Let $\sigma: U \rightarrow \mathbb{R}^{3}$ be a parametrised surface, where $U \subset \mathbb{R}^{2}$ is an open set.

(a) Explain what are the first and second fundamental forms of the surface, and what is its Gaussian curvature. Compute the Gaussian curvature of the hyperboloid $\sigma(x, y)=(x, y, x y)$.

(b) Let $\mathbf{a}(x)$ and $\mathbf{b}(x)$ be parametrised curves in $\mathbb{R}^{3}$, and assume that

$$\sigma(x, y)=\mathbf{a}(x)+y \mathbf{b}(x)$$

Find a formula for the first fundamental form, and show that the Gaussian curvature vanishes if and only if

$$\mathbf{a}^{\prime} \cdot\left(\mathbf{b} \times \mathbf{b}^{\prime}\right)=0$$