# Implicit Functions

• James J. Callahan
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

Given a relation between two variables expressed by an equation of the form f (x,y) = k, we often want to “solve for y.” That is, for each given x in some interval, we expect to find one and only one value y = φ(x) that satisfies the relation. The function j is thus implicit in the relation; geometrically, the locus of the equation f (x,y) =k is a curve in the (x,y)-plane that serves as the graph of the function y = φ(x). The problem of implicit functions—and the aim of this chapter—is to determine the function φ from the relation f, or at least to determine that φ existswhen its exact form cannot be found. There are analogues of this problem in all dimensions;that is, x and y can be vectors, and the relation f (x,y) = k can expand intoa set of equations. However, we begin our analysis with a single equation, becausethe various impediments to finding the implicit function already occur there.

## Keywords

Partial Derivative Tangent Plane Implicit Function Theorem Regular Point Maximal Rank
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.