implicit function theorem

E3650

mathematical theorem

The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.

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Observed surface forms (1)

Surface form	Occurrences
submersion theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	equations F(x,y)=0 ⓘ
assumes	C1 regularity ⓘ Jacobian matrix with nonzero determinant ⓘ continuously differentiable function ⓘ nondegeneracy conditions ⓘ sufficient smoothness conditions ⓘ
concerns	differentiable functions ⓘ implicit equations ⓘ local solvability of equations ⓘ systems of equations ⓘ
concludes	existence of y=g(x) near a point ⓘ
ensures	continuity of the implicit function ⓘ continuous differentiability of the implicit function ⓘ
field	calculus ⓘ differential geometry ⓘ multivariable calculus ⓘ nonlinear analysis ⓘ real analysis ⓘ
generalizes	inverse function theorem ⓘ
guarantees	differentiability of the implicit function ⓘ existence of local solutions ⓘ local representation of variables as functions of others ⓘ uniqueness of the local solution under given conditions ⓘ
hasApplication	comparative statics in economics ⓘ coordinate charts on manifolds ⓘ defining smooth submanifolds as level sets ⓘ local parametrization of solution sets ⓘ
hasVersion	Banach space implicit function theorem ⓘ complex implicit function theorem ⓘ real implicit function theorem ⓘ
implies	inverse function theorem in special cases ⓘ
isUsedIn	differential geometry ⓘ dynamical systems ⓘ economics ⓘ manifold theory ⓘ nonlinear equation solving ⓘ optimization theory ⓘ partial differential equations ⓘ theory of submanifolds ⓘ
logicalForm	local existence and uniqueness theorem ⓘ
relatedTo	constant rank theorem ⓘ rank theorem ⓘ implicit function theorem self-linksurface differs ⓘ surface form: submersion theorem
requires	F(a,b)=0 at a base point (a,b) ⓘ Jacobian with respect to dependent variables invertible at (a,b) ⓘ
typicalAssumption	F is Ck with k≥1 ⓘ
typicalConclusion	implicit function is Ck with k≥1 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

inverse function theorem → isRelatedTo → implicit function theorem ⓘ

implicit function theorem → relatedTo → implicit function theorem self-linksurface differs ⓘ

this entity surface form: submersion theorem

Nash embedding theorem → usesMethod → implicit function theorem ⓘ