0

Maybe an example would help explain. If we sketch this graph it is a bowel shaped object and cut along the horizon at some value of z = constant we get a level curve. I am right so far?

This curve is a circle that is 2 dimensional and sits in 3 D space at the height z?

And if I have all the curves then they form the set ? True.?

imranfat
  • 10,029
Sedumjoy
  • 1,569
  • 1
    Personally, I would say the set of all level curves of $f(x,y)$ or simply the surface of $f(x,y)$ to avoid confusion, but I believe you've got the right idea – mallan Jun 10 '18 at 23:07
  • The level sets are subsets of the $(x,y)$ plane, the don't live in $(x,y,z)$ space. – Hans Lundmark Jun 11 '18 at 07:25

1 Answers1

0

$f(x,y)=c$ gives you the circle of radius $\sqrt{c}$ in $\Bbb{R}^2$. This is the level set (in this case, curve) for the value $c$. Note that the general definition of the level set of a real valued function $f : \Bbb{R}^n \rightarrow \Bbb{R}$ at $c \in \Bbb{R}$ is $ \{ (x_1, \dots, x_n) \in \Bbb{R}^n \mid f(x_1, \dots, x_n)= c \} $. Note how in our case, the points of the level set will be living in $\Bbb{R}^2$.

mathphys
  • 2,899