Let $S = \{(x, y, z) \in \Bbb{R}^3 \ | \ z = x^2 - y^2\}$ be the hyperbolic paraboloid. Our aim is to compute the Gauss curvature at the origin. I thought I knew how to do this, but I keep getting this wrong. Any hints towards the right direction will be the most appreciated.
I proceeded as follows:
We know that the hyperbolic paraboloid is the graphic of a function $f(x, y) = x^2 - y^2$, hence orientable with unit normal field $$ N = \frac{1}{\sqrt{1+|\nabla f|^2}}(-f_x, -f_y, 1) $$ where the subindex denotes partial differentiation. Then, for our function, $$ N(x, y, z) = \frac{1}{\sqrt{1+4(x^2 + y^2)}}(-2x, 2y, 1). $$ I then tried to compute the jacobian matrix of N at the point $(x, y, z)$, which gives us $(dN)_{(x, y, z)}$. At the origin, we have $$ (dN)_{(0, 0, 0)} = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$ what gives zero Gauss curvature since the determinant of this matrix is zero. However, I was supposed to get a negative value. What am I doing wrong?
PS: I chacked my computations with WolframAlpha.
Thanks in advance.