implicit function theorem
E3650
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| implicit function theorem canonical | 3 |
| submersion theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T31649 — 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: implicit function theorem Context triple: [Nash embedding theorem, usesMethod, implicit function theorem]
-
A.
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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B.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
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C.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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D.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
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E.
Differential analyzer
The Differential Analyzer is an early analog mechanical computer designed to solve differential equations using interconnected rotating shafts and wheels.
- 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: implicit function theorem Target entity description: 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.
-
A.
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.
-
B.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
C.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
D.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
E.
Differential analyzer
The Differential Analyzer is an early analog mechanical computer designed to solve differential equations using interconnected rotating shafts and wheels.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo | equations F(x,y)=0 ⓘ |
| assumes |
C1 regularity
ⓘ
Jacobian matrix with nonzero determinant ⓘ continuously differentiable function ⓘ nondegeneracy conditions ⓘ sufficient smoothness conditions ⓘ |
| concerns |
differentiable functions
ⓘ
implicit equations ⓘ local solvability of equations ⓘ systems of equations ⓘ |
| concludes | existence of y=g(x) near a point ⓘ |
| ensures |
continuity of the implicit function
ⓘ
continuous differentiability of the implicit function ⓘ |
| field |
calculus
ⓘ
differential geometry ⓘ multivariable calculus ⓘ nonlinear analysis ⓘ real analysis ⓘ |
| generalizes | inverse function theorem ⓘ |
| guarantees |
differentiability of the implicit function
ⓘ
existence of local solutions ⓘ local representation of variables as functions of others ⓘ uniqueness of the local solution under given conditions ⓘ |
| hasApplication |
comparative statics in economics
ⓘ
coordinate charts on manifolds ⓘ defining smooth submanifolds as level sets ⓘ local parametrization of solution sets ⓘ |
| hasVersion |
Banach space implicit function theorem
ⓘ
complex implicit function theorem ⓘ real implicit function theorem ⓘ |
| implies | inverse function theorem in special cases ⓘ |
| isUsedIn |
differential geometry
ⓘ
dynamical systems ⓘ economics ⓘ manifold theory ⓘ nonlinear equation solving ⓘ optimization theory ⓘ partial differential equations ⓘ theory of submanifolds ⓘ |
| logicalForm | local existence and uniqueness theorem ⓘ |
| relatedTo |
constant rank theorem
ⓘ
rank theorem ⓘ implicit function theorem self-linksurface differs ⓘ
surface form:
submersion theorem
|
| requires |
F(a,b)=0 at a base point (a,b)
ⓘ
Jacobian with respect to dependent variables invertible at (a,b) ⓘ |
| typicalAssumption | F is Ck with k≥1 ⓘ |
| typicalConclusion | implicit function is Ck with k≥1 ⓘ |
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: implicit function theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
submersion theorem