I'm trying to calculate the flux of the field $F = (0, 0, z)$ over the spherical surface $x^2+y^2+z^2 = 9$ where the normal vector is positively oriented.
I have parameterized the sphere in spherical coordinates. $r(\theta, \phi) = (3\sin\phi\cos\theta, 3\sin\phi\sin\theta, 3\cos\phi)$ in the domain $D = \{(\theta, \phi) \in (0, 2\pi)\times(0, \pi)\}$. The positive oriented normal vector is $\vec{n} = (9\sin\phi\cos\theta, 9\sin^2\phi\sin\theta, 9\sin\phi\cos\phi)$.
Now, my resolution and the textbook solutions disagree when calculating the flux over this sphere. My solution:
$$\iint_D F\cdot \vec{n}dS = \iint_D (0, 0, 3\cos\phi)\cdot (9\sin\phi\cos\theta, 9\sin^2\phi\sin\theta, 9\sin\phi\cos\phi) 9\sin\phi d\theta d\phi$$
Textbook solution:
$$\iint_D F\cdot \vec{n}dS = \iint_D (0, 0, 3\cos\phi)\cdot (9\sin\phi\cos\theta, 9\sin^2\phi\sin\theta, 9\sin\phi\cos\phi) d\theta d\phi$$
As you can see, the textbook solution doesn't have the Jacobian for the change in variables from cartesian to spherical coordinates (which should be $3\sin\phi$). But why? I thought it was the $dS = \rho^2 \sin \phi d\phi d\theta$ because a small segment of area $dS$ in sphere from spherical coordinates would have that expression. But it seems that I may be mistaken. Why is it not applicable here?
For the given sphere, the positive oriented normal unit vector is $$\vec{n} = (\sin\phi\cos\theta, \sin\phi\sin\theta,\cos\phi)$$ whereas the infinitesimal area element $dS=R^2\sin\phi d\theta d\phi=9\sin\phi d\theta d\phi$.
Therefore the flux is $$\iint_S \vec{F}\cdot \vec{n}dS = \iint_D (0, 0, 3\cos\phi)\cdot (\sin\phi\cos\theta, \sin\phi\sin\theta,\cos\phi)\, 9\sin\phi d\theta d\phi\\ =\iint_D (0, 0, 3\cos\phi)\cdot (9\sin^2\phi\cos\theta, 9\sin^2\phi\sin\theta,9\sin\phi\cos\phi)\, d\theta d\phi$$ which is your textbook solution (I think you missed a square).