F = 2xz i + yz j + k over the upper half of the sphere

I thought it was dz d\theta z between a and 0 and \theta between \pi and 0 , but im wrong (Crying)

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- May 2nd 2011, 07:26 AMadam_leedsDivergence Theorem limits
F = 2xz i + yz j + k over the upper half of the sphere

I thought it was dz d\theta z between a and 0 and \theta between \pi and 0 , but im wrong (Crying) - May 2nd 2011, 07:40 AMTheEmptySet
I guess I am not sure what you are asking.

Do you want to compute the flux out of the top half of the sphere using the divergence theorem?

If so you end up with the integral

If you convert to spherical coordinates you get

Note that phi is the plane angle and theta is the angle measured from the positive z axis - May 8th 2011, 05:29 AMAnt
Change to spherical polar coordintes would be my best bet. Then integrate wrt theta, phi and r with limits:

theta [0, pi/2]

phi [0, 2pi]

r [0, a]

with the Jacobian being r^2 sin(theta)

that's just how I'd try it though!! - May 8th 2011, 05:31 AMadam_leeds
- May 8th 2011, 05:55 AMAnt
Polar coords are:

The Jacobian is

The Jacobian is used when changing from x, y, z into polar coordinates.

So

becomes

- May 8th 2011, 06:01 AMadam_leeds
- May 8th 2011, 06:54 AMadam_leeds
- May 8th 2011, 07:34 AMTheEmptySet