I need help solving this question from my textbook.
Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , z \geq 0$ oriented upward.
How do I continue? Which theorem do I need to solve this problem?
The flux is given by
$$\text{Flux}=\int_S \vec F\cdot \hat ndS$$
Here, $\hat n=(\hat xx+\hat yy+\hat zz)/\sqrt{x^2+y^2+z^2}$ in spherical coordinates and
$$\vec F\cdot \hat n=(\hat x 4xz+\hat z2y)\cdot (\hat xx+\hat yy+\hat zz)/\sqrt{x^2+y^2+z^2}=2z(2x^2+y)/3$$
Now, we will convert to spherical coordinates with $x=3\sin \theta \cos \phi$, $y=3\sin \theta \sin \phi$, and $z=3\cos \theta$. We can now write
$$\begin{align} \text{Flux}&=\int_S \vec F\cdot \hat ndS\\\\ &=\int_{0}^{2\pi}\int_{0}^{\pi/2}(36\sin^2 \theta \cos^2 \phi \cos\theta +6 \sin \theta \sin \phi \cos \theta)\,9\sin \theta d\theta d\phi\\\\ &=324\pi\left(\int_{0}^{\pi/2}\sin^3 \theta \cos \theta d\theta\right)\\\\ &=81\pi \end{align}$$
NOTE: We tacitly used $\int_{0}^{2\pi} \sin \phi d\phi=0$ and $\int_{0}^{2\pi} \cos^2 \phi d\phi=\pi$ in carrying out the integrations over $\phi$.
