Results 1 to 9 of 9

Math Help - compact torus?

  1. #1
    Member Mauritzvdworm's Avatar
    Joined
    Aug 2009
    From
    Pretoria
    Posts
    122

    compact torus?

    let T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}

    can we show that T^{2} is compact?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Aug 2009
    Posts
    67
    Quote Originally Posted by Mauritzvdworm View Post
    let T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}

    can we show that T^{2} is compact?
    embedd T^{2} inside \mathcal{R}^{3} ... easy to construct such a map. then you can clearly see that it is a closed and bounded subset of \mathcal{R}^{3}, hence by heine-borel theorem it is compact
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Mauritzvdworm's Avatar
    Joined
    Aug 2009
    From
    Pretoria
    Posts
    122
    could you give me an example of such a map?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Aug 2009
    Posts
    67
    i will describe it by words, you write down the expression. call the parameters in the domain a and b. call the corordinate of R^3 as x, y, z.

    in the x-z plane draw a circle of radius 2. call this circle C. parametrize its points by a, which is to say that you choose a reference line segment radius. consider a plane perpendicular to x-z plane, passing through y-axis, such that its intersection with x-z plane makes an angle of a with the refernce radius. this plane rotates along the y-axis, the angle of rotation being a.

    when you have rotated this plane by angle a, draw a circle of radius 1 around the point (2cos a, 0, 2sin a) in this plane. parametrize the points of this circle by b ...


    try to visualize what i am saying ... it is really easy if seen pictotrially.
    i think u can fill in the details. ... convince yourself that this map is smooth embedding.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Mauritzvdworm View Post
    let T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}

    can we show that T^{2} is compact?
    You could use the homeomorphism T^2\cong T\times T, where T = \mathbb{R}/2\pi\mathbb{Z}, together with the fact that a product of compact spaces is compact.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member Mauritzvdworm's Avatar
    Joined
    Aug 2009
    From
    Pretoria
    Posts
    122
    Quote Originally Posted by nirax View Post

    this plane rotates along the y-axis, the angle of rotation being a.
    does the plane rotate about the y-axis or about a point of the circle C which is in the plane with angle a with respect to the reference radius?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Aug 2009
    Posts
    67
    about the y-axis. ok i will give you the final answer. try to work it out. and also if you have access to 3d plots, try to plot it. maybe gnuplot cud help you

     (a,b) \longmapsto (2(1/2 +\cos{b})\cos{a}, \sin{b}, 2(1/2+\cos{b})\sin{a})

    edit :: i am not very sure i have done the computations correctly. plz check it.
    Last edited by nirax; August 24th 2009 at 09:15 PM. Reason: formating
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Aug 2009
    Posts
    67
    is there a list where i can see all the mathematical fonts ?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by nirax View Post
    is there a list where i can see all the mathematical fonts ?
    See the LaTeX help section of the forums (in particular, the tutorials).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 8th 2012, 09:11 PM
  2. Replies: 1
    Last Post: November 19th 2011, 06:32 AM
  3. Finite union of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 8th 2011, 07:43 PM
  4. the intersection of a collection of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 28th 2010, 01:58 PM
  5. Replies: 2
    Last Post: April 6th 2007, 05:48 PM

Search Tags


/mathhelpforum @mathhelpforum