inverse function theorem

E22819

mathematical theorem theorem in calculus theorem in differential geometry

The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.

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Observed surface forms (1)

Surface form	Occurrences
Jacobi determinant	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in calculus ⓘ theorem in differential geometry ⓘ
appliesTo	functions between open subsets of R^n ⓘ smooth maps between smooth manifolds ⓘ
assumes	differentiability of the function ⓘ invertibility of the derivative at a point ⓘ nonzero derivative in the one-dimensional case ⓘ
concerns	differentiable maps between Euclidean spaces ⓘ differentiable maps between manifolds ⓘ local invertibility of functions ⓘ
concludes	continuity of the local inverse ⓘ existence of a differentiable local inverse ⓘ existence of a neighborhood where the function is a diffeomorphism onto its image ⓘ uniqueness of the local inverse near the point ⓘ
field	calculus ⓘ differential geometry ⓘ mathematical analysis ⓘ
formalizes	idea that nonvanishing derivative implies local invertibility ⓘ
generalizationOf	one-dimensional inverse function result from elementary calculus ⓘ
givesConditionFor	differentiability of a local inverse ⓘ existence of a local inverse ⓘ
hasConsequence	existence of local coordinates on manifolds ⓘ local linearization of smooth maps ⓘ
hasVariant	C^k inverse function theorem ⓘ inverse function theorem for Banach spaces ⓘ smooth (C^∞) inverse function theorem ⓘ
historicallyAssociatedWith	Augustin-Louis Cauchy ⓘ Bernhard Riemann ⓘ Karl Weierstrass ⓘ
implies	continuity of the inverse map ⓘ differentiability of the inverse map ⓘ inverse map has derivative equal to the matrix inverse of the original derivative ⓘ local bijectivity near the point ⓘ openness of the map near the point ⓘ
isRelatedTo	Banach inverse mapping theorem ⓘ implicit function theorem ⓘ rank theorem ⓘ
requires	Jacobian determinant to be nonzero at the point ⓘ Jacobian matrix to be invertible at the point ⓘ
typicalAssumption	domain is an open subset of R^n ⓘ function is continuously differentiable (C^1) ⓘ
usedIn	change of variables in multivariable calculus ⓘ coordinate chart constructions ⓘ differential topology ⓘ manifold theory ⓘ nonlinear analysis ⓘ theory of differential equations ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

implicit function theorem → generalizes → inverse function theorem ⓘ

Carl Gustav Jacob Jacobi → notableWork → inverse function theorem ⓘ

this entity surface form: Jacobi determinant