"Let $\Omega$ be the solid upper half ball of radius 2, $$\Omega = \{(x,y,z) : x^2+y^2+z^2<4, z>0\}$$ with the boundary of $\Omega$ being the upper hemisphere $S$, $$S=\{(x,y,z):x^2+y^2+z^2=4,z>0\},$$ and the flat disk $D$ centred at the origin of the $x,y$ plane, $$D=\{(x,y,0):x^2+y^2=4\}.$$ Then for the vector field $$\textbf v(x,y,z)=(e^z-1,3y+\sin z,1-x)$$ letting $\textbf n$ be the unit normal on $D$ and $S$ pointing out of $\Omega$ calculate the flux of $\textbf v$ in the direction of $\textbf n$ and using divergence theorem or otherwise calculate the flux of $\textbf v$ across $S$ in the direction of $\textbf n$."
I was just wondering if the correct way to go about the first part of this problem is to parametrise $D$ by $$\textbf r(r,\theta)=(r\cos \theta, r\sin \theta), 0 \leq r \leq 1, 0 \leq \theta < 2\pi$$ to retrieve the normal $(0,0,-r)$ and compute the calculation $$\int_{0}^{2} \int_{0}^{2\pi} \textbf v(\textbf r(r,\theta))\cdot (0,0,-r) d\theta dr.$$