Surface Integral of a Sphere (non-divergence)

Wildcat04

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

Evaluate: [tex]\int[/tex][tex]\int[/tex]G(r)dA

Where G = z
S: x² + y² + z² = 9 [tex]z \geq 0[/tex]

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
r_u = (r cosu cosv)i + (-r cos u sinv)j + (-r sinu)k
r_v = (-r sinu sinv)i + (r sinu cosv)j + 0k

dA = |r_u x r_v|

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.

Dick

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.

HallsofIvy

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.

Wildcat04

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