Let $g:\mathbb R^3\to \mathbb R$ be smooth and $T$ be a linear isomorphism of $\mathbb R^3$. I'm trying to determine conditions on $g$ such that the fiber $(g\circ T)^{-1}(0)$ is a graph of a smooth function about a point $p$.
The obvious way to do this is to differentiate $g\circ T$ w.r.t $z$ using the chain rule, which gives one answer.
I think there's another possibility, of looking at the right entry of $Dg\cdot A$, where $A$ is the matrix representing $T$ in the standard basis. Thing is, this method gives me a different answer. Can anyone help me out?