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Homework Help: Rate of flow outward through hemisphere

  1. Nov 23, 2009 #1
    1. Seawater has density 1025 kg/m3 and flows in a velocity field v=yi+xj, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2+y2+z2=9, z≥0

    2. surface integral of F over S is ∫∫ F • dS

    3. I parameterized the surface by making r(∅,∂)=<3sin∂cos∅, 3 sin∂sin∅, 3cos∂>. My vector v is <y, x, 0>. Aside from that, i have no idea where to begin. Any help is appreciated.
  2. jcsd
  3. Nov 23, 2009 #2


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    To avoid confusion I would suggest you use the standard letters for the spherical coordinate parameterization:

    [tex]\vec r(\theta,\phi) = \langle 3\sin\phi\cos\theta,3\sin\phi\sin\theta,3\cos\phi\rangle[/tex]

    So your velocity field is

    [tex]\vec v = \langle 3\sin\phi\sin\theta,3\sin\phi\cos\theta,0\rangle[/tex]

    Use the formula

    [tex]\int\int_S \delta\vec v \cdot d\vec S = \int\int_{(\phi,\theta)} \delta\vec v \cdot \vec r_\phi \times \vec r_\theta\ d\phi d\theta[/tex]

    for the spherical surface. For the bottom of the hemisphere in the xy plane you may see a shortcut.
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