---
tag: 0Y5Q8Q9WJ7
kind: definition
name: Level set
msc: [26B99]
defines: 0B2RZ0778N
uses: [0JPCHTXWG0, 01AHG4G9RT]
sources:
  - url: https://en.wikipedia.org/wiki/Level_set
    locator: lead section
---

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$ be a [[function of several variables|0JPCHTXWG0]], and let $c \in \mathbb{R}$.
The level set of $f$ at the value $c$ is the set
$$L_c(f) = \{x \in D : f(x) = c\}.$$
It is the [[preimage|01AHG4G9RT]] $f^{-1}(\{c\})$ of the one-point set $\{c\}$.
For $n = 2$ a level set is also called a level curve, and for $n = 3$ it is also called a level surface.

## Remarks

For $f(x, y) = x^2 + y^2$, the level set at $c > 0$ is the circle of radius $\sqrt{c}$ about the origin. The level set at $c = 0$ is the point $(0, 0)$, and the level set at $c < 0$ is empty.
The level sets of $f$ for different values of $c$ are disjoint, and their union is the domain $D$.
At a point where $f$ is differentiable with a nonzero [[gradient|0ABK9V9Y64]], the gradient is [[orthogonal to the level set|0VXYV3NQQ1]].
A level set of a function of two variables is a [[contour line|0QD83RAN8G]] of its graph.

## Explanations

### L1 human

On a map of a hill, a level set is the set of all places at one fixed height.
For the bowl $f(x, y) = x^2 + y^2$, each level set at a positive height is a circle, and the circles grow as the height grows.