1. ## compact torus?

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

can we show that $\displaystyle T^{2}$ is compact?

2. Originally Posted by Mauritzvdworm
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

3. could you give me an example of such a map?

4. 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.

5. Originally Posted by Mauritzvdworm
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.

6. Originally Posted by nirax

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?

7. 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.

8. is there a list where i can see all the mathematical fonts ?

9. Originally Posted by nirax
is there a list where i can see all the mathematical fonts ?
See the LaTeX help section of the forums (in particular, the tutorials).