definition 13.106 Exact form

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5131 · p. 515

ground object -- no derivation owed

Rests on

depends_on definition 13.103 Exterior derivative ¶
- 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 13.45 Differentiable map on a topological space ¶
      - 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 ¶
depends_on definition 13.98 $k$-form ¶ ↺

Supports

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 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 remark 24.35 What non-squeezing does and does not say about nature ¶
    - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
    - depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
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 ¶

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`	→	Exterior derivative	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5138
`depends_on`	→	$k$-form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5138
`depends_on`	←	Closed but not exact: the angle form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5266
`depends_on`	←	Poincaré lemma	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5146
`depends_on`	←	Product of a closed and an exact form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5370
`depends_on`	←	Converse of the Poincaré lemma on a star-shaped domain	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5181