Editing Implicit function (section)

{{Short description|Mathematical relation consisting of a multi-variable function equal to zero}}
{{Calculus |Differential}}

In [[mathematics]], an '''implicit equation''' is a [[relation (mathematics)|relation]] of the form <math>R(x_1, \dots, x_n) = 0,</math> where {{mvar|R}} is a [[function (mathematics)|function]] of several variables (often a [[polynomial]]). For example, the implicit equation of the [[unit circle]] is <math>x^2 + y^2 - 1 = 0.</math>

An '''implicit function''' is a [[function (mathematics)|function]] that is defined by an implicit equation, that relates one of the variables, considered as the [[value (mathematics)|value]] of the function, with the others considered as the [[argument of a function|argument]]s.<ref name=Chiang>{{cite book |last=Chiang |first=Alpha C. |author-link=Alpha Chiang |title=Fundamental Methods of Mathematical Economics |location=New York |publisher=McGraw-Hill |edition=Third |year=1984 |isbn=0-07-010813-7 |url=https://archive.org/details/fundamentalmetho0000chia_b4p1 |url-access=registration }}</ref>{{rp|204–206}} For example, the equation <math>x^2 + y^2 - 1 = 0</math> of the [[unit circle]] defines {{mvar|y}} as an implicit function of {{mvar|x}} if {{math|−1 ≤ ''x'' ≤ 1}}, and {{mvar|y}} is restricted to nonnegative values.

The [[implicit function theorem]] provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero [[multivariable function]]s that are [[continuously differentiable]].