definition A.308 Integral over the manifold

open in the book · appendices/A-long-proofs.tex:15185 · p. 2942

ground object -- no derivation owed

Rests on

depends_on lemma A.307 Partition of unity on a compact manifold ¶
- depends_on definition A.299 Manifold with boundary ¶
  - depends_on definition A.298 Half-space; smoothness on it ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶ ↺
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶ ↺
    - depends_on equation 3.51 eq:set-indexed ¶ ↺
- depends_on lemma A.306 The standard smooth bump ¶
  - depends_on definition 7.53 $\ee$ ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.4 Convergence ¶
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on proposition 7.46 Geometric series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-29 ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - proves proof ch:05-real-analysis@proof-63 ¶
  - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-21 ¶
  - proves proof app:A-long-proofs@proof-191 ¶
- proves proof app:A-long-proofs@proof-192 ¶
depends_on lemma A.305 The chart integral is well defined ¶
- 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 lemma A.301 Transformation of the top form ¶
      - depends_on definition 13.102 Wedge product ¶
      - 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 ¶
  - 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 theorem 7.25 Heine–Cantor: uniform continuity ¶
      - 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 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`	→	Partition of unity on a compact manifold	declared	appendices/A-long-proofs.tex:15204
`depends_on`	→	The chart integral is well defined	declared	appendices/A-long-proofs.tex:15204
`depends_on`	←	General Stokes theorem	declared	appendices/A-long-proofs.tex:15257