definition 13.103 Exterior derivative

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5046 · p. 514

ground object -- no derivation owed

Rests on

depends_on definition 13.98 $k$-form ¶
- depends_on definition 13.83 Covector ¶
  - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- depends_on definition 13.84 Tensor ¶
  - depends_on definition 13.83 Covector ¶ ↺
  - depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on definition 13.102 Wedge product ¶
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on equation 13.239 eq:mfd-dual-basis ¶

Supports

depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
depends_on definition 13.105 Closed form ¶
- depends_on definition A.67 Symplectic manifold ¶
  - depends_on theorem A.68 Darboux ¶
    - depends_on corollary A.69 No local invariants ¶
      - depends_on remark 24.13 What Darboux's theorem forbids ¶
    - depends_on example A.80 The phase space of one particle in space ¶
    - depends_on theorem 24.12 Darboux ¶
      - depends_on definition 24.20 Liouville volume ¶
      - depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
- depends_on definition 24.8 Symplectic manifold ¶
  - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
    - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
      - depends_on remark 24.11 The SI dimension of every object in this chapter ¶
  - depends_on definition 24.14 Hamiltonian vector field ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on definition 24.50 Prequantum operator ¶
      - depends_on proposition 24.51 Prequantization is a Lie-algebra homomorphism ¶
      - depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
    - depends_on proposition 25.4 Leibniz rule and derivations ¶
      - depends_on lemma A.616 The four identities ¶
    - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
      - depends_on definition 24.37 Casimir function ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶ ↺
    - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
      - depends_on theorem 24.21 Liouville ¶
  - depends_on definition 24.20 Liouville volume ¶ ↺
  - depends_on definition 24.53 Real polarization and polarized sections ¶
    - depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶ ↺
  - depends_on definition 24.49 Prequantum datum ¶
    - depends_on definition 24.53 Real polarization and polarized sections ¶ ↺
    - depends_on definition 24.50 Prequantum operator ¶ ↺
    - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on definition 24.43 Momentum map ¶ ↺
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
  - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
  - depends_on theorem 24.12 Darboux ¶ ↺
  - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
  - depends_on theorem 24.38 Symplectic foliation, quoted ¶
  - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
- depends_on lemma 13.107 Poincaré lemma ¶
  - depends_on proposition 13.113 Product of a closed and an exact form ¶
  - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶ ↺
- depends_on proposition 13.113 Product of a closed and an exact form ¶ ↺
- depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
  - depends_on example 13.110 Closed but not exact: the angle form ¶
  - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
depends_on definition 13.106 Exact form ¶
- depends_on example 13.110 Closed but not exact: the angle form ¶ ↺
- depends_on lemma 13.107 Poincaré lemma ¶ ↺
- depends_on proposition 13.113 Product of a closed and an exact form ¶ ↺
- depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶ ↺
depends_on definition 25.21 First-order action in general coordinates ¶
- depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
  - depends_on example 25.24 A charged particle in a strong magnetic field ¶
depends_on definition 24.9 The canonical form on a cotangent bundle ¶ ↺
depends_on lemma A.312 The computation in one chart ¶
depends_on proposition 13.128 Cartan's magic formula ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶
- depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
- depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
depends_on theorem A.311 General Stokes theorem ¶
- depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
- depends_on lemma A.555 The actions are well defined ¶
- depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶
depends_on theorem 13.156 Cartan structure equations ¶
- depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
- depends_on proposition 13.157 Bianchi identities ¶
  - depends_on proposition 13.154 Symmetries of the curvature ¶
    - depends_on definition 13.111 Linearised curvature of a symmetric field ¶
      - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶ ↺
    - depends_on theorem 44.23 Contracted Bianchi identity ¶
      - depends_on corollary 44.24 Conservation is an integrability condition ¶
      - depends_on proposition 44.41 Four of the ten equations are constraints ¶
      - depends_on proposition 44.30 The cosmological term ¶
      - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
      - depends_on proposition 15.31 Radius and cosmological constant ¶
      - depends_on theorem 48.3 The Robertson–Walker metric ¶
- depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
  - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
  - depends_on theorem 43.3 Equivalence of the two variable sets ¶
    - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	$k$-form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5054
`depends_on`	→	Wedge product	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5054
`depends_on`	←	Frobenius for a Pfaffian system	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6114
`depends_on`	←	Closed form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5128
`depends_on`	←	Exact form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5138
`depends_on`	←	First-order action in general coordinates	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:844
`depends_on`	←	The canonical form on a cotangent bundle	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:269
`depends_on`	←	The computation in one chart	declared	appendices/A-long-proofs.tex:15282
`depends_on`	←	Cartan's magic formula	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5829
`depends_on`	←	General Stokes theorem	declared	appendices/A-long-proofs.tex:15257
`depends_on`	←	Cartan structure equations	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841