I want to know more about the differential geometry of surfaces, especially Gaussian curvature. Obviously, we can get the mean curvature of a surface from the divergence of the unit normal vector of the surface. However, can the Gaussian curvature be derived from the divergence or curl of the unit normal vector of the surface? Perhaps there is also some historical / background information about their importance? Thank you in advance.
Supplement: The mean curvature of a surface specified by an equation $\displaystyle\,\!F(x,y,z)=0$ can be calculated by using the gradient $\displaystyle\nabla F=\left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right)$ and the divergence of the unit normal. A unit normal is given by $\displaystyle\frac{\nabla F}{|\nabla F|}$ and the mean curvature is
$\displaystyle H = -{\frac{1}{2}}\operatorname{div}\left(\frac{\nabla F}{|\nabla F|}\right)$
Best Answer
You can refer to Diferential Geometry Of Three Dimension Vol II by Charles Ernest Weatherburn
https://archive.org/details/diferentialgeome031396mbp/page/n91/mode/2up
$$\displaystyle K_G = \dfrac{1}{2}\operatorname{div}\left[\frac{\nabla F}{|\nabla F|}\cdot\operatorname{div}\left(\frac{\nabla F}{|\nabla F|}\right)+\frac{\nabla F}{|\nabla F|}\times\operatorname{curl}\left(\frac{\nabla F}{|\nabla F|}\right)\right]$$
Weatherburn annotates Gaussian curvature as the second curvature of the surface and the mean curvature as the first curvature of the surface.