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Stokes theorom question with a line

  1. Sep 20, 2015 #1
    1. The problem statement, all variables and given/known data
    F
    =(y + yz- z, 5x+zx, 2y+xy )

    use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x

    C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1)

    2. The attempt at a solution
    I was thinking that I could decide my surface Y to be where the plane is cutting the sphere but I'm not sure how to parametrize this or if it's the right way to do it?
     
  2. jcsd
  3. Sep 20, 2015 #2

    andrewkirk

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    You only have to parameterise the line don't you, not the surface?

    A natural parameterisation might be to use as parameter the angle ##\theta## that the line from the centre of the circle C to ##\mathbf{x}(\theta)## makes with the x-y plane. It should be doable from there.

    If the integration gets too messy you could try rotating the coordinate system by 45 degrees around the z axis so that the circle C has a constant x' coordinate. But I'd leave that as plan B for now.
     
  4. Sep 21, 2015 #3

    Zondrina

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    The limits for ##\theta## given by the curve ##C## would be ##0 \leq t \leq 2 \pi##, where the curve ##C## is the boundary curve of the intersection of ##y = 1 - x## and ##x^2 + y^2 + z^2 = 4##.

    Find a parameterization ##\vec r(t)## such that:

    $$\iint_S \text{curl}(\vec F) \cdot d \vec S = \oint_C \vec F \cdot d \vec r = \int_0^{2 \pi} \vec F( \vec r(t) ) \cdot \vec r'(t) \space dt$$

    Hint: With ##y = 1 - x##, the sphere becomes an elliptic cylinder in the xz-plane: ##x^2 + (1 - x)^2 + z^2 = 4##. The projection of the elliptic cylinder onto the xz-plane produces an ellipse that is not centered at the origin.
     
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