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Stoke's Theorem

  1. Dec 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Use Stoke's Theorem to calculate [itex]\int_C\vec{F}\cdot\, dr[/itex], where
    [itex]\vec{F}=<x^2z\, ,xy^2\,,z^2>[/itex] and C is the curve of the intersection of the plane [itex]x+y+z=1[/itex] and the cylinder
    (C is oriented clockwise when viewed from above.)
    Answer: [itex]\frac{81}{2}\pi[/itex].

    2. Relevant equations Stoke's Theorem

    3. The attempt at a solution

    Okay, let me try to explain where I am getting lost. Firstly, I know that the premise of Stoke's Theorem is that is relates a line integral to a Surface integral.

    When I graph this, I get a cylinder that is symmetrical about the z-axis and it is intersected by a plane which results in an ellipse. (see terrible drawing below)

    Now, I need to parametrize (how do you spell that anyway?) S. Now S is the surface that is bounded by C right? If not, please stop me here.

  2. jcsd
  3. Dec 17, 2008 #2


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    Homework Helper

    Yes. S is any surface that is bounded by C. Usually the easiest surface to work with is the plane surface bounded by C if C can be visualised as lying on a plane.
  4. Dec 17, 2008 #3
    I know form doing this in class that we parametrized S as r=<x, y, 1-x-y>

    but I am not so sure why. Oh wait... is that the surface bounded by the ellipse?

    So now all I need to do is find dr/dx x dr/dy and compute the dot product of the resultant with F(r(x,y) and integrate?
  5. Dec 17, 2008 #4
    You are told that x + y + z = 1 and now you parametrize that by <x,y,f(x,y)> and z = f(x,y) = 1 - x - y by algebra (is that part of your question?)
  6. Dec 17, 2008 #5
    I guess my question is what is S? Is it the the surface of the ellipse? I believe it is, so I guess my question has been answered.
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