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Verifying Stokes' theorem (orientation?)

  1. Apr 24, 2014 #1
    1. The problem statement, all variables and given/known data
    F= <y,z,x>
    S is the hemisphere x^2 + y^2 + z^2 = 1, y ≥ 0, oriented in the direction of the positive y-axis.
    Verify Stokes' theorem.

    2. Relevant equations



    3. The attempt at a solution
    So I completed the surface integral part. I'm trying to do the line integral part of Stokes' theorem and end up with the same answer.
    Where I get confused is there parametrization part.

    I said that r(t) = <cos t, 0, sin t>, 0≤t≤2∏.
    Apparently that's the wrong orientation. But when I "grab" the y-axis with my thumb in the positive y-direction and curl my fingers they go from the z axis to the x-axis counter clockwise. Isn't that the CORRECT orientation?
    I guess what I'm asking is how do I determine the orientation when I'm using Stokes' theorem. I assume I want the same counter clockwise orientation that I do for Green's theorem.
     
  2. jcsd
  3. Apr 24, 2014 #2

    vela

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    Your parameterization goes in the opposite direction. As t goes from 0 to pi/2, r(t) goes from <1, 0, 0> to <0, 0, 1> — in other words, from the x-axis to the z-axis.
     
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