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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
differential 1-form
ⓘ
mathematical concept ⓘ object in differential geometry ⓘ |
| appearsIn |
Carathéodory formulation of thermodynamics
NERFINISHED
ⓘ
Clausius inequality formulation NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
theory of nonintegrable distributions ⓘ |
| typicalLocalExpression | ω = Σᵢ aᵢ(x) dxᵢ ⓘ |
| usedIn |
Pfaffian systems
NERFINISHED
ⓘ
contact geometry ⓘ nonholonomic mechanics ⓘ theory of differential equations ⓘ thermodynamic state space ⓘ |
| usedToFormulate |
integrating factor problems in thermodynamics
ⓘ
nonholonomic constraints in mechanics ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.