Applying Stokes' Theorem in Spherical Coordinates

[BiGyElLoWhAt]

OK, so I know stokes theorem states that I can turn $\int \int_{S}F\cdot dS$ into $\int \int (-P\frac{\partial g(x,y)}{\partial x} -Q\frac{\partial g(x,y)}{\partial y} + R)dxdy$

but I have a problem that I'm working on that's screaming change of variables (to spherical), but I'm not sure if I can apply stokes' this way...

my region is $x^2 +y^2 + z^2 = 64$, and I really want to make my region a spherical region (if possible, as I need to understand this for my final) but I'm not sure how to define g.

What's throwing me off is which partials to take. My F is $<x,-y,z>$ and upon making the proper substitutions, $- \rho <cos(\theta )sin(\phi ), -sin(\theta )sin(\phi ), cos(\ phi)>$
and I suppose g would be $\rho <cos(\theta )sin(\phi ), sin(\theta )sin(\phi ), cos(\ phi)>$ , since that's my region? It's the region in the first quadrant oriented inwards, that's where the negative sign came from, I guess it should probably be in g, but either way it's going make it into the integral.

But now, how do I define my g and my partials? I understand P is my $\hat{i}$ and Q j and R z, but I'm uncertain as to which partials I'm multiplying by which functions.

I'm not sure if I can even do this, I'm just trying to understand this better. any help would be greatly appreciated =]

 

[BiGyElLoWhAt]

Ok, I didn't want to, but I watched the "how to" video, which basically works through the problem, and they ended up solving "g" for z, but I'm still wondering if there is a way to do this in spherical? Or do I need $F = P\hat{\theta} + Q \hat{\phi} + R \hat{\rho}$ in order to make this work like I'm wanting it to?