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Is analysis necessary to know topology and differential geometry?

  1. Nov 17, 2013 #1
    I'm a physics major interested in taking some upper level math classes such as topology, differential geometry, and group theory but these classes are only taught in the math department and are heavy on the proofs. Analysis are recommended and preferred prerequisites but are apparently not necessary. Should physics majors need to know analysis, especially before learning topology and differently geometry? If so, are there any good books you'd recommend on analysis for non-pure math majors and getting introduced to proofs?

    Thanks in advance.
     
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  3. Nov 17, 2013 #2
    Do you have a heavy background in proofs? Topology is very heavy on proofs, but does not necessarily rely on the information you would learn in Analysis. If you are comfortable with proofs and the course does not require you to have analysis then go for it.
     
  4. Nov 17, 2013 #3

    WannabeNewton

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    A lot of topology will lack motivation if you have never studied real analysis. For example, the definition of a continuous function ##f## between two topological spaces ##X## and ##Y## is: ##f## is continuous if for all open sets ##U\subseteq Y##, ##f^{-1}(U)## is open in ##X##. You may very well be asking yourself "how the hell did they even come up with this definition?!". Of course if you've studied basic real analysis you would know that the ##\epsilon##-##\delta## definition of continuity is trivially equivalent to the above definition for metric spaces but the above definition makes no reference to metrics and hence can be lifted to topological spaces. This is a very simple example but the motivation for a lot of concepts in topology comes from real analysis.

    That being said, you don't technically need real analysis to study topology. I studied topology before I did anything more than continuity and convergence in the real analysis setting and I was fine (but to be fair I had a really awesome teacher). I should also add that the way proofs are done in topology is quite different from the way they are done in real analysis if one is to stick to epsilonics/metrics.

    The same goes for differential geometry although to a much lesser extent. Topology is an absolute necessity for differential geometry though (meaning the most general form of differential geometry and not differential geometry of curves and surfaces).

    Regardless, in my opinion real analysis is much, much, much more fun than differential geometry (but not as fun as topology!) so take from this what you will.

    The reason many departments keep real analysis as a pre-req is that real analysis is the first proper proof heavy course that students tend to take and hence serves as a stepping stone into more advanced proof heavy courses (topology, functional analysis, measure theory etc.). If you've had a lot of experience with proofs then I would echo what TheKracken said. Otherwise I would advise against jumping straight into a topology course.

    If you fall into the latter category then there are a lot of great real analysis books out there for you to choose from. My personal favorite is "Real Analysis"-Carothers as it is basically an ~400 page problem solving book. There are also the classics by Apostol and Rudin. Another good choice is "Introduction to Analysis"-Rosenlicht.
     
    Last edited: Nov 17, 2013
  5. Nov 17, 2013 #4
    Differential geometry is locally (multivariable) real analysis, so it is absolutely necessary. For example, many basic results use the inverse and implicit function theorems, and the very definition of a manifold assumes you know basic multivariable real analysis. In addition, the whole point of an introductory course in differential geometry is to lift the machinery of regular real analysis to the manifold level.
    So you should definitely take a real analysis course and a topology course (if possible, in parallell) before diving into differential geometry.
     
  6. Nov 17, 2013 #5

    WannabeNewton

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    espen's post reminded me to point out that you also need to know a very good amount of theoretical linear algebra before delving into the theory of differentiable manifolds.
     
  7. Nov 17, 2013 #6
    Thanks for all the advice. I was planning on getting Rosenlicht introduction to analysis as I like the dover books a lot and they're also well within my budget. Most of the other books that have been recommended are stupendously expensive relatively speaking, but would it be a sufficient introduction to analysis to start tackling topology and differential geometry? I was also looking into getting Velleman's How to Prove It just to get the hang of reading and writing proofs.
     
  8. Nov 17, 2013 #7
    Which books would you recommend, then? I know Axler's "Linear algebra done right" is considered the standard, but it's unfortunately not within my means to buy it at this time.
     
  9. Nov 17, 2013 #8
    You can always use this free book: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
    Another good choice is Lang's linear algebra. Personally, I'm not a fan of Axler since he shuns determinants so much.

    That said, you can do differential geometry without analysis. A very good book here is: https://www.amazon.com/Elementary-Differential-Geometry-Undergraduate-Mathematics/dp/184882890X It just requires some calc 3 and some basic linear algebra. Obviously, you won't be going very deep and you won't do manifolds and bundles. But you can get the taste of some basic differential geometry, and you will find advanced texts much easier.

    For analysis, I recommend Rosenlicht or Berberian: http://books.google.be/books/about/A_First_Course_in_Real_Analysis.html?id=pvI1DFVgP9UC&redir_esc=y You won't need much more than this.

    Of course, to study the advanced texts, you're going to need some topology. Lee's topological manifolds book is excellent. And you can follow-up by his excelent smooth manifolds book.
     
    Last edited by a moderator: May 6, 2017
  10. Nov 17, 2013 #9
    I can understand this view point, but I prefer Axler over a text such as Hoffman and Kunze. It (Hoffman) seemed too wordy, where Axler is generally right to the point. Hoffman and Kunze is a good text, and if you're worried about determinants, you should use it as a secondary source. Never checked out Lang, so I can't really offer anything on that.
     
