Editing First fundamental form (section)

== Definition ==

Let {{math|''X''(''u'', ''v'')}} be a [[parametric surface]].  Then the inner product of two [[tangent vector]]s is
<math display="block">
\begin{align}
& \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\[5pt]
= {} & ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\[5pt]
= {} & Eac + F(ad+bc) + Gbd,
\end{align}
</math>
where {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} are the '''coefficients of the first fundamental form'''.

The first fundamental form may be represented as a [[symmetric matrix]].
<math display="block">\mathrm{I}(x,y) = x^\mathsf{T}
\begin{bmatrix}
E & F \\
F & G
\end{bmatrix}y
</math>