Cartan magic formula
E582441
The Cartan magic formula is a fundamental identity in differential geometry that expresses the Lie derivative of a differential form in terms of the exterior derivative and interior product.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
formula in differential geometry
ⓘ
identity in differential geometry ⓘ |
| appliesTo |
differential k-form
ⓘ
smooth manifold ⓘ vector field ⓘ |
| assumes | smooth structure on the underlying manifold ⓘ |
| characterizes | Lie derivative as graded derivation of the exterior algebra ⓘ |
| clarifies | relationship between flow of a vector field and differential forms ⓘ |
| componentOf | Cartan calculus on differential forms ⓘ |
| domainOfDefinition | smooth differential forms on a manifold ⓘ |
| expresses | Lie derivative of a differential form ⓘ |
| hasAlternativeName |
Cartan formula for the Lie derivative
NERFINISHED
ⓘ
Cartan identity for the Lie derivative NERFINISHED ⓘ |
| hasStandardNotation | L_X ω = i_X dω + d i_X ω ⓘ |
| holdsFor |
any smooth differential form ω
ⓘ
any smooth vector field X ⓘ |
| implies |
Lie derivative is determined by flow of the vector field and exterior derivative
ⓘ
Lie derivative preserves degree of differential forms ⓘ |
| involvesOperator |
Lie derivative L_X
ⓘ
exterior derivative d ⓘ interior product i_X ⓘ |
| logicalForm | L_X = i_X d + d i_X as operators on differential forms ⓘ |
| mathematicalField | geometry ⓘ |
| mathematicalSubfield |
differential geometry
ⓘ
exterior calculus ⓘ tensor calculus ⓘ |
| namedAfter | Élie Cartan NERFINISHED ⓘ |
| property | shows that Lie derivative commutes with pullback along flows up to homotopy ⓘ |
| relatedTo |
Cartan homotopy formula
NERFINISHED
ⓘ
Cartan’s identity for Lie derivative ⓘ |
| relatesConcept |
Lie derivative
ⓘ
differential forms ⓘ exterior derivative ⓘ interior product ⓘ |
| usedFor |
computing Lie derivatives of differential forms
ⓘ
deriving conservation laws in differential form language ⓘ proving Cartan homotopy formula ⓘ |
| usedIn |
Riemannian geometry
NERFINISHED
ⓘ
classical field theory ⓘ differential geometry ⓘ gauge theory ⓘ symplectic geometry ⓘ theory of Lie algebras ⓘ theory of Lie groups ⓘ |
| usedToShow |
Lie derivative commutes with exterior derivative on functions
ⓘ
naturality of exterior derivative under flows ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.