Curl of Normal Vector Field Is Zero on Surface

Question

Let $S$ be a smooth surface in $\mathbb{R}^3$. Let $\vec{N}$ be a unit normal vector field along $S$.

1. Is it true that $\vec{\nabla} \times \vec{N} = 0$ on $S$ for any smooth extension of $\vec{N}$?

2. Is it true that $\vec{\nabla} \times \vec{F} = 0$ on $S$ for any vector field $\vec{F}$ on $\mathbb{R}^3$ such that $\vec{F} = 0$ on $S$?

Thank you.