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.