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DG vs. PDE course

  1. Aug 17, 2009 #1
    I am trying to decide between two different graduate math classes in the fall. These are the only classes that sound really interesting to me, as well as fitting in with the rest of my schedule. My other courses will consist of several physics type courses, notably, Jackson's electrodynamics, intro to plasma physics, and intro to nuclear physics.

    One is called "Differentiable manifolds", using the textbook "Comprehensive introduction to differential geometry, vol. 1", with a description like the following: "This course is the first introduction to differentiable manifolds. We will cover the basics: differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stokes' theorem, De Rham theory, etc. If time allows it, we will branch into Riemannian manifolds."

    The second course is "Partial differential equations", using the textbook "Partial Differential Equations" by L. C. Evans. The description is: "This is an introductory course in partial differential equations. We will follow the textbook by L. C. Evans. The course will consist of the following parts: 1. Basic properties of solutions of Laplace's equation and the heat and wave equations. 2. Second order linear elliptic and parabolic equations: existence, regularity, maximum principles. 3. First order nonlinear PDE: introduction to Hamilton-Jacobi equations and conservation laws."

    Both classes sound interesting and useful, but I'm a little bit unsure which one I'm actually capable of handling. Also, I am unsure which class would be better for my overall mathematical maturity at this point in time. Eventually I will be doing research in physics, but I think both classes would be helpful for my end goals.

    My background is just having graduated with a math degree, completing the following math courses: Real analysis I (Intro to analysis), Real analysis II (Intro to Lebesgue integration, measure theory, limit theorems), complex variables, differential equations, topology, abstract algebra, linear algebra, curves and surfaces (intro to differential geometry)... and that's all. I think I may have a little deficiency in terms of finite dimensional vector spaces, or very rigorous calculus of multiple variables.

    Any advice would be really appreciated...

    Edit: The book for differentiable manifolds is by Spivak.
     
    Last edited: Aug 18, 2009
  2. jcsd
  3. Aug 17, 2009 #2
    I say PDE's, mostly because it will be most immediately useful to you as a physicist. That is not to say that Differential Geometry is not useful, but you will use PDE's before you use DG, at least that is my experience.
    Also to consider is you own research focus in physics. If you are going to be doing theoretical physics, then it is quite possible that DG will be more useful, however if you are focusing on a more engineering physics aspect, then PDE's is a definite must.
    My own personal advice would be to take both. They are very interesting and challenging topics, and will no doubt help you to grow as a mathematician/physicist.
     
  4. Aug 17, 2009 #3
    DG. that sounds like Spivak's work. I haven't formally taken either, but abstract fibre bundles is still very difficult to get through on my own than understanding the applications of the Laplacian. i'd choose DG if I want to understand the published literature better and also the harder class (DG>>PDEs), the easier everything else gets. it looks like you wont have to deal with Lie groups structures, which is where I get messed up. But if you're into QM more than GR, then I'd look at PDE's first. they're both very interesting to read about. goodluck.
     
  5. Aug 18, 2009 #4
    Cody Palmer -- Do you really think the PDE's course will be more useful, considering it will be more about the theory of PDE's and not much about anything practical?

    xaos -- Yes, the differential geometry book is by Spivak. I just corrected that in the post, thanks. Your opinion is interesting because I was under the impression that the PDE course would actually be harder than the DG course.

    Has anybody studied from both these books and can compare their difficulty or informativity, recommending one before the other, etc?
     
  6. Aug 18, 2009 #5
    As far as my experience goes, you are unable to separate PDE's from practicality, as many of the PDE's that we work with (particularly second order) arise from physical problems. But in the end I suppose it is just my opinion. It will largely depend on the instructor how theoretical or practical the course gets.

    Again though, if it was me, I would be taking both. In fact, depending on the instructor, the differential geometry could complement the theoretical portion of the PDE's course very nicely.
     
  7. Aug 18, 2009 #6
    Cody Palmer -- If you believe that even a theoretical PDE's course is practical, I am delighted, because I would like to think the same thing. Given that I haven't taken a PDE course yet, I can't really make an opinion. Real analysis, as an example, has not really been practical for physics/engineering so far, but it has provided a clearer understanding of things like the chain rule, which you use all the time in physics or engineering problems. Also, real analysis has probably been crucial for my success in a course like rigorous PDE's. So in that sense, real analysis has been practical.

    Under this interpretation, if PDE's is more practical then it represents an overall increasing practicality to the study of differential equations than the study of analysis. This seems predictable. However, just flipping through the PDE's book, I am struck by the fact that I just see a lot of existence/uniqueness theorems, and other stuff which is basically impractical. (Not to say disinteresting.) Perhaps there are other elements of the PDE course which would be more practical, but which I am just ignorant about.

