How can one derive surface area Jacobians in spherical coordinates?

[sriracha]

So I've been trying to figure out how to find the surface area Jacobians in spherical coordinates (I know how to use it to find the volume Jacobian). Using the divergence theorem I was able to find these Jacobians top-down, however, I am unsure to how one would derive them in the first place. I have tried playing around with matrix that maps your r, θ, $\phi$ onto x, y, z, without success. I know that there is something to do with a tangent plane here as well, but this is where I get very lost.

Scouring the interwebs, I found that the scale factors matched my surface area Jacobians (well not directly, but they matched what I needed to multiply into my surface integral when I was using d$\phi$ and d$\theta$ to "sweep" across my surface. To explain this better I will list the scale factors:
h_r=1
h_$\theta$=rsin$\phi$
h_$\phi$=r
So if I was trying to find the area of a face with normal vector r I would need to use h_$\theta$=rsin$\phi$ and h_$\phi$=r (since d$\phi and d\theta$ sweep across this face), so r^2sin$\phi$, times d$\theta$d$\phi$ to find the area of this face.)

My question now is how do I find these scales factors in a non-top-down manner. I tried looking here: http://mathworld.wolfram.com/ScaleFactor.html, but it was in Chinese.

I am teaching myself this so please correct me if I am using the wrong terms (in particular, I doubt there are such things called a surface area or volume Jacobian, I just lack a better, more correct descriptor).

 

[arildno]

What, exactly, is a "non-top-down" manner??

 

[mathwonk]

what do spherical coordinates have to do with surface area?