First Fundamental Form

Definition

$g_{i}j{|}_{(1,1,2)}=\left(\begin{array}{cc}6& -2\\ -2& 10\end{array}\right)$ Given $\sigma (u,v)\in {\mathbb{R}}^3$ $\mathrm{I}=Ed{u}^2+2Fdudv+Gd{v}^2:=({\sigma }_{u}du+{\sigma }_{v}dv)\cdot ({\sigma }_{u}du+{\sigma }_{v}dv)=d\sigma \cdot d\sigma$ Here, $E={\sigma }_u^2$, $F={\sigma }_u{\sigma }_v$, $G={\sigma }_v^2$

Facts

$d\sigma$ is defined by Differential 1-Form