It's going to be lengthy anyhow, but here's a slight improvement:
Compute the Gaussian and mean curvatures and in terms of the coefficients in the first and second fundamental forms, and remember that
the principal curvatures satisfy the equation .
Find the equations of the principal curvatures of the surface:
x=u, y=v, z=f(x,y)
I can see a method to do this but it looks like it will be algebraically difficult if not impossible. Can you suggest a simpler way?
What I have come up with is:
I can calculate the surface metric:
And I can calculate the unit normal:
Next I would calculate the curvature tensor:
But the partial derivatives of n are going to get ugly and I still would not be finished. I need the eigenvalues of: