inverse function theorem

Let 𝐟 be a continuously differentiable, vector-valued functionPlanetmathPlanetmath mapping the open set En to n and let S=𝐟(E). If, for some point 𝐚E, the JacobianDlmfPlanetmath, |J𝐟(𝐚)|, is non-zero, then there is a uniquely defined functionMathworldPlanetmath 𝐠 and two open sets XE and YS such that

  1. 1.

    𝐚X, 𝐟(𝐚)Y;

  2. 2.


  3. 3.

    𝐟:XY is one-one;

  4. 4.

    𝐠 is continuously differentiable on Y and 𝐠(𝐟(𝐱))=𝐱 for all 𝐱X.

0.0.1 Simplest case

When n=1, this theorem becomes: Let f be a continuously differentiable, real-valued function defined on the open intervalDlmfPlanetmath I. If for some point aI, f(a)0, then there is a neighbourhood [α,β] of a in which f is strictly monotonic. Then yf-1(y) is a continuously differentiable, strictly monotonic function from [f(α),f(β)] to [α,β]. If f is increasing (or decreasing) on [α,β], then so is f-1 on [f(α),f(β)].

0.0.2 Note

The inverse function theorem is a special case of the implicit function theoremMathworldPlanetmath where the dimension of each variable is the same.

Title inverse function theorem
Canonical name InverseFunctionTheorem
Date of creation 2013-03-22 12:58:30
Last modified on 2013-03-22 12:58:30
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 9
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 03E20
Related topic DerivativeOfInverseFunction
Related topic LegendreTransform
Related topic DerivativeAsParameterForSolvingDifferentialEquations
Related topic TheoryForSeparationOfVariables