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Thread: compact torus?

  1. #1
    Member Mauritzvdworm's Avatar
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    compact torus?

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

    can we show that $\displaystyle T^{2}$ is compact?
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    Quote Originally Posted by Mauritzvdworm View Post
    let $\displaystyle T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$

    can we show that $\displaystyle T^{2}$ is compact?
    embedd $\displaystyle T^{2}$ inside $\displaystyle \mathcal{R}^{3}$ ... easy to construct such a map. then you can clearly see that it is a closed and bounded subset of $\displaystyle \mathcal{R}^{3}$, hence by heine-borel theorem it is compact
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    Member Mauritzvdworm's Avatar
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    could you give me an example of such a map?
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    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.
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  5. #5
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    Quote Originally Posted by Mauritzvdworm View Post
    let $\displaystyle T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$

    can we show that $\displaystyle T^{2}$ is compact?
    You could use the homeomorphism $\displaystyle T^2\cong T\times T$, where $\displaystyle T = \mathbb{R}/2\pi\mathbb{Z}$, together with the fact that a product of compact spaces is compact.
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  6. #6
    Member Mauritzvdworm's Avatar
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    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?
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    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

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

    edit :: i am not very sure i have done the computations correctly. plz check it.
    Last edited by nirax; Aug 24th 2009 at 09:15 PM. Reason: formating
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    is there a list where i can see all the mathematical fonts ?
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  9. #9
    Rhymes with Orange Chris L T521's Avatar
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    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).
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