Confused with a spherical coordinate system surface element I can not understand how a particular surface element is derived in spherical coordinates. 
The equation expressing the surface element vector is given as
$$r_s = (\sqrt{R^2-z_s^2} \cos \phi,\sqrt{R^2-z_s^2}\sin \phi, z_s)$$
And then the differential surface element is given as 
$$ dr_s = R d\phi dz_s $$
The center of the sphere is the origin and its radius is $R$. The z axis coordinate of the surface element is $z_s$.
I can not find a reference on the web that shows this particular surface element.  I figured out that it is the formula for an 'infinitesimal' spherical zone of height $dz_s$ if $d\phi$ can be integrated out by symmetry to yield $2\pi$, (that is a spherical zone area = $2\pi Rh$ where here $h=dz_s$). 
So in an integral where there is symmetry around the z axis, this differential surface area can be used to integrate a function over a sphere by summing up infinitesimal spherical zones over the z axis.  
My question is: How is this surface element derived? 
 A: In the first equation you have the Cartesian coordinates. Let's allow $R$ to vary, and call it $r$. Then you can write the volume element as $$dV=dx\ dy\ dz=|J|dr\ dz\ d\phi$$ 
Here J is the Jacobian of the transformation from $(x,y,z)$ to $(r,z,\phi)$. The surface is perpendicular to $r$ for the sphere, so the surface element will be then just $$ds=\frac{dV}{dr}=|J|dz\ d\phi$$
The Jacobian is
$$J=\begin{vmatrix}
\frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r} \\ 
\frac{\partial x}{\partial z} & \frac{\partial y}{\partial z} & \frac{\partial z}{\partial z} \\ 
\frac{\partial x}{\partial \phi} & \frac{\partial y}{\partial \phi} & \frac{\partial z}{\partial \phi}
\end{vmatrix}=\begin{vmatrix}
\frac{r}{\sqrt{r^2-z^2}}\cos\phi & \frac{r}{\sqrt{r^2-z^2}}\sin\phi & 0\\ 
\frac{-z}{\sqrt{r^2-z^2}}\cos\phi & \frac{-z}{\sqrt{r^2-z^2}}\sin\phi & 1\\
-\sqrt{r^2-z^2}\sin\phi&  \sqrt{r^2-z^2}\cos\phi &0
\end{vmatrix}=-r
$$
Now plugging in $r=R$, you get $ds=R\ dz\ d\phi$