    Although I agree with you that the equations in PDE's may be physically analogous, I don't think this in and of itself gives PDE's any practicality. It arguably only provides just the barest correspondence between a course like PDE's and any physics course, just something to say "Oh, this appears in both places," and nothing more. However, I am guessing your observation goes deeper than this. Overall, it is very interesting to discuss the practicality of PDE's and I would appreciate more of your thoughts on the subject.

    Actually, I probably will take both courses, but I can only afford to take roughly one math course per semester, so I feel it is crucial that I arrange my courses in a good order. Especially if PDE's provided a better background for DG, or vice versa, or something like that. However, nobody has seemed to say anything to that effect so far.
     
  8. Aug 19, 2009 #7
    mordechai9-
    To me, and again this is my own opinion, the true practicality of PDE's (and ODE's for that matter) comes from the fact that once we are capable of solving these equations we are able to more accurately model a cornucopia of real life situations. For example in basic mechanics when we consider a pulley system, we assume the rope has no mass for the sake of simplicity, and hence as the rope moves through the pulley, the mass remains the same on both sides of the pulley. However if we want to account for the mass of the rope changing on each side of the pulley as it moves through, we will need ODE's or PDE's. In this sense PDE's might be considered more practical the DG.

    However, we require theoretical results to justify our uses i.e. existence and uniqueness. In fact beyond existence and uniqueness ODE's and PDE's are of very little theoretical value.

    There is a good chance that if you are taking the course through the Math Department, then it will have a very theoretical bent. Again though, that depends on the instructor. When I took PDE's my instructor, an applied mathematician, spent maybe 4 or 5 class sessions on the existence and uniqueness, and then spent the rest of the semester solving PDE's, and our final consisted almost entirely of solving physical problems, one of them being and underground water flow problem.

    Probably the most likely place to take a practical (and I suppose by "practical" I mean "not as theoretical") PDE's course would be through the Physics Department. In fact this semester I am going to be taking the PDE's course through our Physics department, and I am eager to see how practical it will be, versus my Math Dept PDE's course.
     
  9. Aug 20, 2009 #8
    maybe you're right after reviewing the topics you'll be covering. but once you get to looking at Lie group structures or symmetry groups solution methods, both subjects become very different creatures and very difficult (at least to me). Spivak is pretty good at explaining all the careful details. in my experience, i've always found analytic methods to be easier to cope with than topological methods.
     
  10. Aug 20, 2009 #9
    I was under the impression that "differentiable manifolds" and differentiable geometry were pretty analysis oriented. At least, I took an intro to differential geometry class last spring, and it seemed like it was mainly reminiscient of vector calculus and other stuff that I had seen. It sounds like you are saying it is much more topological/algebraic flavored, which sounds interesting. I haven't done any advanced topology and so I am a bit unfamiliar with those sort of mechanics. Algebra in my experience has also been much more challenging than analysis, but I hear about other people who say the opposite, so it seems fairly random.
     
  11. Aug 21, 2009 #10
    well you're putting analytic (differential) structure onto a topological object, then you're putting additional algebraic (Lie group) structure on top of that, but you will not likely be seeing algebra in your DG class. Chapter 1 of Spivak is almost pure topology, so heavy use of intuition will be necessary. the rest of the course will have more analysis, but its really a geometry text, so the idea of mappings play a strong role (rather than the rule of assignment, e.g. in calculus).
     
  12. Aug 23, 2009 #11
    I was hoping somebody would come out and say "You really ought to take geometry before PDE's in order to understand how this fundamentally works..." or vice versa... I guess it must not matter that much.
     
  13. Aug 23, 2009 #12
    Well for what it is worth the university I am getting my bachelors at does not even have a differential geometry class and it's the best math school in the state. Granted that state is Nevada...
     
  14. Aug 24, 2009 #13
    Both subjects are huge, so what you will be seeing in either course will only be a slice of what's out there (which is why they say 'introduction' although neither are introductory). DG will tell you why you can use Stokes theorem in your PDE derivation. PDE will tell you how to apply that strange looking object called an inverted nabla in various settings (which by the way also has roots in DG!).

    One subject will enlighten the other maybe is the point of not giving you a straight answer.
     
  15. Aug 24, 2009 #14

    morphism

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    Why not sit in on both classes and pick the one with the more interesting professor?
     
  16. Aug 24, 2009 #15
    If you are an applied physicist you will live and breathe PDE's. But you will also learn to solve them in your physics classes. For instance, the second book of Feynmann's Lecutres on Physics can be viewed as an exercise in solving Maxwell's equations.

    Differential geometry is useful for Relativity, String Theory, Gauge theory and Mechanics. If you think you will need these areas of Physics then you better get going on the geometry.
     
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