I am a bit confused about how to solve this exercise. I can't use Stokes Theorem since both sides are open right?
Let S be the surface of the paraboloid: $ z=3-x^2-y^2 $ for $-1\leq z\leq 1$ oriented with the outward unit normal. Let
$\vec F(x,y,z)=x\vec i+z\vec j-y\vec k $ be the velocity field of the fluid
The paraboloid is $x^2 + y^2 = 3-z$, so you always must have $z\leq 3$ (the vertex is at $z=3$).
So I wonder if you can do something like $$\int_{z\in[-1,1]} = \int_{z\in[-1,3]} - \int_{z\in[1,3]}$$
so that in the two integral on the RHS the boundary is connected?