PlanetMath: inverse function theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

inverse function theorem

(Theorem)

Let $\mathbf{f}$ be a continuously differentiable, vector-valued function mapping the open set $E \subset \mathbb{R}^{n}$ to $\mathbb{R}^{n}$ and let $S = \mathbf{f}(E)$ . If, for some point $\mathbf{a} \in E$ , the Jacobian, $\vert J_{\mathbf{f}}(\mathbf{a}) \vert$ , is non-zero, then there is a uniquely defined function $\mathbf{g}$ and two open sets $X \subset E$ and $Y \subset S$ such that

$\mathbf{a} \in X$ , $\mathbf{f}(\mathbf{a}) \in Y$ ;
$Y = \mathbf{f}(X)$ ;
$\mathbf{f}:X \to Y$ is one-one;
$\mathbf{g}$ is continuously differentiable on and $\mathbf{g}(\mathbf{f}(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x} \in X$ .

Simplest case

When

, this theorem becomes: Let

be a continuously differentiable, real-valued function defined on the open interval

. If for some point $a \in I$ , $f'(a) \neq 0$ , then there is a neighbourhood $[\alpha, \beta]$ of

in which

is strictly monotonic. Then $y \to f^{-1}(y)$ is a continuously differentiable, strictly monotonic function from $[f(\alpha), f(\beta)]$ to $[\alpha, \beta]$ . If

is increasing (or decreasing) on $[\alpha, \beta]$ , then so is $f^{-1}$ on $[f(\alpha), f(\beta)]$ .

Note

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

"inverse function theorem" is owned by vypertd.

(view preamble)

Attachments:: proof of inverse function theorem (Proof) by paolini

Log in to rate this entry.
(view current ratings)

Cross-references: variable, dimension, implicit function theorem, decreasing, increasing, strictly monotonic, neighbourhood, open interval, function, Jacobian, point, open set, mapping, vector-valued function, continuously differentiable
There are 6 references to this entry.

This is version 6 of inverse function theorem, born on 2002-08-24, modified 2002-12-28.
Object id is 3346, canonical name is InverseFunctionTheorem.
Accessed 24394 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact