Divergence theorem integral

Question

I'm stuck on this question. For part (a), I chose g(x,y,z) = (9-x^2-y^2)/(9+x^2+y^2) - z. Then I calculated the unit normal vector which ended up being a mess. How do I use the divergence theorem for part (a)?

Answer 1

Over the Surface $D, z = 0 \implies 9 - x^2 - y^2 = 0$

or $D$ is a disk of radius $3$

$n = (0,0,-1)\\ F\cdot n = -y^2$

$\iint -y^2\ dD$

b) $\nabla \cdot F = x+2$

$\iiint x\ dV + 2V$ and $\iiint x\ dV=0$ as it is an odd function integrated around a symmetric region.

c) by the divergence theorem $\iint F\ dD + \iint F\ dS = \iiint \nabla \cdot F \ dV$