Let $S_1$ be the lateral surface of $x^2 + y^2 \leq 4$ for $-3 \leq z \leq 3$, and $S_2$ be the sphere $x^2 + y^2 + (z-3)^2 = 4$. Assuming $S$ to be $S_1 \cup S_2$ and $\hat{N}$ to be unit outward normal on $S$, for a given vector field $\overrightarrow{F}$ for which $div(\overrightarrow{F}) = 0$, how can we evaluate $\iint_S\overrightarrow{F}.\hat{N}dS$ using divergence theorem?
So the solution offers adding a disk with equation $x^2 + y^2 \leq 4$ for $z = -3$, which encloses the region and then divergence theorem can be used. But what about the half spherical surface which lies inside the cylindrical surface? By adding this disk, don't we exclude the flux passing through this inner half-spherical surface inside the region?
By the way, the answer suggested is $\iint_S\overrightarrow{F}.\hat{N}dS = - \iint_{S'}\overrightarrow{F}.(-\hat{k})dS$, where $S'$ represents the disk.