page_title: Level set
section_title: Examples
hierarchical_section_title: Level set / Examples
context_page_description: In mathematics, a level set of a real-valued function f of n real variables is a set of the form
  that is, a set where the function takes on a given constant value c.
  When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. So a level curve is the set of all real-valued solutions of an equation in two variables x₁ and x₂. When n = 3, a level set is called a level surface, and for higher values of n the level set is a level hypersurface. So a level surface is the set of all real-valued roots of an equation in three variables x₁, x₂ and x₃, and a level hypersurface is the set of all real-valued roots of an equation in n variables.
  A level set is a special case of a fiber.
context_section_description: Consider the 2-dimensional Euclidean distance: 
   A level set  of this function consists of those points that lie at a distance of  from the origin, otherwise known as a circle. For example, , because . Geometrically, this means that the point  lies on the circle of radius 5 centered at the origin. More generally, a sphere in a metric space  with radius  centered at  can be defined as the level set .
  A second example is the plot of Himmelblau's function shown in the figure to the right. Each curve shown is a level curve of the function, and  they are spaced logarithmically: if a curve represents , the curve directly "within" represents , and the curve directly "outside" represents .
caption: Log-spaced level curve plot of Himmelblau's function [1]