(a) Let $S \subset \mathbb{R}^{3}$ be a surface. Give a parametrisation-free definition of the first fundamental form of $S$. Use this definition to derive a description of it in terms of the partial derivatives of a local parametrisation $\phi: U \subset \mathbb{R}^{2} \rightarrow S$.

(b) Let $a$ be a positive constant. Show that the half-cone

$$\Sigma=\left\{(x, y, z) \mid z^{2}=a\left(x^{2}+y^{2}\right), z>0\right\}$$

is locally isometric to the Euclidean plane. [Hint: Use polar coordinates on the plane.]

(c) Define the second fundamental form and the Gaussian curvature of $S$. State Gauss' Theorema Egregium. Consider the set

$$V=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 x y-2 y z=0\right\} \backslash\{(0,0,0)\} \subset \mathbb{R}^{3}$$

(i) Show that $V$ is a surface.

(ii) Calculate the Gaussian curvature of $V$ at each point. [Hint: Complete the square.]