Level Set

The level sets of a function are all the input values that result in some defined output value $c$. Another way to interpret a level set is that in a topographical map of, say, a mountain, the level sets are all the different areas in the mountain that are of the same height.

Definition §

Level Set ^definition

For any function $f(x_1,x_2,\dots ,x_n)$ of arity $n$ (i.e. a function with $n$ arguments) with domain $D$, the level set is defined as the set:

$$L_c=\{(x_1,x_2,\dots ,x_n)\in D:f(x_1,x_2,\dots ,x_n)=c\}$$

where $c$ is some value in $\mathbb{R}$.

Note the gradient vector is always perpendicular to the level set.

Contour §

Contour ^definition-contour

A level set in 2 dimensions, i.e. $z=f(x,y)$ is called a contour:

$$L_c=\{(x,y)\in D:f(x,y)=c\}$$

That is, the contour defines all the points that have the same $z$-value i.e. all inputs which have the same output $c$.

The $L_0$ contour is sometimes called a level curve.

Visualisation §

%%🖋 Edit in Excalidraw, and the dark exported image%%

Examples §

1: Level sets of a paraboloid

Sketch the level sets for the function $f(x,y)=x^2+y^2$

Solution

Note that technically these are contours, but contours are level sets as well. We let:

$$x^2+y^2=c$$

This is the equation of a circle with radius $\sqrt{c}$. So we have level curves: %%🖋 Edit in Excalidraw, and the dark exported image%%