Editing Implicit function theorem (section)

{{short description|On converting relations to functions of several real variables}}
{{Calculus |expanded=multivariable}}
In [[multivariable calculus]], the '''implicit function theorem'''{{efn|Also called '''[[Ulisse Dini|Dini]]'s theorem''' by the Pisan school in Italy. In the English-language literature, [[Dini's theorem]] is a different theorem in mathematical analysis.}} is a tool that allows [[relation (mathematics)#Definition|relations]] to be converted to [[functions of several real variables]]. It does so by representing the relation as the [[graph of a function]]. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the [[domain of a relation|domain]] of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

More precisely, given a system of {{mvar|m}} equations {{math|1=''f<sub>i</sub>''{{space|hair}}(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'', ''y''<sub>1</sub>, ..., ''y<sub>m</sub>'') = 0, ''i'' = 1, ..., ''m''}} (often abbreviated into {{math|1=''F''('''x''', '''y''') = '''0'''}}), the theorem states that, under a mild condition on the [[partial derivative]]s (with respect to each {{math|''y<sub>i</sub>''}} ) at a point, the {{mvar|m}} variables {{math|''y<sub>i</sub>''}} are differentiable functions of the {{math|''x<sub>j</sub>''}} in some [[neighborhood (mathematics)|neighborhood]] of the point. As these functions generally cannot be expressed in [[closed form expression|closed form]], they are ''implicitly'' defined by the equations, and this motivated the name of the theorem.<ref>{{Cite book |last=Chiang |first=Alpha C. |author-link=Alpha Chiang |title=Fundamental Methods of Mathematical Economics |publisher=McGraw-Hill |edition=3rd |year=1984 |pages=[https://archive.org/details/fundamentalmetho0000chia_b4p1/page/204 204–206] |isbn=0-07-010813-7 |url=https://archive.org/details/fundamentalmetho0000chia_b4p1/page/204 }}</ref>

In other words, under a mild condition on the partial derivatives, the set of [[zero of a function|zeros]] of a system of equations is [[local property|locally]] the [[graph of a function]].