1. For each of the surfaces below calculate (i) their first and second fundamental forms and (ii) their Gaussian and Mean curvatures.
(a) $z = x^2 - y^2$.
(b) $x(u, v) = (\cosh u \cos v, \cosh u \sin v, \sinh u)$.
(c) $x^2 + y^2 - z^2 = 1$.
(d) $x(u, v) = (u - \frac{u^3}{3} + uv^2, v - \frac{v^3}{3} + vu^2, u^2 - v^2)$.
(e) $x(u, v) = ((R + r\cos u)\cos v, (R + r\cos u)\sin v, r\sin u)$, $R > r > 0$.
(f) $x(u, v) = \alpha(u) + v\beta(u)$, where $\alpha(u)$ and $\beta(u)$ are smooth curves in $\mathbb{R}^3$.
