lemma A.305 The chart integral is well defined

open in the book · appendices/A-long-proofs.tex:15036 · p. 2940

Rests on

depends_on definition A.304 Integral over a chart ¶
- depends_on definition 7.125 Multiple integral ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
- depends_on definition A.302 Orientation ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
  - depends_on lemma A.301 Transformation of the top form ¶
    - 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 equation 13.241 eq:mfd-antisimb ¶
    - depends_on proposition 13.100 The space of $k$-forms ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 13.84 Tensor ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-20 ¶
    - proves proof app:A-long-proofs@proof-189 ¶
- depends_on proposition 13.100 The space of $k$-forms ¶ ↺
depends_on lemma A.301 Transformation of the top form ¶ ↺
depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
- depends_on definition 7.125 Multiple integral ¶ ↺
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
proves proof app:A-long-proofs@proof-190 ¶

Supports

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 ¶

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`	→	Integral over a chart	declared	appendices/A-long-proofs.tex:15040
`depends_on`	→	Transformation of the top form	declared	appendices/A-long-proofs.tex:15040
`depends_on`	→	Change of variables for multiple integrals; quoted	declared	appendices/A-long-proofs.tex:15040
`depends_on`	←	Integral over the manifold	declared	appendices/A-long-proofs.tex:15204
`proves`	←	app:A-long-proofs@proof-190	declared	appendices/A-long-proofs.tex:15044