Level Set

Topics: Calculus - Topology

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(definition)

Let ${\mathbb{R}}^n\to \mathbb{R}$. We define the level set of level $k$ for $f$ as:

$$C_k=\{\overline{x}\in {\mathbb{R}}^n\mid f(\overline{x})=k\}$$

Contour lines are an instance of level sets.

Example

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be defined by $f(x,y,z)=x^2+y^2+z^2$.

The level set of level $1$ for $f$ is:

$$C_1=\{(x,y,z)\in {\mathbb{R}}^3\mid x^2+y^2+z^2=1\}$$

…or in other words, the sphere of radius $1$ with centre $(0,0,0)$.

Similarly, the level set of level $k$ is the sphere of radius $k$ with centre $(0,0,0)$:

$$C_k=\{(x,y,z)\in {\mathbb{R}}^3\mid x^2+y^2+z^2=k\}$$