Torus Definition and 145 Threads

What is the simplest way to calculate the surface area of a region of a torus? Please see this diagram: https://www.dropbox.com/s/73eics7x43bgiwm/surface-area-torus.png?dl=0 This is a cross section through a torus, the dashed line is the central axis. I am interested in the external surface...

Concerning Quantum Phenomenon: I understand the effects of liquid nitrogen on the magnetic fields. However If you ran a tube of liquid nitrogen in the center of a torus magnet or toroidal inductor - would the magnetic field increase, toroidal magnetic vector potential increase. (Zero-point...

I came across this strange relationship when deriving the degree-4 equation for a torus. First thing that comes to mind is the 'Freshman's Dream'. Apparently, it was pure coincidence that they are equal. But, I don't believe in coincidences when it comes to a math expression. There is something...

I know that this has been discussed several times, but I was not able to find a fully convincing conclusion. Suppose there are two twins traveling with relative velocity v on a torus. 1st question: is it possible to find one unique inertial frame which can serve as a global rest frame? my...

Is it true that the atlas for a torus can consist of a single map while the atlas for a sphere requires at least two maps? Can we ever get by with a single map for some Calabi–Yau manifolds assuming that question makes sense? If not is there some maximum number required? Thanks for any help!

Hello , please help me out , I can find the E field of a sphere on google and read that there is no field on the inside of the sphere , but what is the e field of a torus ? Not on the inside but on the outside surfaces also in the inner loop or the middle ?

I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus. Munkres shows that H_1 (T) \simeq \mathbb{Z} \oplus \mathbb{Z} and H_2 (T) \simeq \mathbb{Z} . After some work I now (just!) follow the...

I am reading Martin Crossley's book, Essential Topology. Example 5.43 on page 74 reads as follows: I am really struggling to get a good sense of why/how/wherefore Crossley came up with the maps f and g in EXAMPLE 5.43. How did he arrive at these maps? Why/how does f map S^1 \times...

Homework Statement Use cylindrical shells to find the volume of a torus with radii r and R. Homework Equations V= ∫[a,b] 2πxf(x)dx y= sqrt(r2 - (x-R)2) The Attempt at a Solution V= ∫ [R, R+r] 2πx sqrt(r2 - x2 - 2xR + R2) dx I feel like this isn't going in the right direction...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Here is the question: I have posted a link there to this thread so the OP can see my work.

Hi! I am trying to compute the natural frequency of a system and I also need to add the added mass term to my equation. The object I have is a torus and is basically in heaving motion. I found a good paper, which gives a formula for 3d added mass and 2d added mass. I would figure that I...

Homework Statement Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral. Homework Equations Formula for the semi-circle: y=\sqrt{4-x^2} Solving for x give x=\pm\sqrt{4-y^2} The Attempt...

Hello, I am approximating a helix by parts of torus, to build an optical fiber wrapped around a cylinder simulation. Due to the software limitations, there is no easier way. So I take a part of the torus, rotate it so the one end points slightly up, connect similar part of the torus to the...

Hi all, Perhaps I'm asking the wrong question but I am wondering about the relationship between different definitions of, for the sake of argument, the torus. We can define it parametrically (or as a single constraint) and from there work out the induced metric as with any surface. But...

Hello, I'm trying to construct an explicit map that takes the 4D torus to the 4-sphere such that the wrapping is non-trivial (i.e. homotopically, i.e. you can't shrink it continuously to zero). More concretely, I'm looking for \phi: T^4 \to S^4: (\alpha,\beta,\gamma,\delta) \mapsto (...

Hi, If i use a spoon with a hole in it in a cup of water, bubbles emerge. Does that mean that air got in? Regards, Justin

Homework Statement Homework Equations spacetime interval ##Δs^2=-Δt^2+Δx^2+Δy^2+Δz^2## The Attempt at a Solution We know that the straight path in spacetime diagram is the one with maximum proper time, however we get the same value after going around the universe and this is not consistent of...

I am just trying to figure out how to make a CW complex for this. For the n-genus orientable manifold (connect sum of n-tori) I feel like a lot of things make sense, fundamental group, CW complex, etc. But in the infinite case, things seem to fall apart. For example, I can not figure out how...

How does the mass of a toroidal top load on a Tesla coil affect performance, assuming all dimensions are similar except for mass? You see, I have one hollow aluminum torus and one torus made of heavy, mostly solid metal. Both same size, but different weights. Thanks!

Ok, so this relates to my homework, but I really can't find an answer anywhere, so this is more of a general question. First off, what does a "chart" of a manifold look like? Is it a set, a function, a drawing, a table, what?! I have found so many things about charts, but nothing shows what...

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

Homework Statement When calculating the difference in pressure inside a spherical raindrop, the force exerted by the surface tension is calculated to be 2pi*R*gama, where R is the radius of the drop and gamma is dE/dS (dyne/cm). When the shape of the raindrop is said to be that of a torus...

Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we...

This problem is really making my head spin. I'm in university calculus II and my teacher loves giving us extra homework that is supposed to really challenge us. This is problem 1 of 5. "Let 0<r<R and x2+(y-R)2=r2 be the circle centered at (0,R) of radius r. Revolving the disk enclosed by that...

