theorem 13.62 Regular value theorem in codimension $k$

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2863 · p. 487

Rests on

depends_on definition 13.52 Differential; pushforward ¶
- depends_on definition 13.49 Smooth map between manifolds ¶
  - depends_on definition 13.44 Atlas ¶
    - depends_on definition 13.43 Coordinate system ¶
  - depends_on definition 13.45 Differentiable map on a topological space ¶
    - 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 13.81 Vector on a manifold ¶
  - depends_on definition 13.71 Differentiable curve ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
depends_on definition 13.54 Embedded submanifold ¶
- depends_on definition 13.53 Immersion, submersion, embedding ¶
  - depends_on definition 13.52 Differential; pushforward ¶ ↺
  - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 3.47 Bijective map ¶ ↺
      - depends_on definition 3.49 Composition map ¶
      - depends_on definition 3.50 Identity map ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- depends_on definition 13.48 Differentiable manifold ¶ ↺
depends_on theorem A.286 Implicit function theorem ¶
- depends_on definition 7.99 Differentiability 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 definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on equation 6.12 eq:top-euclidean-metric ¶
- 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 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - 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 ¶
  - proves proof ch:05-real-analysis@proof-63 ¶
- depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
  - 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 7.99 Differentiability at a point ¶ ↺
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - 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-19 ¶
  - proves proof ch:05-real-analysis@proof-62 ¶
- proves proof app:A-long-proofs@proof-185 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-11 ¶

Supports

depends_on corollary 13.64 The image of a constant-rank map, locally ¶

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`	→	Differential; pushforward	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879
`depends_on`	→	Embedded submanifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879
`depends_on`	→	Implicit function theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879
`depends_on`	←	The image of a constant-rank map, locally	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3002
`proves`	←	ch:11-manifolds-tensors-curvature@proof-11	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2883