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Can you apply Stoke's theorem to this?

  1. Jul 14, 2011 #1
    1. The problem statement, all variables and given/known data

    What is the surface area of a hemi-sphere of radius 6 centered at the origin above the xy-plane lying outside the cylinder r2 = 9


    3. The attempt at a solution

    I did a lot of algebra using the hard way by setting [tex]z = \sqrt{6^2 - x^2 - y^2}[/tex] and use the formula

    [tex]A(S) = \iint \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} dA[/tex]

    Did a lot of messy algebra and got 324π

    Is there an easier way to do this?
     
  2. jcsd
  3. Jul 14, 2011 #2
    If you could, wouldn't that imply that the surface area of any figure with the same bounding curve be 324π? Also I don't see how you would get the surface area in terms of curl of a vector field.
     
  4. Jul 14, 2011 #3
    Surface area somehow relates to surface integral...

    Never mind, I will trsut in my own answer.
     
  5. Jul 14, 2011 #4
    Yes, but not necessarily to the curl of a vector field.
     
  6. Jul 14, 2011 #5
    Stoke's theorem says that the surface integral of the curl of a vector field is equal to the loop integral of the vector field over the bounding curve, right?
     
  7. Jul 14, 2011 #6
    Yeah....?
     
  8. Jul 14, 2011 #7
    Never mind, it seems Hurkyl deleted that post.
     
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