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Schrodinger Group Title

Could someone try and explain a Manifold to me? I've looked on Wolfram and elsewhere but I can't really find a uniform, concise definition of what it exactly is.

  • 2 years ago
  • 2 years ago

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  1. estudier Group Title
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    In simple terms, a manifold is just a generalization (and therefore includes) the simpler geometric spaces you already know, odd shaped spaces if you like, which if you just look at a small piece, are still Euclidean.

    • 2 years ago
  2. Schrodinger Group Title
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    What makes Geometry Euclidean. I mean, could you provide me a specific example of a manifold object, or a non-manifold one and give the explicit reasons why they are so? I'm looking fr a comprehensive definition.

    • 2 years ago
  3. Schrodinger Group Title
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    Thanks regardless, but i'm really trying to understand this.

    • 2 years ago
  4. Schrodinger Group Title
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    *the Geometry of the objects Euclidean

    • 2 years ago
  5. estudier Group Title
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    "What makes Geometry Euclidean" Euclidean space is the space we are in and the rules that apply there (usually modelled as R^3). Other sorts of spaces include hyperbolic space or spacetime. Say you take a sphere (I already said it is a generalization and includes the familiar). OK, you know it is a sphere but if you were standing on it (like on the Earth) it would not be immediately obvious that it was anything other than flat (ie Euclidean). Also have a look here http://en.wikipedia.org/wiki/Thurston_conjecture

    • 2 years ago
  6. Schrodinger Group Title
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    I understand that but i've heard of several other qualifying "features" that make a figure Manifold. All I understand is that single point, and i've hear other very, very different things, which is why i'm trying to find everything I can to really, really nab it.

    • 2 years ago
  7. Schrodinger Group Title
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    (I'm also not sure who's right. I've seen/heard so many different things on this. Also, thanks so much for this stuff so far.)

    • 2 years ago
  8. nbouscal Group Title
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    If you're trying to think about it non-rigorously, the easiest way to think about a manifold is to think about Earth. Earth is a 3-dimensional object, but we live on a 2-dimensional plane on the "outside" of that object. That's the easiest example of a manifold.

    • 2 years ago
  9. nbouscal Group Title
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    If you want more rigorous than that it kinda does require a decent grounding in topology.

    • 2 years ago
  10. Schrodinger Group Title
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    If so, then where would I learn more about topology in this sense? When I hear about topology the few times it's been commonly spoken, lol, it's generally referring to less rudimental, fundamental geometrical concepts and more so to maps and geography.

    • 2 years ago
  11. nbouscal Group Title
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    You might be thinking of topography. Topology is a pretty important area of mathematics, if you're looking for an introductory course I think usually recommended is the text Topology by JR Munkres

    • 2 years ago
  12. nbouscal Group Title
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    Topology is generally pretty advanced stuff, though, so keep that in mind going in.

    • 2 years ago
  13. estudier Group Title
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    Did you look at the link I gave you? (and the linked Uniformization theorem). Understanding this stuff requires some background in topology and classical geometries plus Riemannian geometry (actually the more geometry/topology the merrier). Instead of asking what is a manifold (which I think I have explained) you need to ask what do I want to do with it? Do you want to do calculus on it, for example. Then you need your manifold to be differentiable.In what dimension do you want to work? 2,3, infinite?

    • 2 years ago
  14. Schrodinger Group Title
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    This one? http://www.amazon.com/Topology-First-Course-James-Munkres/dp/0139254951 I looked at the link, but I understood little at first glance. I'm willing to try to read that but it won't make much sense to me at the moment, lol. Yeah, I have no background in any of that, i'm just a curious high school student.

    • 2 years ago
  15. Schrodinger Group Title
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    I've done precalc and a very minimal amount of calculus, so, yeah...lol.

    • 2 years ago
  16. nbouscal Group Title
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    Pretty sure that's the right book, yeah. To be honest though, I would wait until you've done a lot more math.

    • 2 years ago
  17. nbouscal Group Title
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    Topology usually isn't even introduced until relatively late in the course of undergraduate studies, and it will be very helpful to have taken some analysis first.

    • 2 years ago
  18. Schrodinger Group Title
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    Okay. At what point would you think I would start to comprehend this stuff or at least study it? I'm not going for a math major, but I am doing Physics for my undergrad and might pursue math later at a separate time. Oh, okay.

    • 2 years ago
  19. estudier Group Title
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    If you are that way inclined you could try to get to grips with Special Relativity as a way in...

    • 2 years ago
  20. estudier Group Title
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    Back later

    • 2 years ago
  21. Schrodinger Group Title
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    I understand Special Relativity to a decent degree, but I don't see how it could be a way in other than the analogy between the relativity of frames of reference to an object being two dimensional or Manifold, unless that's what it was. Thanks so much man. I would medal both you guys if I could, lol.

    • 2 years ago
  22. nbouscal Group Title
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    I don't remember exactly the ordering of courses in most undergrad math programs, but topology is definitely after a good amount of calculus, differential equations, linear algebra, and i'm pretty sure it comes after real analysis and modern algebra also. It requires a pretty decent mathematical grounding. As for special relativity, I think both special and general relativity rely on topology to a decent degree, but I have not studied them in-depth so that's just a guess.

    • 2 years ago
  23. Schrodinger Group Title
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    Yeah. I've only studied the conceptual components of Relativity overall to a basic degree and worked with the already derived equations of stuff so I didn't have to go through the calculus or complex math of any of it. Well, in general, thanks. Looks like i've got a long way to go...

    • 2 years ago
  24. nbouscal Group Title
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    Haha we all have a long way to go, use it as an inspiration :)

    • 2 years ago
  25. estudier Group Title
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    http://en.wikipedia.org/wiki/Minkowski_space

    • 2 years ago
  26. Schrodinger Group Title
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    Oh, i've seen this before. I've never seen it referred to as that before, though.

    • 2 years ago
  27. estudier Group Title
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    Well worth finding a history of the events around that time and the people involved.

    • 2 years ago
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