Answer to Question #138615 in Differential Geometry | Topology for anjali g

Question

Sketch the level curves f−1(c) for the following functions:
a) f(x ,y ,z) = x−y^2 −z^2, c = −1,0,1.

Answer 1

Given,

$$f(x,y,z)=x-y^2-z^2$$

Now, level curve is defined as

$$G_f=\{(x,y,z)\in \mathbb{R}^3|f(x,y,z)=c\}$$

That is, here, $f^{-1}(c)=G_f$

Now, set

$$f(x,y,z)=c\implies (x-c)=y^2+z^2$$

Clearly, the level curves are family of paraboloid which passes through the center $(c,0,0)$

Here, blue, green and red are level curve of $f^{-1}(-1),f^{-1}(0)\& f^{-1}(1)$ respectively.