Homework Statement I can't figure out why there is no bound current in the problem 6 (very subtle hint boldfaced) is the pdf below: http://astronomy.mnstate.edu/cabanela/classes/phys370/homework/ps10.pdf Can anybody give me a hint as to why there should be no bound current The Attempt at a...

Homework Statement I need to derive the prarametric equation of a certain torus. defined by a unit circle on xz plane with center (a,0) and revolving about z-axis. Homework Equations * I don't know if this is relevant but here is something from wikipedia. Surfaces of revolution give another...

Homework Statement To calculate the moment of inertia of a solid torus through the z axis(the torus is on the xy plane), using the parallel and perpendicular axis theorem. Homework Equations The Attempt at a Solution Well, first I divided the torus into tiny little disks and...

For each 2x2 integer matrix with determinant +/-1 is there a homeomorphism of the 2 torus so that the the induced map on the first homology group is multiplication by this matrix?

So the first question is to find the surface area of a torus generated by rotating the circle (or shall I say semi circle) y=√r-x2 around y=r if the idea is to find the surface are of the half torus and then multiply by 2, wouldn't it be the same for the circle (or shall I say semi circle...

Homework Statement The question is attached in the picture. I did part (a) without much problems.. But I have no clue what part (b) is about at all! Even the solutions don't make much sense to me. The Attempt at a Solution I tried to work out how the diagram would look like...

A torus is generated by rotating the circle x2[/SUP+(y-R)2=r2 Find the volume enclosed by the torus. Well, I don't know what to do! I thought that rewritting it as \sqrt{r2-x[SUP]2}+R would help, but I am not sure. Thanks. PD: Is this solid revolutions? because I forgot how...

How do I compute Morse homology of a tilted torus? Thanks.

I am trying to prove the following result: Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a...

How does one compute the modular group of the torus? I see how Dehn twists generate the modular group, and I see how Dehn twists are really automorphisms of isotopy classes. Based on this, it seems that the modular group should be Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z). But I've read that the modular...

Hi I know that the sphere has positive curvature everywhere, and the torus has positive an negative curvature. Is there a space homeomorphic to the sphere such that this space has negative curvature everywhere? And, is there a space homeomorphic to the torus such that the curvature is always...

Problem solved I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :( Homework Statement Let a and b be constants, with a > b > 0.A torus is formed by rotating the circle (x - a)^2 + y^2 = b^2 about the y-axis. The cross-section at y =...

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Hello! I'm new here. I've been seeing this forum for a long time, but i never registered. I'd like to begin to contribute to this forum, i'll try it (even if my english doesn't help me). So now, i ask you for help: I've to find an isometric immersion of the flat torus on \mathbb{R}^4. I know...

A question:- Is it possible to create a magnetic field inside a perfectly superconducting torus if you did not have any to start with? I am talking about a field that goes all the way around the torus, not one that goes partway and doubles back. I am having trouble visualizing how the field...

Hi, I want to do a similar statistics analysis as in the next paper: http://www.phy.bris.ac.uk/people/Berry_mv/the_papers/Berry340.pdf But the boundary conditions are on a two dimensioanl torus, so a solution will be of the form u(R)=\sum_{j=1}^{\infty} \sum_{m,n=0}^{\infty} (A_{mn}...

Homework Statement We have a metal conducting torus and a point charge that is located on the torus' axis (location on the axis is arbitrary). Calculate the (influenced) charge distribution on the torus and the electrostatic force on the point charge. Homework Equations Equation for...

Hi, All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to SL(2,Z). The method: every homeomorphism h: <\tex> T^2 -->T^2<tex> gives rise to, aka, induces an isomorphism g: <\tex> \mathbb pi_1(T^2)-->\mathbb...

Hi, All: I'm a bit confused about this: the 1st homology of the torus T(over Z) is Z(+)Z, so that elements of the form (a,b) =/ (0,0) are non-trivial , meaning these are cycles (closed curves) that do not bound subsurfaces of the torus, and every cycle that does bound is...

Homework Statement I'm trying to calculate singular homology groups of the torus and Klein's bottle using the Mayer-Vietoris sequence. The Attempt at a Solution I represent both spaces as a rectangle with identified edges. Then I take the sets: U=rectangle without the boundary...

I have a real world problem. To design a calculator for a special machining process, I need to determine the volume of material removed when making a groove in the shape of a semicircle inside a tube on a lathe. The volume removed looks like this...

Hi, everyone: I am trying to understand why the mapping class group of the torus T^2 (i.e., the group of orientation-preserving self- diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just realized this is the name of the group of orientation-preserving...

Hi, Could someone help to explain that why for the 2 D system, like 2 dimensional electronic gas, the crystal momentum k in 2D band structure is defined on a torus? Thanks a lot.:)

The problem is with calculating the force of a particle by another particle while both particles lie on a torus. By this I mean, if particle i and j are in a box (LxL) and particle i is on the left edge of the box and particle j is on the right edge of the box, the distance used in the force...

So I'm working on a problem where you have 100 argon atoms in a box that obey the Lennard Jones potential. We're using the Verlet method to calculate the position of each particle at the next time step. The issue I'm having has to do with our boundary conditions. We assume that atoms are on a...