differential geometry (which may require topology...which may require real analysis). But complex analysis is fun, will increase your mathematical maturity, and is usually significantly easier than a course in real analysis.
     
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