I am watching a video:
https://www.youtube.com/watch?v=DRBNp7SZCvU
In the second page, I am confused about the following:
- why it requires "$F(x,y,z)=f(x,y)-z = 0$"
- And why it says the unit normal vector to $f(x,y)$ instead of to $F(x,y,z)$?
In short, I am a bit confused about the difference between $f(x,y)=z$ and $F(x,y,z)$ in geometric meaning.
What I know is
- $(x,y,f(x,y)=z)$ is a surface.
The normal vector to the surface can then be calculated as
$$(f_x(x,y),f_y(x,y),-1)$$
Thanks!
Given a differentiable function $f(x,y,z)$, its gradient will always be normal to any level surface $f(x,y,z)=c$. If you want to say something about a normal vector to the graph $z=f(x,y)$ of a function of two variables $f(x,y)$, you need to express the graph of this function as a level surface of a function of three variables.
The trick is then to define $F(x,y,z)=f(x,y)-z$. Then the graph $z=f(x,y)$ is precisely the level surface $F(x,y,z)=0$, and we can use the above fact to conclude that a normal vector to the graph will be the gradient of $F$. We can then compute $$\nabla F = \langle F_{x}, F_{y}, F_{z}\rangle = \langle f_{x}, f_{y}, -1\rangle$$