Let $\mathcal{D}$ be a cylinder: $\mathcal{D}=B(0,R)\times[-a,a]$ where $a > R > 0$ and let $\mathcal{D}$ be oriented by outer normal $n$. Parametrize $S(0,R) \times [-a,a]$.
I am not sure how to go about this. Why is $\mathcal{D}$ a cylinder? I do not see the general picture. Would $S(0,R) \times [-a,a]$ be a cylinder without the boundary? Any suggestions appreciated.