Principal curvatures of a surface

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: