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Basic stokes theorem

  1. Jun 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Use the surface integral in stokes theorem to find circulation of field F around the curve C.
    F=x^2i+2xj+z^2k
    C: the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from above



    2. Relevant equations
    stokes theroem: cirlulation=double integral of nabla X F.n d(sigma)


    3. The attempt at a solution
    i got nabla cross F is 2k
    for the normal, aint it just k? coz im getting confused by if i let g(x,y,z)=4x^2+y^2-4=0 (the elispe)
    isnt n=grad(g)=8xi+2yj
    im confused with this

    also should i parameterize the ellipse?
    im not sure how im meant to set the double integral out?
    im really lost, any help please?
     
  2. jcsd
  3. Jun 11, 2009 #2

    Dick

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    Right, curl(F)=2k and n=k. What's the dot product? You want to integrate that dx*dy over the interior of the ellipse 4*x^2+y^2=4. From here on the problem is not that different than finding the area of an ellipse or a circle using a double integral. Take a deep breath and try it. If you're clever, you'll notice the integrand is a constant so you don't have to integrate at all if you know a formula for the area of the region.
     
  4. Jun 11, 2009 #3
    thanks, its just isnt the normal grad(g), or am i getting this confused with somethig else?
     
  5. Jun 11, 2009 #4

    Dick

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    You are getting it confused with something else. You want the normal to the region in the x-y plane, which is k, as you said. grad(4x^2+y^2-4) is normal to the elliptical cylinder 4x^2+y^2-4=0.
     
    Last edited: Jun 11, 2009
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