equation 13.239 eq:mfd-dual-basis

Rests on

No declared or derived dependency edges point away from this node yet.

Supports

• depends_on definition 13.102 Wedge product ¶
  • 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 definition 24.8 Symplectic manifold ¶
        • depends_on definition 24.9 The canonical form on a cotangent bundle ¶
        • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on definition 24.20 Liouville volume ¶
        • depends_on definition 24.53 Real polarization and polarized sections ¶
        • depends_on definition 24.49 Prequantum datum ¶
        • depends_on definition 24.17 Symplectomorphism ¶
        • 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 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 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 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 lemma A.305 The chart integral is well defined ¶ ↺
  • depends_on theorem 13.156 Cartan structure equations ¶ ↺