Definition:First Fundamental Form

Definition
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:


 * $\operatorname I: T_pS \times T_pS \longrightarrow \R$

induced from the dot product of $\R^3$:


 * $\operatorname I \left({x, y}\right) = \left\langle x, y \right\rangle$

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 $R \subset S$.

Also see

 * Definition:Second Fundamental Form