definition 13.102 Wedge product

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4954 · p. 513

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 equation 13.239 eq:mfd-dual-basis ¶

Supports

depends_on definition 13.103 Exterior derivative ¶
- 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 example A.80 The phase space of one particle in space ¶
      - depends_on theorem 24.12 Darboux ¶
  - 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 definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.43 Momentum map ¶
      - depends_on definition 24.50 Prequantum operator ¶
      - depends_on proposition 25.4 Leibniz rule and derivations ¶
      - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
      - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
    - depends_on definition 24.20 Liouville volume ¶
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 24.21 Liouville ¶
      - depends_on theorem 24.24 Poincaré recurrence ¶
    - 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 definition 24.17 Symplectomorphism ¶
      - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.43 Momentum map ¶ ↺
      - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
    - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - 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 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 44.23 Contracted Bianchi identity ¶
      - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
  - 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 ¶ ↺
depends_on definition 13.114 Volume form ¶
- depends_on definition 13.116 Hodge dual ¶
depends_on definition 24.20 Liouville volume ¶ ↺
depends_on lemma A.301 Transformation of the top form ¶
- depends_on definition A.302 Orientation ¶
  - depends_on definition A.309 The induced orientation of the boundary ¶
    - depends_on theorem A.311 General Stokes theorem ¶ ↺
  - depends_on definition A.304 Integral over a chart ¶
    - depends_on lemma A.305 The chart integral is well defined ¶
      - depends_on definition A.308 Integral over the manifold ¶
- depends_on lemma A.305 The chart integral is well defined ¶ ↺
depends_on theorem 13.156 Cartan structure equations ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	$k$-form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4964
`depends_on`	→	eq:mfd-dual-basis	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4964
`depends_on`	←	Exterior derivative	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5054
`depends_on`	←	Volume form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5393
`depends_on`	←	Liouville volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:653
`depends_on`	←	Transformation of the top form	declared	appendices/A-long-proofs.tex:14969
`depends_on`	←	Cartan structure equations	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6841