Surface integral over a hemisphere

[LyraDaBraccio]

I was having a looks at multiple integrals, line/surface/volume integrals and the like the other week, and decided to try some problems, but this one stumped me:

 

[math] \int \int_S xz\mathbf{i} + x\mathbf{j} + y\mathbf{k}\: \textrm{d} S [/math]

 

where S is the unit hemisphere of radius 9 for y >= 0

 

I thought I could change the variables to spherical co-ordinates, but I don't see how that would work with the particularly nasty stuff you'd get for the [math] \sqrt{\left( \frac{\partial z}{\partial x} \right) ^2 + \left( \frac{\partial z}{\partial y} \right) ^2 +1} [/math] along with the square roots necessary in writing z in terms of x and y.

 

Basically this confused the heck out of me and I'd appreciate any help

[MindShadowfax]

I'm not 100% about what I'm going to say as this is what i'm studying myself but take a look at this..

When you have the first int you posted then you would get IntInt(F(r(u,v) |N|) dS right? that's doing F=<xz;x;y> and looking at the surface with a parametric equation. Then you can use the Gauss Theorem

 

Gauss

 

int intF n dS = int int int div F dV

 

And that is way easier. Sorry about the way i wrote the equations I don't know how to do it in a proper way.

Anyway I'd do that

Good luck

Edited December 3, 2012 by MindShadowfax