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Easier to self-teach: differential geometry or complex analysis

  1. Jul 22, 2007 #1
    Hi all,

    I'm torn between taking complex analysis or differential geometry at the advanced third year level.

    Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics?

    I know that differential geometry shows up in general relativity but I'm not sure about relativistic quantum mechanics and other advanced theories.

    Are there actually any serious uses of complex analysis other than contour integration?? It seems like complex analysis is all the hype but I've never actually seen it USED for anything other than this.

    Thanks.
     
  2. jcsd
  3. Jul 22, 2007 #2

    CompuChip

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    I can't really defend complex analysis, as in my physics courses, I've also used it only (explicitly, at least) for contour integration. Anyway, though I have only tasted a little of differential geometry, I think it is hard enough that you will want to follow a course on it. Complex analysis on the other hand, is mainly about a systematic build-up and running through lots of proofs, which I think you can also do yourself.

    Though you can probably better take a course instead of self-study, if you really want to thoroughly understand a subject, I tend to recommend doing differential geometry and self-learn complex analysis, which was easier in my opinion (though my exam grade shows it can still be trickier than you think :tongue2:)
     
  4. Jul 22, 2007 #3
    I am just starting to sink complex analysis in. It is pretty amazing to extend the real numbers. I'd say its worth all the hype. It systematically extends our system but gives non trivial results. I wonder if the complex system is it or does it lie inside an even larger system?
     
  5. Jul 22, 2007 #4

    cristo

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    I've taken classes in both courses, and to me complex analysis was a lot easier to learn than differential geometry. As to which will benefit you more, I would suggest that, whilst complex analysis may be useful later on, you will definitely need differential geometry. Therefore, I would advise you to take DG and self-learn complex analysis.
     
  6. Jul 22, 2007 #5
    The system we have now, the complex one, has been proven by mathematicians to be "it" no need for more generalization
     
  7. Jul 22, 2007 #6

    cristo

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    Huh? :confused:
     
  8. Jul 22, 2007 #7
    ^ me too...
     
    Last edited: Jul 22, 2007
  9. Jul 22, 2007 #8
    I think he means that the complex numbers are closed, unlike the sets within the complex numbers.
     
  10. Jul 23, 2007 #9
    Hi,

    I agree with the previous posters. My brief exposure to differential geometry was frightening, while I found complex analysis to be natural, fascinatingly beautiful and relatively easier. It is something everyone should learn simply to appreciate the beauty of pure mathematics. I think there are some applications of complex analysis in QFT, like the Feynman path integral formulation, but differential geometry is used more extensively throughout physics. Though GTR is the place it is traditionally used, I believe it is also be in the coordinate free formulation of classical mechanics based on calculus on manifolds. This is all second-hand knowledge though.

    Molu
     
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