Pfaffian form
E506851
A Pfaffian form is a type of differential 1-form on a manifold that encodes constraints or relations between variables, widely used in thermodynamics and differential geometry.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Pfaffian form canonical | 1 |
| Pfaffian system of equations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5256336 — 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: Pfaffian form Context triple: [Carathéodory’s formulation of the second law of thermodynamics, usesConcept, Pfaffian form]
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A.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
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B.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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C.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
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D.
Fuchsian differential equation
A Fuchsian differential equation is a type of linear ordinary differential equation characterized by having only regular singular points, extensively studied in complex analysis and the theory of special functions.
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E.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
- 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: Pfaffian form Target entity description: A Pfaffian form is a type of differential 1-form on a manifold that encodes constraints or relations between variables, widely used in thermodynamics and differential geometry.
-
A.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
B.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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C.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
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D.
Fuchsian differential equation
A Fuchsian differential equation is a type of linear ordinary differential equation characterized by having only regular singular points, extensively studied in complex analysis and the theory of special functions.
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E.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
differential 1-form
ⓘ
mathematical concept ⓘ object in differential geometry ⓘ |
| appearsIn |
Carathéodory formulation of thermodynamics
ⓘ
Clausius inequality formulation ⓘ
linked to:
Clausius theorem
|
| associatedWith |
constraints on admissible motions
ⓘ
hyperplane field in the tangent bundle ⓘ |
| canBe |
closed 1-form
ⓘ
exact 1-form ⓘ non-exact 1-form ⓘ |
| canBeAnalyzedBy |
exterior derivative
ⓘ
wedge product ⓘ |
| canDefine |
Pfaffian equation ω = 0
ⓘ
Pfaffian system {ω¹ = 0,…,ωᵏ = 0} ⓘ |
| closelyRelatedTo |
Pfaffian differential equation
ⓘ
Pfaffian system of equations ⓘ
linked to:
Pfaffian form
|
| codomain | real numbers ⓘ |
| definedAs | differential 1-form encoding constraints between variables ⓘ |
| definedOn | smooth manifold ⓘ |
| domain | tangent bundle of a manifold ⓘ |
| encodes |
constraints
ⓘ
relations between variables ⓘ |
| example |
work 1-form in mechanics
ⓘ
δQ in classical thermodynamics ⓘ |
| field |
differential geometry
ⓘ
mathematical physics ⓘ thermodynamics ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| is | 1-form that may define a distribution of hyperplanes ⓘ |
| mathematicalNature | linear functional on each tangent space ⓘ |
| namedAfter | Johann Friedrich Pfaff ⓘ |
| relatedTo |
Frobenius integrability condition
ⓘ
contact forms ⓘ exterior differential systems ⓘ integral curves ⓘ integral manifolds ⓘ |
| specialCaseOf | differential form of degree 1 ⓘ |
| studiedIn |
Cartan’s method of equivalence
ⓘ
linked to:
theory of G-structures
theory of nonintegrable distributions ⓘ |
| typicalLocalExpression | ω = Σᵢ aᵢ(x) dxᵢ ⓘ |
| usedIn |
Pfaffian systems
ⓘ
contact geometry ⓘ nonholonomic mechanics ⓘ theory of differential equations ⓘ thermodynamic state space ⓘ |
| usedToFormulate |
integrating factor problems in thermodynamics
ⓘ
nonholonomic constraints in mechanics ⓘ |
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: Pfaffian form Description of subject: A Pfaffian form is a type of differential 1-form on a manifold that encodes constraints or relations between variables, widely used in thermodynamics and differential geometry.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pfaffian form