lemma A.307 Partition of unity on a compact manifold

open in the book · appendices/A-long-proofs.tex:15138 · p. 2941

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - 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 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - 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 definition 7.2 Absolute value ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶ ↺
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - 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 3.43 Map ¶
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - 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 definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - 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 ¶
    - 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 ¶

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`	→	Manifold with boundary	declared	appendices/A-long-proofs.tex:15148
`depends_on`	→	Compact set	declared	appendices/A-long-proofs.tex:15148
`depends_on`	→	The standard smooth bump	declared	appendices/A-long-proofs.tex:15148
`depends_on`	←	Integral over the manifold	declared	appendices/A-long-proofs.tex:15204
`proves`	←	app:A-long-proofs@proof-192	declared	appendices/A-long-proofs.tex:15151