A very good review question for the final exam:
If is the upper hemisphere () of a unit sphere oriented upward, and is the unit disk in the -plane oriented upward, is ?
What if I claimed that . Would you think I was lying? If not, what could you conclude about the divergence of ?
Answer in the comments if you like; I’ll post a further discussion tomorrow.
[EDIT: Haven’t had any comments or questions about this one. It is designed to remind you that you can’t always change one surface into another with the same boundary, but it requires that divergence is zero.]