(i) Give the definition of the surface area of a parametrized surface in $\mathbf{R}^{3}$ and show that it does not depend on the parametrization.

(ii) Let $\varphi(u)>0$ be a differentiable function of $u$. Consider the surface of revolution:

$$\left(\begin{array}{l} u \\ v \end{array}\right) \mapsto f(u, v)=\left(\begin{array}{c} \varphi(u) \cos (v) \\ \varphi(u) \sin (v) \\ u \end{array}\right)$$

Find a formula for each of the following: (a) The first fundamental form. (b) The unit normal. (c) The second fundamental form. (d) The Gaussian curvature.