proof of implicit function theorem

We state the Theorem with a different notation:

Theorem 1.

Let Ω be an open subset of Rn×Rm and let fC1(Ω,Rm). Let (x0,y0)ΩRn×Rm. If the matrix Dyf(x0,y0) defined by


is invertiblePlanetmathPlanetmathPlanetmath, then there exists a neighbourhood URn of x0, a neighbourhood VRm of y0 and a functionMathworldPlanetmath gC1(U,V) such that

f(x,y)=f(x0,y0)y=g(x)  (x,y)U×V.



Consider the function F𝒞1(Ω,n×m) defined by


One finds that


Being Dyf(x0,y0) invertible, DF(x0,y0) is invertible too. Applying the inverse function TheoremMathworldPlanetmath to F we find that there exist a neighbourhood U of x0 and V of y0 and a function GC1(U×V,n+m) such that F(G(x,y))=(x,y) for all (x,y)U×V. Letting G(x,y)=(G1(x,y),G2(x,y)) (so that G1:V×Wn, G2:V×Wm) we hence have


and hence x=G1(x,y) and y=f(G1(x,y),G2(x,y))=f(x,G2(x,y)). So we only have to set g(x)=G2(x,f(x0,y0)) to obtain


Differentiating with respect to x we obtain


which gives the desired formulaMathworldPlanetmathPlanetmath for the computation of Dg. ∎

Title proof of implicit function theorem
Canonical name ProofOfImplicitFunctionTheorem
Date of creation 2013-03-22 13:31:23
Last modified on 2013-03-22 13:31:23
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 8
Author paolini (1187)
Entry type Proof
Classification msc 26B10