Question

Sketch the level curves f−1(c) for the following functions: (a) f(x,y) = x2 + y2, c = 0,1,2,3,4.

Answer 1

Given, $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a smooth map such that $f$ is defined as below

$$f(x,y)=x^2+y^2$$

Now, when c=0 then

$$f(x,y)=0\implies x^2+y^2=0\implies x=y=0$$

Thus, $f^{-1}(c=0)=\{(0,0)\}$ .

Now, for positive $c$ ,

$$f(x,y)=c=x^2+y^2$$

Represents locus of radius $\sqrt{c}$ and center is (0,0), thus

$$f^{-1}(c)=\{(x,y)\in \mathbb{R}^2:f(x,y)=c\}$$

is the set of points on the circle mentioned above.

Thus, level curves for c=0,1,2,3,4 will be

Where, origin is $f^{-1}(0)$ , Blue, Green, Violet and black circles are respectively $f^{-1}(1),f^{-1}(2),f^{-1}(3)\&f^{-1}(4)$ respectively.