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Homework Help: Flux of a vector field over an elliptical region

  1. Jun 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the Flux of the Vector Field <-1, -1, -y> where the surface is the part of the plane region z + x = 1 that is on the ellipsoid [tex]{x}^{2}+2\,{y}^{2}+{z}^{2}=1[/tex]
    (oriented in the +ve z direction)


    2. Relevant equations
    Surface Integral

    3. The attempt at a solution
    Parametrize the Surface:
    <u, v, 1 - u>

    The intersection of the plane and the ellipsoid is:
    [tex]{u}^{2}+2\,{v}^{2}+ \left( 1-u \right) ^{2}=1[/tex]
    [tex]{u}^{2}+{v}^{2}=u[/tex]

    Which is a circle of radius 1/2 centered at (1/2,0)
    Or the polar region [tex]0\leq r\leq \cos \left( \theta \right) [/tex] and [tex]0\leq \theta\leq 2\,\pi [/tex]

    Then, ru x rv = <1, 0, 1>
    Then dotting the vector field with the above vector = -1 - v

    So the integral becomes:
    [tex]\int \!\!\!\int \!-1-v{dv}\,{du}[/tex]
    After converting to polar and limits for the circle:
    [tex]\int _{0}^{2\,\pi }\!\int _{0}^{\cos \left( \theta \right) }\!-r-{r}^{
    2}\sin \left( \theta \right) {dr}\,{d\theta}
    [/tex]
    Which gives me [tex]-1/2\,\pi [/tex]

    But, when I try to find the flux with maple by using the Flux command, it gives me -pi/4

    Am I doing it wrong? Could someone point out where I went wrong please?

    Thank you!
     
    Last edited: Jun 18, 2010
  2. jcsd
  3. Jun 18, 2010 #2
    Ooh nevermind, I found out why! Such a stupid mistake. I gave theta the range of 0 to 2pi, while the circle isn't defined after pi/2. I shouldve gone from -pi/2 to pi/2. Works then!
     
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