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Area of square in spherical geometry

  1. Dec 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Please see the attached.
    It is a badly drawn sphere :tongue:
    By common sense,the area of the shaded region in the sphere = area of square = r^2
    But can anyone show me the mathematical proof?
    Moreover,does it apply to the reality?
    Imagine when you bend a square sheet with length = r,does the length of curve = r after you bend it?When you bend a substance(with a small force),its molecular structure will change slightly,which means the length of side of the substance will change slightly?
    I don't know whether I should post this here.If I post it at the wrong place,please move this
    thread to the correct position.Thx :)

    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Dec 2, 2012 #2
    Your square is delimited by lines with constant azimuthal angle (parallels), and lines with constant polar angle (meridians) in spherical coordinates. The element of area is given by:
    [tex]
    dA = R^2 \, \sin \theta \, d\theta \, d\phi
    [/tex]
    Therefore, you get the area by doing the multiple integral:
    [tex]
    A = R^2 \, \int_{\phi_1}^{\phi_2} \int_{\theta_1}^{\theta_2} \sin \theta \, d\theta \, d\phi
    [/tex]
     
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