inverse function theorem
E22819
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| inverse function theorem canonical | 3 |
| Jacobi determinant | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T179409 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: inverse function theorem Context triple: [implicit function theorem, generalizes, inverse function theorem]
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A.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
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B.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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D.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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E.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: inverse function theorem Target entity description: 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.
-
A.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
-
B.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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D.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
E.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: inverse function theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: inverse function theorem