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How much math should a theoretical physicist know?

  1. Jun 2, 2007 #1
    i bet it's written down somewhere in the "so you wanna be a physicist thread" but im asking anyway? i like math, a whole lot, so the question is not what do i have to overcome but what do i have to look forward to? i wouldn't mind taking every single math and physics class offered at my uni but i don't think my financial aid will cover all that unfortunately ( if anyone has any ideas how i could do that, i would love to know ). so i would like to know what i should definitely get covered before i go off having fun.

    i've taken calc 1-3 and one elementary course in DE
  2. jcsd
  3. Jun 2, 2007 #2
    For a list of topics that (theoretical) physicists must know, you can look at any standard book on Mathematical Physics, for instance the one by Weber/Arfken. At your stage, perhaps taking more courses in differential equations (PDEs mainly), linear algebra, real analysis, complex analysis seems to be a good idea. Which year of univ are you in?

    (PS--I'm not a physicist..)
  4. Jun 2, 2007 #3

    Dr Transport

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    A theoretical physicist cannot have a deep enough math background. As Hilbert said "Our current mathematics will be theoretical physics in the future".
  5. Jun 2, 2007 #4
    The general statement seems to be: the more math you know, the better (but some areas might be more "useful" than others at some point in time). With mathematics, you cannot really say which part can be left out--because someone can come up with a practical (physics) use for some specific/abstract field in mathematics which was earlier not in use by physicists.
  6. Jun 2, 2007 #5
    I think you should know upto algebraic topology for String theory
  7. Jun 2, 2007 #6
    any ideas about how i could pay for like a master's in math and a phd in physics?
  8. Jun 2, 2007 #7

    Chris Hillman

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    The OP's question certainly seems to be a FAQ

    In general terms, not a bad suggestion, although Weber/Arfken is kid stuff: a good physics student should know all that by the second year undergraduate. I was going to suggest the multivolume work by Landau and Lifschitz, Course of Theoretical Physics for some indication of what a second year physics graduate student should have known back in 1940 or so. Then multiply that by three or four.

    As someone already said, you can't possibly know too much math. That's true regardless of what profession you seek, but its particularly true of theoretical physics.

    Well, the kind of degree you purchase by mail is unlikely to open any doors, and certainly won't help you learn any physics. Or did you mean something else? :wink:
    Last edited: Jun 2, 2007
  9. Jun 2, 2007 #8
    i meant if i were rich i would do as i said and just take every course in physics in math that is available in my university. how can i do this even though im not rich
  10. Jun 3, 2007 #9
    Yes, well there are lots of other math-phy books, but if you've done only some calculus and ordinary differential equations, you need to do some of the stuff mentioned in my second post below which would roughly get you started with some mathematics of quantum mechanics. If you have already done most of these topics and want more, you could also have a look at Hilbert's two volumes (Mathematical Physics). Griffiths, in his QM book, says that the more tools you know the better. Surely, you will discover as you do more physics that you need to digress to do some mathematics and then return to the physics. Somehow, I think discovering what you don't know during the process of reading physics books (as for instance the Landau-Lifschitz course) would be a good idea rather than take all mathematics courses and read all mathematics books and then do the physics. But again, I'm no physicist.

    To recapitulate this seems to be a list of basic topics which you cannot do without at least for classical mechanics, quantum mechanics, electromagnetic theory: series, sequences (convergence, divergence criteria), differential and integral calculus of multiple variables, vector calculus, (everything there is in "Calculus and Analytic Geometry" by Thomas and Finney), linear algebra (matrices, determinants, finite and infinite dimensional vector spaces, linear transformations), complex analysis (complex numbers, complex differential and integral calculus, residue theorem, power series), differential equations (ordinary and partial), Fourier Series/Integrals/Transforms, generalized and special functions...probability/statistics...

    There may be other topics too, someone can shed more light on this. It goes without saying that the subtopics in the above list are not exhaustive. We do all this (or most of this) in the first two years and this holds for all undergraduate students (engineering and science). BTW, what is the content of the three calculus courses you have done?

    [PS--Then there are things like tensor algebra, differential geometry, etc. which would be needed for General Relativity]

    For String Theory, superstringtheory.com lists some topics in mathematics here:


    Maybe some string theorists can shed light on the mathematics required in string theory, here.
  11. Jun 3, 2007 #10
    From what I've seen, a good mathematical physicist must know more math than any other person. A mathematician can get away with only the branches of mathematics that are relevent to his field. A mathematical physicist studying fundamental physics would be hard pressed to find any branch of mathematics that is not relevent to his field.
  12. Jun 3, 2007 #11
    I've found that the math that every physicist must know is generally what is covered in two semesters of single-variable calculus, one semester of multivariable calculus, and one semester of differental equations/linear algebra. Differential geometry and (especially) complex analysis are also helpful for physicists. Anything beyond that is superfluous for an undergrad physics major. But then, math is pretty interesting in its own right, so it never hurts to study more math.
  13. Jun 3, 2007 #12

    Chris Hillman

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    Interestingly enough, I've heard mathematicians agree with this, with one critical caveat: often, physicists can get away with a broad but comparatively shallow knowledge, while mathematicians need to be intimately familiar with all the gory details, technical issues, and conceptual subtleties. I think there's a kernel of truth in that.

    There's also a kernel of truth in the somewhat contradictory assertion that one of the more interesting phenomena in the early 21st century is that math and physics almost seem to be merging after having separated sometime in the 18th century, after a period of unification under the controversial :rolleyes: leadership of Isaac Newton. The best way to see what I mean is perhaps to look at recent arXiv eprints and to observe how many of the math papers are directly inspired by technical issues in contemporary physics, then to look at the physics papers to see how many follow "definition, theorem, proof" style.

    I don't think either of these assertions tells the whole story, but as I say, they each have a kernel of truth.
  14. Jun 4, 2007 #13


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    in order to do what? clean out the fridge? how much math does witten know?
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