proving that this map is not an isometry between the surfaces Consider the parametrized surfaces
  $$\phi(u,v)= (u\cos(v),u\sin(v),\log(u))$$
  $$\psi(u,v)= (u\cos(v),u\sin(v),v)$$
  Prove that the Gaussian curvature equals in both surfaces in each $(u,v)$. And then prove that $\psi \circ \phi^{-1}$ is not an isometry (i.e  the differential is not an isomorphism).


I proved that the Gaussian curvature are the same for both surfaces, but I'm struggling with the computation of the differential (or maybe there is another way to do it).
Compute the corresponding first fundamental forms.
Compute the corresponding first fundamental forms.