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Stokes's Theorem showing 2 surface integrals are equal

  1. Jul 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Let F = <z,x,y>. The plane D1: z = 2x +2y-1 and the paraboloid D2: z = x^2 + y^2 intersect in a closed curve. Stoke's Theorem implies that the surface integrals of the of either surface is equal since they share a boundary (provided that the orientations match).


    2. Relevant equations

    show each of the following integral and show them that they are equal.
    ∫∫[itex]_{D1}[/itex](∇[itex]\wedge[/itex]F).N dS
    ∫∫[itex]_{D2}[/itex](∇[itex]\wedge[/itex]F).N dS

    3. The attempt at a solution

    I first found that the boundary is given by ]r[/U]=(cosθ +1, sinθ +1, 2(sinθ + cosθ) +3), 0<θ<2pi. Then ∫F.dr on this boundary = -3pi. Then I will try to show the 2 surface integrals equal to this.

    But I have trouble parameterising the surfaces.
    for D1, r = (u cosθ +1, u sinθ +1, 2r(sinθ + cosθ) +3), 0<r<1, 0<θ<2pi
    for D2, r = (x, y, x^2+y^2), ,0<x^2+y^2<2x+2y-1

    I tried to solve both but got really complicated integrals which I could not solve. Could someone please tell me if I have parameterise them correctly?
     
  2. jcsd
  3. Jul 14, 2014 #2

    LCKurtz

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    I presume the ##\wedge## symbol is supposed to be ##\times##, right?

    Don't use r for two different things. And I suppose the u is supposed to be r? So$$
    D_1:~~\vec R = \langle 1 + r\cos\theta, 1 + r\sin\theta, 2r(\sin\theta+\cos\theta)+3\rangle$$

    You have done a good job choosing your parameters for ##x## and ##y## (for which you didn't show your steps). I would use the same ##r,\theta## parameters for ##D_2##. You might find it easier to calculate the curl in terms of the x and y variables, then make the substitution. Or better yet, use$$
    \iint_S \nabla \times \vec F\cdot d\vec S =\pm \iint_{(r,\theta)}\vec F\cdot \vec R_r\times \vec R_\theta~drd\theta$$
     
    Last edited: Jul 14, 2014
  4. Jul 14, 2014 #3
    Yes thats a mistake, u is the radius

    So for D2 is (r cosθ, r sinθ, r^2), 0<θ<2pi, 0<r^2<2r(cosθ + sinθ) -1 correct?
     
  5. Jul 14, 2014 #4

    LCKurtz

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    That is not the same parameterization of ##x## and ##y## as you used for ##D_1##.

    [Edit, added]. For your problem ##\nabla \times \vec F = \langle 1,1,1\rangle##. There is a typo in my last formula in post #2 which is too late for me to edit. All you have to do now is calculate$$
    \pm\iint_{(r,\theta)} \langle 1,1,1\rangle \cdot \vec R_r\times \vec R_\theta~drd\theta$$with the proper sign to verify your (correct) answer of ##-3\pi##.
     
    Last edited: Jul 15, 2014
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