definition A.298 Half-space; smoothness on it

open in the book · appendices/A-long-proofs.tex:14878 · p. 2939

ground object -- no derivation owed

Rests on

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 definition 3.43 Map ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
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 ¶

Supports

depends_on definition A.299 Manifold with boundary ¶
- depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
  - 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 lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on definition A.308 Integral over the manifold ¶
    - depends_on theorem A.311 General Stokes theorem ¶ ↺

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`	→	Functions of class $C^{1}$	declared	appendices/A-long-proofs.tex:14894
`depends_on`	→	Topological space	declared	appendices/A-long-proofs.tex:14894
`depends_on`	←	Manifold with boundary	declared	appendices/A-long-proofs.tex:14908