here is a problem im trying to solve for a few days, and Im not getting sucess.
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $f\in C^1$ and $F(x,y,z)=f(y/x,z/x)$. Consider a level surface $S$ defined by $F(x,y,z)=0$ and $(x_0,y_0,z_0)\in S$. What are the conditions, in a neighborhood of $(x_0,y_0,z_0)$, that we can write $S$ in the form $z=g(x,y)$ ? Also, show that at this conditions, its true that $x\frac{\partial g}{\partial x}(x,y)+y\frac{\partial g}{\partial y}(x,y)=g(x,y)$
Well, here is what I've done: To use the Implicit Function Theorem I choose $F(x_0,y_0,z_0)=0$ and I have to show that $\frac{\partial F}{\partial z}(x_0,y_0,z_0)\neq 0$. $\frac{\partial F}{\partial z}(x_0,y_0,z_0)=\textrm{lim}_{t\rightarrow 0}\frac{F(x_0,y_0,z_0+t)-F(x_0,y_0,z_0)}{t}=\textrm{lim}_{t\rightarrow 0}\frac{F(x_0,y_0,z_0+t)}{t}=\textrm{lim}_{t\rightarrow 0}\frac{f(y_0/x_0,(z_0+t)/x_0)}{t}\neq 0$
Here I tried something that I dont know if is right, and even if is right, i couldnt conclude the right answer from this. $\textrm{lim}_{t\rightarrow 0}\frac{f(y_0/x_0,(z_0+t)/x_0)}{t}=\textrm{lim}_{t\rightarrow 0}\frac{f(y_0/x_0,(z_0/x_0)+(t/x_0))}{t}=\frac{1}{x_0}\frac{\partial f}{\partial y}(y_0/x_0,z_0/x_0)$ I did this variable change $u=t/x_0$ with $u\rightarrow 0$ and calculate the limit. But Im really not sure about it and I didnt found the last equality they are asking.
Thank you everyone one more time.