Tag Archives: torus

vector calculus examples


the following hieroglyph

\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} \cos V\cos W\\ \cos V\sin W\\ \sin V\end{array}\right)

represent a mapping \mathbb{R}^2\to\mathbb{R}^3. It is enough to take

-\frac{\pi}{2}< V<\frac{\pi}{2}\quad,\quad 0< W<2\pi

to G.P.S -ing (almost) every point in \mathbb{R}^3 which are at distance one from the origin. That is the sphere.

Now the partial derivatives \partial_1=\frac{\partial\phi}{\partial V} and \partial_2=\frac{\partial\phi}{\partial W} are pictorially as:

These derivative are:

\partial_1=\left(\begin{array}{c}-\sin V\cos W\\ -\sin V\sin W\\ \cos V\end{array}\right) and \partial_2=\left(\begin{array}{c}-\cos V\sin W\\ \cos V\cos W\\ 0\end{array}\right)

An easy calculation give that for the inner products \langle\phi,\partial_1\rangle=\langle\phi,\partial_2\rangle=0, so the \partial_i are orthogonal to the position \phi. Then the product \partial_1\times\partial_2, which is orthogonal to the plane determined by the couple of vectors \{\partial_1,\partial_2\}, hence colinear to the position \phi. In fact, after normalization N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}.

A good exercise is to unfold the same program for the torus by employing the parameterization given by:

\left(\begin{array}{c}V\\ W\end{array}\right)\stackrel{\phi}\to\left(\begin{array}{c} (2+\cos V)\cos W\\ (2+\cos V)\sin W\\ \sin V\end{array}\right)

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Filed under differential geometry, geometry, math

torus in chains


more than thousand googolplex parole

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seven circles in a torus


this right graffiti uses the  initial GPS of a torus to mark 7 copies of S^1 (circles) and between them being parallel and/or transversal… how?

which is the maximum open domain to maintain injectivity?

What about sending \left(\begin{array}{c}\!\!v\\\!\!2v\end{array}\!\!\right)?.. Or \left(\begin{array}{c}\!\!3v\\\!\!4v\end{array}\!\!\right)?

 

 

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can you see a solid torus?


dcoato5The image on the right is a solid torus with four lateral annulus in colours: orange, green, yellow and blue.

Solid tori are important elementary type of 3-manifolds. Also called orientable genus one handlebodies and can be fibered -in the sense of Seifert- by longitudinal circles, and in many different ways.

Among three dimensional technologeeks, they are customed to see a solid torus that can be fibered as the M\ddot{o}\stackrel{\sim}\times I, the twisted interval bundle over the Möbius strip.  In constrast M\ddot{o}\times I is the genus one non orientable handlebody: the solid Klein bottle. Isn’t difficult to prove that they are the only two I-bundles over M\ddot{o}

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Filed under fiber bundle, topology