Calculate the flux of the vector field $F(x, y, z) = (y + 2xz, y + z, −2x − z^2)$ through the surface given by $x^2 + y^2 + z^2 = 4$ and $x, y, z > 0$. The normal direction to the surface points away from the origin.
What I did was I first changed my coordinates and then basically used the divergence theorem on the $1/8$ of the sphere. $$\iint F(x,y,z) \cdot ds = \iiint_{k} r^2\sin(\phi) dr d\phi d\theta = \frac{4\pi}{3}$$
But in the answers this is what they did:
I understand they added the remaining 2D sides of the open $1/8$ of a sphere but is it really necessary? Doesn't the divergence already take care of that? That brings me to my next question does the divergence actually take care of that since these three surfaces isn't even in integral? Last but not least, how does telling me the that the normal vector points away from the origin help me? Is it just to tell kind of tell us that the surface is oriented?