lemma 13.107 Poincaré lemma
open in the book ·
parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5144
· p. 515
Rests on
-
depends_on
definition 13.105
Closed form
¶
-
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 13.84
Tensor
¶
- depends_on definition 13.83 Covector ¶ ↺
- depends_on definition 13.81 Vector on a manifold ¶ ↺
-
depends_on
definition 13.83
Covector
¶
-
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
¶
- depends_on definition 13.98 $k$-form ¶ ↺
-
depends_on
definition 13.103
Exterior derivative
¶
-
depends_on
definition 13.106
Exact form
¶
- depends_on definition 13.103 Exterior derivative ¶ ↺
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on equation 13.245 eq:mfd-nilpotency ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-21 ¶
Supports
- 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 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 24.33 Symplectic capacity ¶ ↺
-
depends_on
definition 24.28
Ball and cylinder
¶
-
depends_on
remark 24.11
The SI dimension of every object in this chapter
¶
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 |
→ | Closed form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5146 |
depends_on |
→ | Exact form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5146 |
depends_on |
→ | eq:mfd-nilpotency | 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 |
← | The canonical form is symplectic and intrinsic | declared | parts/03-classical-mechanics/07-symplectic-geometry.tex:280 |
proves |
← | ch:11-manifolds-tensors-curvature@proof-21 | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5149 |