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Verifying Stoke's theorem

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

    S = [(x,y,z): x2+y2+z2=1, z≥0
    ∂S = [(x,y): x2+y2=1
    F=<x,y,z>

    I did this problem and checked the answer - I chose the wrong integral bounds and I am wondering why they are wrong.

    2. Relevant equations

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

    3. The attempt at a solution
    I was able to do the left hand of the solution easily. I knew that I should get zero for the cross product, so that's all I had to do.

    For the right hand side:

    ∫F⋅dS = ∫xdx+ydy

    x= cosΘ
    y=sinΘ
    dx=-sinΘ
    dy=cosΘ

    0≤Θ≤2π

    Apparently this is the correct range for theta:

    0≤θ≤π

    Why is it just π?
     
  2. jcsd
  3. Dec 17, 2014 #2

    vela

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    You should write ##dx=-\sin\theta\,d\theta## and ##dy=\cos\theta\,d\theta##.

    0 to ##2\pi## is the correct interval.
     
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