Generally compute Gauss curvature

Question

For a surface $F(x,y,z)=0$, we want to compute its Gauss curvature. I tried to suppose $z=f(x,y)$ locally and get a complicated expression. Is there any direct way to compute this? Thanks for your help.

Answer 1

More or less.There is a formula which you can try to apply in some cases. I'll state here lemma $5.1$ in O'Neill's Elementary Differential Geometry, page $236$:

Let $Z$ be a nonvanishing vector field on $M$. If $V$ and $W$ are tangent vector fields such that $V \times W = Z$ then: $$K =\frac{Z \cdot \nabla_VZ \times \nabla_WZ}{\|Z\|^4}\qquad H = -Z\cdot\frac{\nabla_VZ\times W + V\times \nabla_WZ}{2\|Z\|^3}.$$

You can take $Z$ to be $\nabla F$, but finding $V$ and $W$ can still be a problem, though. Usually one tries to find $V$ and $W$ geometrically, knowing something about the surface beforehand.