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Multivariable Calculus: Geodesic problem

  1. Nov 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the parametrization of a torus:
    [itex] \tau(u,v)=((2+cos(v)cos(u),(2+cosv)sinu,sinv) [/itex]
    The distance from the origin to the center of the tube of the torus is 2 and he radius of the tube is 1. Let the coordinates on [itex] \mathbb R^3 [/itex] be [itex] (x,y,z) [/itex]. If [itex] p = \tau(u,v) [/itex] then u is the angle between the x-axis and the line connecting the origin to the projection of p onto the xy-plane. A slice of the torus through the origin and p by a plane perpendicular to the xy-plane cuts out two circles on the torus, one of which contains p. The angle v is the angle from the center of the circle containing p out to p.

    (a) Compute the first fundamental form matrix for [itex] \tau [/itex]

    (b) Find the lengths of (1,0) and (0,1) as elements of the tangent space at an arbitrary point (u,v). Explain the varying of these lengths by appealing to the picture above (textbook has a picture of a torus).

    (c) What is the angle between (1,0) and (0,1) as elements in the tangent space at an arbitrary point (u,v)?

    (d) Show that the circles [itex] t \mapsto (c,t) [/itex] where c is a constant are geodesics in [0,2∏] X [0,2∏] with the metric induced by [itex] \tau [/itex]

    (e) Show that the circles [itex] t \mapsto (t,c) [/itex] where c is a constant are not generally geodesics. When are they?


    2. Relevant equations
    I can't really think of any relevant equations, but the question itself is in this PDF:
    http://people.reed.edu/~davidp/homepage/211.pdf [Broken]

    Problem is Chapter 6, #3 (Page 130)
    Of course, throughout chapter 6, there are some definitions and such. I'll edit this post to include them if wanted.

    Notes, from prof. on geodesics is here: http://people.reed.edu/~davidp/211.2012/lectures/41lecture.pdf [Broken]

    And this is a first semester multivariable calculus class for reference.

    3. The attempt at a solution

    So, I understand a-c, completed them, no trouble, I'll still write out my answers below (and edit with the work for them if necessary). But I missed the lecture on geodesics due to illness, and neither the textbook or notes that the professor puts up online aren't particularly helpful. I'm less worried about the specific problem and more about understanding geodesics themselves, so it's a more general question, I guess.

    (a)
    [itex] \mathbb I =
    \begin{pmatrix}
    (cos(v)+2)^2 & 0\\
    0 & 1
    \end{pmatrix}
    [/itex]

    (b) [itex] |(1,0)|_{(u,v)} = 10 [/itex]
    [itex] |(0,1)|_{(u,v)} = (2+cos1)^2 [/itex]

    Haven't done the explanation part for this yet, not sure why it's the case. I'll have to think about it some more.

    (c) [itex] \theta = cos^{-1} ((3+7cos1)/(10(2+cos1)^2))≈ 1.4655 radians [/itex]

    (d) This is where I'm lost. I've read through the notes and textbook many times, and I have no clue where to begin. Could anyone give me a tip on where to begin?

    (e) Same as d

    Thanks for any help.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
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