# Scale Factors and Jacobians

1. Mar 12, 2012

### 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).

2. Mar 12, 2012

### arildno

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

3. Mar 12, 2012

### mathwonk

what do spherical coordinates have to do with surface area?