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:

$\operatorname I: T_p S \times T_p S \longrightarrow \R$

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 $R \subset S$.

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The term Definition:Tangent Space as used here has been identified as being ambiguous. In particular: tangent space for the surface seems not yet defined If you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Disambiguate}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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Also see

• Definition:Second Fundamental Form

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