# Surface Integral of a Sphere (non-divergence)

1. ### Wildcat04

34
1. The problem statement, all variables and given/known data

Evaluate: $$\int$$$$\int$$G(r)dA

Where G = z
S: x2 + y2 + z2 = 9 $$z \geq 0$$

2. Relevant equations

Parameterization
x = r sinu cosv
y = r sinu sin v
z = r cos u

3. The attempt at a solution

r(u,v) = (r sinu cosv)i + (r sinu sinv)j + (r cosu)k
ru = (r cosu cosv)i + (-r cos u sinv)j + (-r sinu)k
rv = (-r sinu sinv)i + (r sinu cosv)j + 0k

dA = |ru x rv|

I am not sure if I am approaching this correctly or if I am way off base. My next step was to complete the dot product of z with dA but this does not seem right and I can't find any good examples in my text.

Thank you in advance.

2. Science news on Phys.org
3. ### Dick

25,887
You are doing it ok. There's a simpler way to get dA. You know that dV in spherical coordinates is just r^2*sin(u)*du*dv*dr, right? dA over a sphere is just that without the dr. But you should get the same thing by finding the norm of your cross product.

Last edited: Nov 11, 2008
4. ### HallsofIvy

40,674
Staff Emeritus
r= 3 in this problem and you don't use "the dot product of z with dA" because neither is a vector! Just multiply and integrate.

5. ### Wildcat04

34
Thank you very much for the help! I believe that I have figured it out now.

Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?