Torus Definition and 145 Threads

Hi I'm trying to solve this exercise "Prove that if C is a circular cylinder with S_1 and S_2 as its boundary circles and S_1 and S_2 are identified by mapping them both homeomorphically onto some third circle S, giving a map f: S_1 \cup S_2 \rightarrow S then (C - S_1 \cup S_2) \cup S...

Homework Statement A torus is a surface obtained by rotating a circle about a straight line. (It looks like a doughnut.) If the z-axis is the axis of rotation and the circle has radius b, centre (0, a, 0) with a > b, and lies in y − z plane, the torus obtained has the parametric form r(u, v) =...

The fundamental group of a torus is Z*Z,then the fundamental group of a punctured torus is Z*Z*Z. But I've ever done a problem,it said a punctured torus can be continuously deformed into two cylinders glued to a square patch.Really? If that is right,then the fundamental group of punctured...

The problem reads(from Stewart Calculus Concepts and Contexts 4th edition, Ch.6 section 2 pg. 447 #45 a)Set up an integral for the volume of a solid torus(the donut-shaped solid shown in the figure) with radii r and R b)By interpreting the integral as an area, find the volume of the torus...

In four dimensions, a flat torus is an object that has zero curvature but still has closed geodesic curves. What this means is that if you try to measure geometry locally, you will find that it is perfectly Euclidean. Nevertheless, if you travel on a straight line, you'll eventually end up...

Homework Statement Consider the parametrization of torus given by: x=x(ø,ß)=(3+cos(ø))cos(ß) y=y(ø,ß)=(3+cos(ø))sin(ß) z=z(ø,ß)=sin(ø), for 0≤ø,ß≤2π What is the radius of the circle that runs through the center of the tube, and what is the radius of the tube, measured from the...

I am having trouble visualizing when a 2 form is exact and have a specific case that I am struggling with at the moment. Any help is welcome. Take an oriented 2 torus and divide it ,using parallel circles, into an even number of tube shaped regions. In each tube, assign a 2-form that fades...

Hello, Im trying to learn about string theory in toroidal compactification on an undergraduate level. I am mostly using Zwiebach's "A first course in string Theory" but now I am trying to do something that doesn't seem to be covered in the book or any other literature explicitly. Perhaps...

Greetings, I am trying to figure out how to modify the equations for a trefoil knot to make it toroidal. A trefoil knot is: x = (2 + cos 3t)cos 2t y = (2 + cos 3t)sin 2t z = sin 3t A torus is: x(u, v) = (R + r cos v) cos u y(u, v) = (R + r cos v) sin u z(u,v) = r sin v where...

Define the mapping torus of a homeomorphism \phi:X \rightarrow X to be the identification space T(\phi)= X \times I / \{ (x,0) \sim (\phi(x),1) | x \in X \} I have to identify T(\phi) with a standard space and prove that it is homotopy equivalent to S^1 by constructing explicit maps f:S^1...

Hi So I've been using Seifert-Van Kampen (SVK) to calculate the fundamental group of the torus. I haven't done any formal group theory, hence my problem ... I have T^2=(S^1)x(S^1) If A= S^1, B=S^1, A intersection B is 0. And T^2 = union of A and B. Then fundamental group of (A...

Homework Statement Find an atlas and coordinates for a torus T^2 = S^1 X S Homework Equations The Attempt at a Solution I know that an atlas on a manifold M is a collection of charts whose domains cover M, but i am not sure how to start this one mathematically.

Consider, if it were the case, that the plasma torus left in Io's path as it orbits Jupiter followed the same orbital path as the moon and was not deformed by Jupiter's magnetosphere...thus a space station orbiting Jupiter could effectively orbit in Io's plasma torus. Yes, this is going...

The physics of a 1D string with fixed end points is found here: http://www.uio.no/studier/emner/matnat/ifi/INF2340/v05/foiler/sim04.pdf Now imagine a string under tension T and of mass density rho confined to the surface of a cylinder of radius r. I posit that this string will act just as...

I am confused with trivialization of a tangent bundle. Can anyone can help me solve the problem of finding a trivialization of the tangent bundle of the torus S^1*S^1 in R^4. Thanks.

how does the formula for volume of a torus work. is there a proof with integration?? could you use an ellipse?

Homework Statement this is from ch9 (functions of several variables)of baby rudin a,b real b>a>0 define a mapping f=(f1,f2),f3) of R2 into R3 by f1(s,t)=(b+acos(s))cos(t) f2(s,t)=(b+acos(s))sin(t) f2(s,t)=asin(s) I showed that there are exactly 4 points p in K=image(f) such that...

Homework Statement Find the moment of inertia of a torus if mass is m and density \rho is constant. The cross-sectional radius is 'a' and the radius is R. Homework Equations I= \int r^2 dm The Attempt at a Solution Well I looked up the answer to be I_z= m(R^2 +...

Homework Statement http://faculty.tcu.edu/richardson/Calc2/H20090323torusVolume.htm This is a link to the homework. 2. The attempt at a solution I did number 1 by doing 2 * integral from -2 to 2 of sqrt(4-x^2) = 4pi The second problem is where I am completely confused and don't know...

I am trying to show that any smooth map F from the 2-sphere to the 2-torus has degree zero. The definition of the degree of F can be taken to be the (integer) degF such that \int_{\mathbb{S}^2}F^*\omega=(\deg F)\int_{\mathbb{T}^2}\omega For omega any 2-form on T². Another definition would be...

Hello all. Consider the torus T^2 as a subset of R^3, for example the inverse image of 0 by the function f(x,y,z)=(\sqrt{x^2+y^2}-1)^2+z^2-4. I need to obtain a example of a vector field X defined in the whole R^3, such that: 1) X is invariant in the torus 2) the orbits of X in the torus are...

