Theorem of local inversion

The theorem of local inversion is one of the most important theorems of the theory of the differentiable applications continuously . It makes it possible to prove the existence, locally, of a reverse for any function realizing certain conditions. It extends to the nonlinear field of the properties of the linear systems of Cramer.

It is formally equivalent to the Théorème of the implicit functions. The whole of these two theorems constitutes the base on which the theory of the varieties is built.

Statement of the theorem

In the continuation, one will consider E and F , two vector spaces of the same dimension N ( $n \ geq 1$ ) on $\ R$ and F an application of class $C^k$ ( $k \ geq 1$ ) of a Ouvert U of E in F . One places oneself finally in a point has an element of U .

If the Differential of F: df applied to the point has is a Isomorphisme then it exists a Voisinage of has , V (a) , such as the restriction of F on V (a) is a $C^k$ Difféomorphisme (injective function of class $C^k$ to value in open and whose reciprocal application is also of class $C^k$ ).

Moreover, in this case, the differential of reciprocal is reciprocal differential

d (f^ {- 1}) (F (a)) = \ left^ {- 1}

One of the simplest applications of this theorem relates to the functions of the real variable: if F is derivable on U open of $\ R$ , of continuous derivative, and if $f' (A) \,$ is nonnull, then it exists a vicinity of has , V (a) , on which the function F is a bijection of V (a) on F (V (a)) .

In the case of a function defined on E , the condition " df applied to the point has is an isomorphism " is equivalent to " the jacobien of F in has is not nul".

Source

I integrate. Course of mathematics of 2nd year . ED. Dunod .ISBN 2100054120

See too

Theorem of the implicit functions
Theorem of the immersion

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