A multivariate function of Toruses - tangent vectors

In summary, the torus has a map that sends a tangent vector X to Y as follows: X=α∂∂θ+β∂∂ϕ, Y=(α+β)∂∂ζ+(α−β)∂∂η.
  • #1
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Homework Statement
this map multiplies all lengths of tangent vectors by the same factor. what is the factor?
Relevant Equations
p^2+q^2=1
r^2+s^2=1

w^2+x^2=1
y^2+z^2=1

w=pr+qs
x=qr-ps
y=pr-qs
z=qr+ps
Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c.

Screen Shot 2020-10-09 at 10.01.51 PM.png


I think this map multiplies tangent vectors by a factor of sqrt(2). We can think of tangent vectors as velocity while traveling along a curve on the surface. From simply plotting points, we know if travel around the pq plane or the rs plane once, we travel around both the wx and yz planes. We also know the two planes are orthogonal to each other because it's the definition of the torus under this equation. Under this map, the velocity gains an additional orthogonal component of equal magnitude, and from Pythagorean theorem the net magnitude is sqrt(1+1) = sqrt(2).

Is my intuition correct?

I've calculated the metric in part a and solved part b. but, I don't think these parts will help with c. thanks
 
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  • #2
The metric tells you how to compute the length of tangent vectors. Maybe it can help?
 
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If you set [itex](p,q,r,s) = (\cos \theta, \sin \theta, \cos \phi, \sin \phi)[/itex] you will find that in appropriate charts the map sends [itex](\theta, \phi)[/itex] to [itex](\zeta, \eta) = (\theta - \phi, \theta + \phi)[/itex]. Thus a tangent vector [itex]X = \alpha \frac{\partial}{\partial \theta} + \beta\frac{\partial}{\partial \phi}[/itex] is mapped to [tex]Y = (\alpha + \beta)\frac{\partial}{\partial \zeta} + (\alpha - \beta)\frac{\partial}{\partial \eta}.[/tex] You can assume that all of the basis vectors are orthogonal with unit norm.
 
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Office_Shredder said:
The metric tells you how to compute the length of tangent vectors. Maybe it can help?

Thank you. I ended up using the metric and found the answer to be 2.
pasmith said:
If you set (p,q,r,s)=(cos⁡θ,sin⁡θ,cos⁡ϕ,sin⁡ϕ) you will find that in appropriate charts the map sends (θ,ϕ) to (ζ,η)=(θ−ϕ,θ+ϕ). Thus a tangent vector X=α∂∂θ+β∂∂ϕ is mapped to Y=(α+β)∂∂ζ+(α−β)∂∂η. You can assume that all of the basis vectors are orthogonal with unit norm.

Sorry I'm having trouble visualizing these tangent vectors. How does one understand tangent vectors in polar coordinates?

Thanks,
 

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  • #5
I realized my solutions are wrong. How do I delete my previous post?
 
  • #6
Here are the complete solutions to the problem. Feel free to let me know if you see an error :)
Screen Shot 2020-10-13 at 3.16.23 PM.png
Screen Shot 2020-10-13 at 3.16.31 PM.png
Screen Shot 2020-10-13 at 3.16.38 PM.png
Screen Shot 2020-10-13 at 3.16.45 PM.png
 

Related to A multivariate function of Toruses - tangent vectors

1. What is a multivariate function?

A multivariate function is a mathematical function that depends on more than one variable. This means that the output of the function is determined by multiple input values, rather than just one.

2. What is a torus?

A torus is a three-dimensional geometric shape that resembles a donut or a tire. It is created by rotating a circle around an axis in three-dimensional space.

3. What is a tangent vector?

A tangent vector is a vector that is tangent to a curve or surface at a specific point. It represents the direction and rate of change of the curve or surface at that point.

4. How are toruses and tangent vectors related in a multivariate function?

In a multivariate function of toruses, the input variables are the coordinates of points on the surface of a torus, and the output is a tangent vector at that point. This means that the function describes the direction and rate of change of the torus at different points on its surface.

5. What are the applications of studying multivariate functions of toruses and tangent vectors?

Studying multivariate functions of toruses and tangent vectors has many applications in fields such as physics, engineering, and computer graphics. It can be used to model and analyze the behavior of physical systems, design and optimize structures, and generate realistic 3D graphics.

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