Level set

Builds on Function of several variables Preimage of a set

Definition

Let $D \subseteq \mathbb{R}^n$ , let $f \colon D \to \mathbb{R}$ be a function of several variables, 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 $f^{-1}(\{c\})$ 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, the gradient is orthogonal to the level set. A level set of a function of two variables is a contour line of its graph.

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Tag	`0Y5Q8Q9WJ7`
URI	`m2://0Y5Q8Q9WJ7`
Kind	definition
Writing	W0 stub
MSC	26B99
Defines	Level set
Uses	Function of several variables, Preimage of a set

Level set

Definition

Remarks

Reading path

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