(i)

Consider the surface

$$z=\frac{1}{2}\left(\lambda x^{2}+\mu y^{2}\right)+h(x, y)$$

where $h(x, y)$ is a term of order at least 3 in $x, y$. Calculate the first fundamental form at $x=y=0$.

(ii) Calculate the second fundamental form, at $x=y=0$, of the surface given in Part (i). Calculate the Gaussian curvature. Explain why your answer is consistent with Gauss' "Theorema Egregium".