Homework Statement Find the volume enclosed by the torus rho = sin theta. Homework Equations The Attempt at a Solution I tried setting the limits as phi from 0 to pi, theta from 0 to 2 pi, and rho from 0 to sin theta. However, if i do that, i get a volume of 0. How should i...

Hi, Consider a cone with height H and radius of base circle R. Take a point on the circular edge of the cone and make that the center of another circle of radius r whose normal points at the apex of the cylinder. Rotate this circle around the axis of the cone to create a surface. Given an...

Homework Statement Write a vector field equation which describes fluid flowing around a pipe of radius r whose axis is a circle of radius R in the (x,y)-plane. Homework Equations x2+y2=r2 Equation of a torus? The Attempt at a Solution What I've gathered from the question: the pipe...

Right now I'm trying to design a floatation device for putting 1000m under the sea (~1500 psi external pressure). The nicest looking and most convenient shape would be a torus but I'm having trouble estimating stress on the thing. I'm looking for around 20lb of buoyancy or more and dimension are...

I noticed somewhere the line element of a two-dimensional torus is written in the form ds^2=r^2(d\theta^2_1+d\theta^2_2) The author only states that he assumes same radius parameter for simplicity and no further explanation is given. But I do not understand how that form is possible. I...

gravity of a rotating torus While Gm/r^2 seems to apply only for gravitational acceleration towards spheroids, what would be the equation for gravitational acceleration towards a rotating torus? I'm sure the equation would be the same for spheroids and toroids at large distances but what would...

I have been goven the task of finding all the stresses on a Torus (i.e. donut) Pressure vessel. Any help would be much appreciated!

Hey folks, Does anyone know how to get the line element on a torus? The ds^2 term. I want to find the metric for toroidal geometry. Any help appreciated! Richard

Hi everyone. As you may know, it is possible to create a torus or toroidal vortex of air. Some toys such as the air zooka create a torus of air that travels across a room to knock down a light object. Do you think it is possible to create a device that produces a continuous torus of air...

Can anybody help? Mathematical Physics. I'm seeking an analytical expression for the path length of a point that follows a helical path with the helix wound about an axis to form a torus. The arc path length of a helix is simple to compute, but when its formed into a torus there is a...

Homework Statement What is the boundary of a torus? What is the boundary of a sphere? The Attempt at a Solution Both 0?

I was playing with a 2D model with point particles and gravitational attraction. To avoid particles going off to infinity, I thought I might turn the 2D space into a topological torus: connect x=L = x=0 and y=L to y=0. But now a problem occurs: What is the distance between 2 points? Because of...

Homework Statement Two graphs defined for a two dimensional torus: f_1(t) = (\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),b\ cos\ t),\ t \in (-\frac{\pi}{2},\frac{\pi}{2}) f_2(t) = (a\ cos(t+\frac{\pi}{4}),a\ sin(t+\frac{\pi}{4}),b),\ t \in...

Homework Statement What is the net electric flux through the torus (i.e., doughnut shape) of the figure Homework Equations net flux= E*A I believe is needed The Attempt at a Solution I don't know how to do this problem at all. I feel like I don't have enough information to...

Homework Statement Volume of Torus: using Shell's mehod 4\pi \int^{1}_{-1}((R-x) \sqrt{r^2 - x^2})dx Homework Equations The Attempt at a Solution I don't know how to integrate this at all. I cannot use any conventional methods...or I can't think of a way... i.e. use isolate a function as...

I know I am making a stupid mistake but I am not sure what it is... Find an equation for the plane tangent to the torus X(s,t)=((5+2cost)coss, (5+2cost)sins, 2sint) at the point ((5-(3)^1/2)/(2)^1/2, (5-(3)^1/2/(2)^1/2, 1). First I have to find what s and t are in order to sub them in for...

I am a novice at stress analysis. For my competition, I need to calculate the stress on a torus in vacuum filled with air at pressure P. Here's is what I have managed to do. I have attached an image below. The torus section is an angular section of a curved cylinder. Its angular dimensions are...

hi, for most of you this might be a simple question: Is it possible to embed the flat torus in Euclidean space? If we, for example, take a rectangle and identify the left and the right sides we get a cylinder shell, that can be embedded easily in R^3. If we construct the...

Hi, I need to find the volume of the solid that lies above the cone with equation (in spherical coordinates) \phi = \frac{\Pi}{3} and inside the torus with equation \rho = 4\sin\phi . I thought that the bounds are: 0\leq\rho\leq4\sin\phi, \frac{\Pi}{3}\leq\phi\leq\frac{\Pi}{2}, and...

Which one has more magnetic field Torus or Bolt when current is going though? :eek:

I'm asked to consider regular networks on a torus. I'm given that V - E + F = 0. I need to show it is impossible to have a regular network on a torus where the faces are pentagons; I don't understand that at all. Surely it is easy to have pentagons as faces… All you would need to is draw a...

Hi! This may not be the right place for it but I have a question about the torus. In the centre point, the exact middle of the hole in the torus if a let's say, a perfect sphere was placed there, would it simply stay in the one place if everything was stationary? Also could you walk all...

Hi, A small but exceptionally annoying algebraic topology question: I'm trying to find the Hodge numbers (from the Hodge-de Rham cohomology) for a 2n-dimensional torus (that is, n complex dimensions). Anyone have any ideas? It's a rather technical question, but I don't really want to...

There is a developing theory that the Universe may be shaped like a three-torus, the mathematical equivalent of a rubber cube that's bent so that all opposing sides are connected. This would mean that the Universe is finite, but does not have the problematic edge that's included in most...