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Length of sinusoid on a sphere

  1. Apr 7, 2017 #1
    1. The problem statement, all variables and given/known data
    Originally the statement:
    Find a length of two points on sphere. It was easy.
    ##\int \sqrt{g_{\phi\phi}}d\phi##
    I hope you agree :-) But I have idea, how to find a length of path which is NOT a part of arc (circle). For example sinusoid. Is possible to find length of sinusoid on the sphere and how?

    2. Relevant equations
    ##ds^2=g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\phi\phi}d\phi^2##

    3. The attempt at a solution
    My attempt hit the snag very early :-)
    A took ##\theta=\pi/2+\sin{\phi}##
    ##d\theta=\cos{\phi}d\phi##
    ##ds^2=0+r^2d\theta^2+r^2\sin^2{\theta}d\phi^2##
    ##ds=r\sqrt{\cos^2{\phi}+\sin^2{(\pi/2+\sin{\phi})}}d\phi##
    And now I don't know. I'm not sure if my procedure is so naive, and it exists better, or such problem doesn't have an analytical solution.
    Please advice.
     
  2. jcsd
  3. Apr 7, 2017 #2
    I don't agree until you tell me what does those symbols mean.

    What does length mean here ? arc length or something else ?
     
  4. Apr 7, 2017 #3
    ##r,\theta,\phi## are spherical coordinates and ##\theta=\pi/2## is equator. ##g_{ij}## is metric tensor in these coordinates. By length I mean arc length (I hope it is same number when you take a ruler and measure sinusoid on a ball).
     
  5. Apr 7, 2017 #4

    Mark44

    Staff: Mentor

    Thread moved.
    @Vrbic, please post questions involving integrals and tensors in the Calculus & Beyond section. These concepts are well beyond the Precalculus level.
     
  6. Apr 7, 2017 #5
    @Vrbic Do you really require calculus here ? I think this question is perfectly feasible without calculus.
     
  7. Apr 7, 2017 #6
    Yes, I would like calculations (maybe both:-) ). I believe it is a training for work in general relativity. No?
     
  8. Apr 7, 2017 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Calculus is definitely needed. Even the problem of the length of a sinusoid in a plane involves the (non-elementary) elliptic function.
     
  9. Apr 7, 2017 #8
    Not that problem. I was talking about,

     
  10. Apr 8, 2017 #9
    Ok, elliptic function are needed for final solution, it seems not trivial. But how to get to them?
    1) Is all right my procedure for finding length between two points on a sphere?
    2) How to find length of path between two points connected by sinusoid (or sinusoid along all equator?)
     
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