Cartan formula
E582440
The Cartan 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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
formula in differential geometry
ⓘ
mathematical identity ⓘ |
| alsoKnownAs | Cartan magic formula NERFINISHED ⓘ |
| appliesTo |
differential k-form
ⓘ
smooth manifold ⓘ vector field ⓘ |
| compatibleWith |
Cartan magic formula for Lie derivative
NERFINISHED
ⓘ
graded Leibniz rule for exterior derivative ⓘ |
| context |
Cartan calculus
NERFINISHED
ⓘ
exterior calculus ⓘ |
| expresses | Lie derivative of a differential form in terms of exterior derivative and interior product ⓘ |
| field | differential geometry ⓘ |
| holdsFor |
all degrees of differential forms
ⓘ
smooth vector fields ⓘ |
| implies | Lie derivative commutes with pullback along flows ⓘ |
| involvesConcept |
Lie derivative
NERFINISHED
ⓘ
contraction operator ⓘ differential form ⓘ exterior derivative ⓘ interior product ⓘ |
| isToolFor |
computing Lie derivatives in coordinates
ⓘ
relating symmetries to conserved quantities in geometric mechanics ⓘ studying invariance of forms under flows ⓘ |
| mathematicalDomain |
smooth manifolds
ⓘ
tensor calculus on manifolds ⓘ |
| namedAfter | Élie Cartan NERFINISHED ⓘ |
| property |
is linear in the differential form
ⓘ
is linear in the vector field ⓘ |
| relatesOperator |
Lie derivative
ⓘ
exterior derivative ⓘ interior product ⓘ |
| requiresStructure |
exterior algebra of differential forms
ⓘ
smooth structure on the manifold ⓘ tangent bundle ⓘ |
| standardForm | L_X ω = i_X dω + d i_X ω ⓘ |
| usedIn |
Riemannian geometry
NERFINISHED
ⓘ
de Rham cohomology NERFINISHED ⓘ differential topology NERFINISHED ⓘ gauge theory ⓘ symplectic geometry ⓘ theory of flows on manifolds ⓘ |
| usedToDefine | Lie derivative of differential forms ⓘ |
| validityCondition |
differential form is smooth
ⓘ
vector field is smooth ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.