I don't have enough reputation to comment on the answer to the question asked here so I had to make a new question to ask a follow-up.
I know that the cross product gives an area but I'm having trouble visualizing $r_r \times r_\theta$.
Below is a link to a diagram I made of the top hemisphere of sphere in the x-z plane. (I can't yet embed images either apparently).
Sphere in cylindrical coordiantes
In the answer by kuifje, they state: $A=2\iint_D \|r_r\times r_{\theta}\| dA$
I believe that $r_r$ is r in the r direction (radially outward) and thus is the green arrow I drew; and $r_\theta$ is r in the $\theta$ (tangential to the radius) and thus is the purple dotted circle I drew for a vector coming out of the page. However, the proper cross product to produce the differential areas for this surface seem to me to require the orange arrow crossed with the purple dotted circle. Where is my thinking going wrong?
Is it that the surface integral of region D can be thought of as the hemisphere projected onto a plane and it looks like a circle? Then it would make sense to me that $r_r$ and $r_\theta$ form the differential areas.
Like so: Region D
Any help would be appreciated.