How Do You Calculate the Flux of a Vector Field Through a Hemisphere?

[vela]

Try calculating the flux through the bottom of the hemisphere. It should be obvious to you and your friends when you calculate $\vec{F}\cdot{\hat{n}}$ that the flux through that surface is 0. Since that surface is bounded by the same contour, Stokes' theorem tells you it's equal to $\iint \vec{A}\cdot d\vec{r}$ as well.

Alternatively, by the divergence theorem, you know that the sum of the flux through the bottom and the flux through the top is equal to the volume integral of the divergence, which you found was 0. Therefore, the flux through the top must also vanish.