Yes since grad(F) is not zero we can assume that( without loss of generality) , according to the implicit function theorem where f is a smooth function. Then the remaining is trivial.

To see the relationship between mean curvature and the normal vector field, still let be the unit normal vector of the hyper-surface M in . be the second fundamental form of M, where is the ordinary differential operator of . The mean curvature is the mean value of the principal curvatures, that is, eigenvalues of .

So .

Let be an orthonormal basis of the tangent plane, we have

.

The last term is nothing but . You may find a proof in some standard text book. But if you couldn't find one let me know, and I'll post one here.