Let $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the Gaussian curvature $K$ of $S$ in terms of the coefficients of the first and second fundamental forms, computed with respect to a local parametrization $\phi(u, v)$ of $S$.

Prove the Theorema Egregium, i.e. show that the Gaussian curvature can be expressed entirely in terms of the coefficients of the first fundamental form and their first and second derivatives with respect to $u$ and $v$.

(b) State the global Gauss-Bonnet theorem for a compact orientable surface $S$.

(c) Now assume that $S$ is non-compact and diffeomorphic to $\mathbb{S}^{2} \backslash\{(1,0,0)\}$ but that there is a point $p \in \mathbb{R}^{3}$ such that $S \cup\{p\}$ is a compact subset of $\mathbb{R}^{3}$. Is it necessarily the case that $\int_{S} K d A=4 / \pi ?$ Justify your answer.