I have the regular surface given by the zeros of $$f: \mathbb{R}^3\longrightarrow \mathbb{R} $$ $$ f(x) = (4x-2y-z)^2-1$$ I'm supposed to calculate the Gaussian Curvature of this surface, defined as $K=k_1k_2$ being $k_1$ and $k_2$ the principal curvatures.
To calculate the principal curvatures, I need the derivative of the Normal Unit Vector along the surface $\mathcal{N}$. In other words, I need a chart $(U,\varphi)$ of the surface so that I can calculate $\mathcal{N}$.
The only way I have of calculating a chart for $f$ is trying to view it as the graph of a function, hence I should try to obtain an expression for either $x$, $y$ or $z$ from the expression of $f$. This is not possible as the expression is difficult to manipulate and it really doesn't seem to be the way.
Then how am I supposed to calculate the Gaussian Curvature? Is there any other method not involving an expression of a chart $(U,\varphi)$? I understand that for most $f$ is not easy to find an explicit chart.
The zero set of your function is given by:
$$S = \{(x, y, z) \in \mathbb{R}^3 \ \vert \ |4x - 2y-z| = 1 \}$$
This is a regular surface whose connected components are clearly planes. Therefore the Gaussian curvature of $S$ vanishes everywhere, i.e $K_S \equiv 0$.