Compute the flux of $(z \sin x, yz \cos x, x^2 + y^2)$ through the paraboloid. Given the vector field
$$F(x, y, z) = \langle z \sin x, yz \cos x, x^2 + y^2 \rangle,$$
calculate the flux $\int_S F \cdot \hat{n} \; dS$ through the paraboloid
$$S = \{(x,y,z) : z = -3(x^2 + y^2) + 3, x^2 + y^2 \leq 1\},$$
where $\hat{n}$ is the unit normal pointing in the positive $z$-direction.
Question: What does the integral evaluate to? Can this be computed by hand?
This is an exam question, and I don't know if it's written correctly. Stokes' theorem doesn't work since it's not divergence free. I tried to compute the integral explicitly by both the divergence theorem and explicitly taking the dot product with the normal, but it seems that there is no way to evaluate the integral by hand.
If one of the $x$ or $y$ components has a sign change, then the flux integral becomes easy to evaluate, so I don't know if that's what they meant. I have no way to contact the exam writer directly. I am asking this question to confirm that the problem may be incorrect before bringing it up.

Hint to find $\hat{n}:$
If we have some function $z=f(x,y)$, then the outer normal vector to the graphic of $f(x,y)$ is denoted by $(-f_x,-f_y,1)$.
