Contour Line

Topics: Calculus - Topology

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(definition)

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a vectorial function. We define the contour line of level $k$ as:

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

Contour lines are an instance of level sets.

Example

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

The contour line of $k=1$ is:

$$C_i=\{(x,y)\mid x^2+y^2=1\}$$