proposition 13.100 The space of $k$-forms
open in the book ·
parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4817
· p. 511
Rests on
-
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 13.47 Differentiable structure ¶
-
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 13.43 Coordinate system ¶ ↺
-
depends_on
definition 13.44
Atlas
¶
- depends_on definition 13.48 Differentiable manifold ¶ ↺
-
depends_on
definition 13.71
Differentiable curve
¶
-
depends_on
definition 13.81
Vector on a manifold
¶
- depends_on definition 13.98 $k$-form ¶
- depends_on definition 13.84 Tensor ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-20 ¶
Supports
-
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
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 A.311
General Stokes theorem
¶
-
depends_on
definition A.308
Integral over the manifold
¶
-
depends_on
lemma A.305
The chart integral is well defined
¶
-
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
definition A.309
The induced orientation of the boundary
¶
- depends_on lemma A.305 The chart integral is well defined ¶ ↺
-
depends_on
definition A.302
Orientation
¶
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 |
→ | Covector | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4829 |
depends_on |
→ | $k$-form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4829 |
depends_on |
→ | Tensor | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4829 |
depends_on |
← | Integral over a chart | declared | appendices/A-long-proofs.tex:15032 |
depends_on |
← | Transformation of the top form | declared | appendices/A-long-proofs.tex:14969 |
proves |
← | ch:11-manifolds-tensors-curvature@proof-20 | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4832 |