I'm trying to find $dz$ and $d^2z$ for the function defined by the equation $F(\frac{x}{z},\frac{y}{z}) = k$, where $F \in C^2$.
To solve the problem, the most natural way to proceed in my mind is to apply the Implicit Function Theorem to find $z(x,y)$ around $z \in \mathbb{R}, z \neq 0$ and then compute the total differentials.
If that's correct, my next question is how I should obtain $z(x,y)$. I don't know how to think about the arguments of $F$ being expressed in terms $z$. Is a change of variables necessary?
If that's incorrect, what is the best strategy here?
Thank you.