Editing First fundamental form (section)

==Gaussian curvature==

The [[Gaussian curvature]] of a surface is given by 
<math display="block"> K = \frac{\det \mathrm{I\!I}_p}{\det \mathrm{I}_p} = \frac{ LN-M^2}{EG-F^2 }, </math>
where {{mvar|L}}, {{mvar|M}}, and {{mvar|N}} are the coefficients of the [[second fundamental form]].

[[Theorema egregium]] of [[Carl Friedrich Gauss|Gauss]] states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that {{mvar|K}} is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the [[Gaussian curvature#Alternative_formulas|Brioschi formula]].