Principal Curvature

Definition

Given a graph $S\in {\mathbb{R}}^3$ of a smooth function $z=f(x,y)$, take any point $p\in S$ (we may assume $p=(0,0,0)$). We call the Eigenvalues ${\lambda }_1,{\lambda }_2$ of $\mathrm{Hess}(f):=\left(\begin{array}{cc}{f}_{xx}(0)& f_{xy}(0)\\ f_{yx}(0)& f_{yy}(0)\end{array}\right)$ by principal curvatures

The maximum and minimum values of curvature at a point $p\in S$

Facts

The Gaussian Curvature $K={\lambda }_1{\lambda }_2=\det (\mathrm{Hess}(f))$

The Mean Curvature $K=\frac{1}{2}({\lambda }_1+{\lambda }_2)=\frac{1}{2}\mathrm{tr}(\mathrm{Hess}(f))$