Level sets

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 (see also isosurface), 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 (n > 3) variables.

A level set is a special case of a fiber.

Level sets show up in many applications, often under different names.

For example, an implicit curve is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an implicit equation. Analogously, a level surface is sometimes called an implicit surface or an isosurface.

The name isocontour is also used, which means a contour of equal height. In various application areas, isocontours have received specific names, which indicate often the nature of the values of the considered function, such as isobar, isotherm, isogon, isochrone, isoquant and indifference curve.

Given a specific radius r, the equation of a circle defines an isocontour:

If we choose ${\displaystyle r=5}$ then our isovalue is ${\displaystyle c=5^{2}=25.}$ All points (x,y) that evaluate to 25 constitute the isocontour. This means that they are members of the isocontour's level set. If a point evaluates to less than 25 the point is on the inside of the isocontour. If the result is greater than 25, it is on the outside.

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