I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following question;
Question: Find all the regular value of the map $g : \mathbb{R}^2 \rightarrow \mathbb{R}$ given by: $g(x,y) = x^2 - y^2$
My Answer: Well, $dg(x,y) = [2x, -2y]$, and, in order for $z \in \mathbb{R}$ to be a regular value of $g$ we need that for every $(x,y) \in g^{-1} (z)$ for rank $dg(x,y) =1$. So in this case we just need rank $dg(x,y) \neq 0$, i.e. $(0,0) \notin g^{-1}(z)$. So since $g(0,0) =0$ we have that $0$ is not a regular value of $g$ and every $z \in \mathbb{R} - \{0\}$ is a regular value of $g$.
Any advice here would be appreciated as I am finding things a little confusing.