Variation of the normal direction across a surface

Divide the square $\left\{ 0 \leqslant x \leqslant 1, 0 \leqslant y \leqslant 1\right\}$ into finitely many parts $\sigma$ of diameter $\leqslant \delta$. If the surface

is such that at any two points $P(x,y)$ and $P_{1}(x_{1}, y_{1})$ belonging to the same part $\sigma$ the normal directions differ by less than $1^{\circ }$, find an upper bound for the number $\delta$.

Let

be the equation of a surface, and let $\varphi (P_{1}, P)$ be the angle between the normals of the surface at the two points $P(x,y) \in D$ and $P_{1}(x_{1}, y_{1}) \in D$.

Prove that if the domain $D$ is bounded and closed, and the function $f(x,y)$ has bounded second-order derivatives in $D$, then the Lyapunov inequality

holds, where $C$ is a constant and $\rho (P_{1}, P)$ is the distance between the points $P$ and $P_{1}$.