2. [11 points] Let $\sigma$ be a surface patch given by
$\sigma(u, v) = (u, v, e^{uv})$,
where $u, v \in \mathbb{R}$.
1. [3 points] Show that $\sigma$ is a regular surface and find its first fundamental form.
2. [4 points] At the point $P(0, 0, 1)$, find the principal curvatures $k_1$ and $k_2$ of $\sigma$ at
P and then the mean and Gaussian curvatures of $\sigma$ at P.
3. [4 points] Find the normal and geodesic curvatures of the curve $\gamma(t) = \sigma(1, t)$.
