# Graph einer Funktion/Zweidimensional/EFG-Formel/Bemerkung

Es sei

${\displaystyle {}M={\left\{(u,v,\psi (u,v))\mid (u,v)\in W\right\}}\subseteq W\times \mathbb {R} \,}$

der Graph von

${\displaystyle \psi \colon W\longrightarrow \mathbb {R} ,}$

wobei ${\displaystyle {}W\subseteq \mathbb {R} ^{2}}$ eine offene Teilmenge sei. In diesem Fall stehen Fakt und Fakt wie folgt miteinander in Beziehung. Die partiellen Ableitungen sind ${\displaystyle {}{\begin{pmatrix}1\\0\\{\frac {\partial \psi }{\partial u}}\end{pmatrix}}}$ und ${\displaystyle {}{\begin{pmatrix}0\\1\\{\frac {\partial \psi }{\partial v}}\end{pmatrix}}}$. Daher ist ${\displaystyle {}E=1+{\left({\frac {\partial \psi }{\partial u}}\right)}^{2}}$, ${\displaystyle {}F={\frac {\partial \psi }{\partial u}}{\frac {\partial \psi }{\partial v}}}$, und ${\displaystyle {}G=1+{\left({\frac {\partial \psi }{\partial v}}\right)}^{2}}$. Somit ist

${\displaystyle EG-F^{2}={\left(1+{\left({\frac {\partial \psi }{\partial u}}\right)}^{2}\right)}{\left(1+{\left({\frac {\partial \psi }{\partial v}}\right)}^{2}\right)}-{\left({\frac {\partial \psi }{\partial u}}\right)}^{2}{\left({\frac {\partial \psi }{\partial v}}\right)}^{2}=1+{\left({\frac {\partial \psi }{\partial u}}\right)}^{2}+{\left({\frac {\partial \psi }{\partial v}}\right)}^{2}\,.}$