TITLE: Intuitive Understanding About the Implicit Function Theorem
QUESTION [6 upvotes]: In multivariable calculus, given a function like $F(x,y,z) =0$, the implicit function $z=f(x,y)$ exists if and only if $\frac{\partial F}{\partial z} \not = 0$. And the implicit function is given by $\frac{dz}{dy}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$. 
The theorem can be proved by mathematical reasoning. But I want to know whether there is some intuitive understanding of the theorem, say geometry intuition in the three dimensional space, etc.

REPLY [3 votes]: If you question ask for the intuition concerning the existence of the implicit function, let me suggest the following. 
I prefer to simplify by restricting to two variables. The theorem states that if at $(x_0,y_0)$ the curve  $\{(x,y)\in\mathbb R^2 : F(x,y)=0\}$ does not look  like a vertical line (i.e if $\partial F/\partial y(x_0,y_0)\neq 0$) then the curve is locally the graph of a function $x\mapsto y(x)$. A good example to have in mind is the circle, $F(x,y)=x^2+y^2-1$. The same reasoning works for higher dimensions. 
If you ask about the form of the implicit function, the answer of copper.hat looks nigh enough to me.