Niveausæt

Niveausæt, or level set, refers to a fundamental concept in mathematics for a real-valued function. If f is defined on a domain D with values in the real numbers, the niveausæt corresponding to a chosen level c is the set of all points x in D for which f(x) = c. This set is often written as f−1({c}) and is also described as the contour at value c.

A level set should be distinguished from a sublevel set, which consists of points where f(x) ≤ c. While a sublevel set aggregates all points up to a given value, a level set isolates the exact value c. In multidimensional contexts, level sets for different c become level curves (in two dimensions) or level surfaces (in three dimensions). The gradient of f is orthogonal to the level set at points where ∇f ≠ 0, reflecting the notion that movement along the level set does not change the function value.

Examples help illustrate the idea. For f(x, y) = x^2 + y^2, the level set f(x, y) = c is the circle with radius sqrt(c) when c ≥ 0, and it is empty for c < 0. In one dimension, level sets are the points where f(t) = c; in higher dimensions, they form curves or surfaces.

Key properties include topology and smoothness. If f is continuous, a level set is closed, and if ∇f ≠ 0 on the set, the implicit function theorem implies the level set is a smooth (n−1)-dimensional manifold locally. Level sets have broad applications in geometry, physics (equipotential or isosurfaces), computer graphics (contours and isosurfaces), and image processing.

Related concepts include isocontours, iso-values, and the level-set method, a computational approach for tracking evolving interfaces.