Problem :
Find equations of the tangent plane and the normal line to a surface at a point.
Suppose a surface $S$ is given implicitly by $\color{red}{F(x,y,z)=k}$ for a differentiable function $F$, i.e., the level surface of a differentiable function of three variables.
Any surface of the form $\color{red}{z = f (x, y)}$ can be put in this form as $\color{red}{f (x, y) − z = 0}$ for a differentiable $f$ , i.e., $\color{red}{F (x, y, z) = f (x, y) − z}$ and $\color{red}{k = 0}$.
I don't understand the $\color{red}{\text{red}}$ part. Doesn't the function $F(x,y,z)$ require $4$-axis for its full representation? $F$ is essentially $4$-dimensional dependent on $3$-dimension independent variables $x,y,z$. Why then is $F (x, y, z) = f (x, y) − z$? When $f (x, y) − z = 0$ for $f(x,y)$?
I am probably missing some knowledge somewhere. Can anyone suggest what it is?
The function does, but the equation $F(x, y, z) = k$ does not, right? The surface $S$ is not representing the function $F$, but the solution of the equation.