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Homework Help: Another Stokes' Theorem Problem

  1. Dec 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Verify Stokes' theorem for the given surface S and boundary dS and vector fields F

    S = x2+y2+z2, z≥0
    dS= x2+y2=1

    F = <y,z,x>

    2. Relevant equations

    Stokes' theorem:
    ∫∫(∇×F)dS = ∫F⋅ds

    3. The attempt at a solution
    1. Curl of F:

    ∇×F = <-1,-1,-1>

    2. After getting the curl, I just treated this as a surface integral, ∫∫F⋅dS = ∫∫F⋅(Tu×Tv)dudv

    I parametrized the sphere thusly,

    x = sinφcosθ

    Tφ= <cosφcosθ, cosφsinθ, -sinφ>
    Tθ= <-sinφsinθ, sinφcosθ, 0>

    Tφ×Tθ = <sin2φcosθ, sin2φsinθ, sinφcosθ>

    Am I doing this right so far? I asked a friend what he would do, and this is what he had for dS:
    <dydz, dxdz, dxdy> (Idk how to get this).

    So then he got:

    ∫∫(∇×F)⋅dS = ∫∫(<-1, -1, -1>⋅<dydz, dxdz, dxdy>) = -∫∫dydz+dxdz+dxdy, but I don't know how to integrate that...
  2. jcsd
  3. Dec 17, 2014 #2


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    I would ignore your friend's comments. Just continue what you are doing. You are now going to calculate$$
    \iint \langle -1,-1,-1\rangle \cdot T_\phi \times T_\theta~d\phi d\theta$$with appropriate limits. Be sure to check whether you need a minus sign for orientation or not.
  4. Dec 17, 2014 #3


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    There's a common factor of ##\sin\phi##. If you pull it out front, you have ##\sin\phi \langle \sin\phi \cos\theta, \sin\phi \sin\theta, \cos\phi \rangle##. If you recognize that vector, you should be able to convince yourself you're on the right track.
    Last edited: Dec 19, 2014
  5. Dec 23, 2014 #4

    rude man

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    This defines a surface?
    A half-sphere of radius sqrt(S)?
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