definition A.299 Manifold with boundary

open in the book · appendices/A-long-proofs.tex:14897 · p. 2939

Rests on

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 definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- 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 ¶ ↺

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Half-space; smoothness on it	declared	appendices/A-long-proofs.tex:14908
`depends_on`	→	Atlas	declared	appendices/A-long-proofs.tex:14908
`depends_on`	→	Differentiable manifold	declared	appendices/A-long-proofs.tex:14908
`depends_on`	←	The boundary is well defined, and is a manifold	declared	appendices/A-long-proofs.tex:14917
`depends_on`	←	Partition of unity on a compact manifold	declared	appendices/A-long-proofs.tex:15148