  11. Nov 18, 2013 #10
    I second Lee's topological manifolds as a good book for topology. Follow up with his smooth manifolds book and you have a solig grounding in differential topology and you can dive head-first into differential geometry.

    For linear algebra, Steven Roman's book "Advanced linear algebra" is the gold standard, but it may be too advanced for you if you haven't seen any abstract algebra before.

    @R136a1: Looks like Presseley's book gets mixed reviews on amazon. Are you sure the classic https://www.amazon.com/Differential...athematics/dp/0486667219/ref=pd_bxgy_b_text_y isn't a better choice?
     
    Last edited by a moderator: May 6, 2017
  12. Nov 18, 2013 #11
    Whatever book you choose, don't get Kreyszig. The book is horribly outdated. I'm sure the book was good in the 18th century, but now things should be done very differently. Kreyszig's functional analysis text is brilliant, but his differential geometry text is now too outdated and the notation is old-fashioned (and trust me, notation matters a lot in differential geometry!).

    Good books on elementary differential geometry are:

    https://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897 A bit of a leisurely introduction to the topic. Very nice, and great exercises (with hints which tend to solve the entire problem).

    https://www.amazon.com/Elementary-Differential-Geometry-Revised-Second/dp/0120887355 Very beautiful book which does things with forms. It does require some more maturity than Do Carmo and Pressley.

    https://www.amazon.com/dp/0132641437/?tag=stackoverfl08-20 Is nice too.
     
    Last edited by a moderator: May 6, 2017
  13. Nov 19, 2013 #12
    I recently bought the analysis book by Rosenlich, it's at a very good price for a book that appears to be really good, which is hard to find in a math or physics book these days.

    However, it's quite compact at only 248 pages. Is this really all the real analysis I'd need to take on higher texts in topology and differential geometry? Does this include group theory as well?

    Also, what about any advanced linear algebra (more advanced and rigorous than physics' majors usually cover such as in Axler's book)? Or is basic linear algebra good enough?
     
    Last edited by a moderator: May 6, 2017
  14. Nov 19, 2013 #13
    Yes, it's really everything you need, although it might be helpful to read up a bit on metric spaces, I don't think Rosenlicht covers that.
    It doesn't do group theory, but there's not really much group theory you need anyway. The thing is that many group theory books start of explaining finite groups, while the groups in differential geometry are usually infinite. So group theory texts aren't really all that useful.

    If you know things like Axler, then you're all set. You don't need more than that. You certainly do need to be acquainted with vector spaces and linear transformations though.
     
  15. Nov 19, 2013 #14
    Thanks. Just one last question, I've been looking at that differential geometry book you recommended that only requires calc III and basic linear algebra. Is there an equivalent book for topology that you'd recommend?

    Edit: The Rosenlicht book actually does cover metric spaces.
     
  16. Nov 20, 2013 #15
    Not OP but do I need to go to all of this trouble to do GR? Will a grad-level GR class cover the necessary math or will I need to study topology first?
     
  17. Nov 20, 2013 #16

    WannabeNewton

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    You will not have to go to all this trouble. Tensor calculus and tensor algebra are done very differently in most GR texts so most of what you learn from the aforementioned texts won't even be of much help to you in the end as far as solving GR problems goes.
     
  18. Nov 21, 2013 #17
    From my experience, a first grad-level GR course will only deal with local phenomena, so you can pretend to be working on a topologically trivial manifold.

    I suppose if you want to work in more exotic contexts, like on a non-orientable manifold, or in general any case where the problem of patching together local tensor fields into a global ones is non-trivial, you would need to bring more sophisticated tools to the table.
     
  19. Nov 21, 2013 #18
    Completely contrary to what 2 of my physics professors told me, and they're well respected cosmologists/relativists in the field. They have told me the best investment I can do as an undergrad in learning GR is taking topology and differential geometry, and that would put me well ahead of the competition when it comes to graduate school.
     
  20. Nov 21, 2013 #19

    WannabeNewton

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    Go ahead and learn topology and differential geometry from the aforementioned math texts, open up a GR text like Wald, and see how much of what you learned actually has any relevance in solving the end of chapter problems. I can tell you from experience that not even the first 4 chapters of Lee's text on topological manifolds (which I went through thoroughly) had much if any relevance except for some basics that showed up in chapter 8 of Wald's text (global causal structure). I don't think you realize how different the math in a math textbook is from what is presented in a physics textbook. You have to take a look yourself to see.

    Even better, after you spend all your time on Lee and/or Rosenlicht go ahead and attempt the problems in MTW (which are far superior to the problems in Wald) and see just how inconsequential what you learned from the aforementioned math texts actually is in solving the problems.

    As a side note, I like to learn pure math for the sake of pure math. Thinking that learning pure math will somehow help you understand a physics text better than someone who doesn't have a background in pure math is absolutely ridiculous.

    EDIT: In fact what going through various pure math texts has really done for me is make it really hard to go through physics textbooks without being nitpicky about every single detail that the physics textbooks get wrong from a mathematical standpoint. Go through a functional analysis text like Conway and try to read a standard QM text (like Sakurai or Shankar) and take note of how impossible it is not to burn the text because of how badly it butchers the math.
     
    Last edited: Nov 21, 2013
  21. Nov 21, 2013 #20
    I hope that one day, I too will annoy mathematicians. I think I'll use the approximation e=pi since they're close enough. Maybe I'll say L'Hospital a LOT.
     
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