Inverse function theorem: Difference between revisions

Revision as of 02:02, 31 October 2023

Revision as of 05:41, 2 December 2023

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{{shortShort description|Theorem in mathematics}}

{{Use dmy dates|date=December 2023}}

{{Calculus}}

In [[mathematics]], specifically [[differential calculus]], the '''inverse function theorem''' gives a [[Necessity and sufficiency|sufficient condition]] for a [[function (mathematics)|function]] to be [[Invertible function|invertible]] in a [[Neighbourhood (mathematics)|neighborhood]] of a point in its [[___domain of a function|___domain]]: namely, that its ''derivative is continuous and non-zero at the point''. The theorem also gives a [[formula]] for the [[derivative]] of the [[inverse function]].

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is bijective with the smooth inverse. That is to say, <math>g</math> gives a local parametrization of <math>f^{-1}(b)</math> around <math>a</math>. Hence, <math>f^{-1}(b)</math> is a manifold. <math>\square</math> (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)

More generally, the theorem shows that if a smooth map <math>f : P \to E</math> is transversal to a submanifold <math>M \subset E</math>, then the pre-image <math>f^{-1}(M) \hookrightarrow P</math> is a submanifold.<ref>{{cite web|website=northwestern.edu|title=Transversality
|url=https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf {{Bare URL PDF|date=May 2022}}</ref>

== Global version ==