\begin{definition}[Definition:Normal Vector]
Let $S$ be a surface in ordinary $3$-space.
Let $P$ be a point of $S$.
Let $\mathbf n$ be a vector whose initial point is at $P$ such that $\mathbf n$ is perpendicular to $S$ at $P$.
Then $\mathbf n$ is a '''normal vector''' to $S$ at $P$.
\end{definition}