Rate of flow outward through hemisphere

[aw90]

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.

 

[LCKurtz]

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

\vec r(\theta,\phi) = \langle 3\sin\phi\cos\theta,3\sin\phi\sin\theta,3\cos\phi\rangle

So your velocity field is

\vec v = \langle 3\sin\phi\sin\theta,3\sin\phi\cos\theta,0\rangle

Use the formula

\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

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