(a) For each of the following subsets of $\mathbb{R}^{3}$, explain briefly why it is a smooth embedded surface or why it is not.

$$\begin{aligned} S_{1} &=\{(x, y, z) \mid x=y, z=3\} \cup\{(2,3,0)\} \\ S_{2} &=\left\{(x, y, z) \mid x^{2}+y^{2}-z^{2}=1\right\} \\ S_{3} &=\left\{(x, y, z) \mid x^{2}+y^{2}-z^{2}=0\right\} \end{aligned}$$

(b) Let $f: U=\{(u, v) \mid v>0\} \rightarrow \mathbb{R}^{3}$ be given by

$$f(u, v)=\left(u^{2}, u v, v\right),$$

and let $S=f(U) \subseteq \mathbb{R}^{3}$. You may assume that $S$ is a smooth embedded surface.

Find the first fundamental form of this surface.

Find the second fundamental form of this surface.

Compute the Gaussian curvature of this surface.