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A fluid has density 2 and velocity field .

Find the rate of flow outward through the sphere .

So far I've found n, which is

1/2x+1/2y+1/2z

and F dot n gives z^2.

I converted to spherical coordinates and z^2 is equal to 4cos^2[phi].

My integral is set up as:

4*(int[0-2pi] int[0-pi] (cos^2[phi]*sin[phi]dphi dtheta.

The first integral is -1/3cos^3[phi] from 0-pi which is 1/3 - - 1/3 = 2/3

The second integral gives 2/3*2*pi, so the entire thing is 4*2/3*2*pi.

I thought I was just supposed to multiply that by 2 (the density) but that's not the right answer. Can someone tell me what I did wrong or what I'm supposed to do with the density?

Thanks a lot.

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# Flux through a sphere

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