\begin{definition}[Definition:First Fundamental Form]
Let $S$ be a surface of a three-dimensional euclidean space.
Let $p$ be a point of $S$ and $T_pS$ be the tangent space to $S$ at the point $p$.
The '''first fundamental form''' is the bilinear form:
:$\operatorname I: T_p S \times T_p S \longrightarrow \R$
induced from the dot product of $\R^3$: 
:$\map {\operatorname I} {x, y} = \innerprod x y$
The '''first fundamental form''' is a way to calculate the length of a given line $C \subset S$ or the area of a given bounded region $R \subset S$.
{{Disambiguate|Definition:Region}}
{{expand}}
\end{definition}