lemma A.301 Transformation of the top form
open in the book ·
appendices/A-long-proofs.tex:14957
· p. 2939
Rests on
-
depends_on
definition 13.102
Wedge product
¶
-
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 13.81
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 equation 13.239 eq:mfd-dual-basis ¶
-
depends_on
definition 13.98
$k$-form
¶
- depends_on equation 13.241 eq:mfd-antisimb ¶
- depends_on proposition 13.100 The space of $k$-forms ¶
- proves proof app:A-long-proofs@proof-189 ¶
Supports
-
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 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.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
definition A.308
Integral over the manifold
¶
-
depends_on
lemma A.305
The chart integral is well defined
¶
-
depends_on
definition A.309
The induced orientation of the boundary
¶
- depends_on lemma A.305 The chart integral is well defined ¶ ↺
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 |
→ | Wedge product | declared | appendices/A-long-proofs.tex:14969 |
depends_on |
→ | eq:mfd-antisimb | declared | appendices/A-long-proofs.tex:14969 |
depends_on |
→ | The space of $k$-forms | declared | appendices/A-long-proofs.tex:14969 |
depends_on |
← | Orientation | declared | appendices/A-long-proofs.tex:15001 |
depends_on |
← | The chart integral is well defined | declared | appendices/A-long-proofs.tex:15040 |
proves |
← | app:A-long-proofs@proof-189 | declared | appendices/A-long-proofs.tex:14972 |