theorem 13.63 Constant rank theorem

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2925 · p. 487

Rests on

depends_on corollary A.292 Inverse function theorem ¶
- 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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 ¶
      - 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 A.286 Implicit function theorem ¶
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - 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 7.97 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 theorem 7.34 Rolle ¶
      - 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 app:A-long-proofs@proof-186 ¶
depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
depends_on theorem A.286 Implicit function theorem ¶ ↺
proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶

Supports

depends_on corollary 13.64 The image of a constant-rank map, locally ¶
depends_on lemma A.533 The orbit map has constant rank $d$ ¶
- depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
- depends_on proposition A.535 Existence of a slice ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶
    - depends_on theorem A.537 The smooth structure on the orbit space ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶
depends_on lemma A.538 Submersions have smooth local sections ¶
- depends_on proposition A.539 Universal property ¶
  - depends_on corollary A.540 Uniqueness of the smooth structure ¶
depends_on theorem 13.67 Quotient manifold theorem ¶
- depends_on example 13.68 Why each hypothesis is there ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
- depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
  - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
    - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- depends_on theorem A.563 Marsden–Weinstein reduction ¶
  - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
  - depends_on example A.572 Rotational reduction of the central-force problem ¶

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`	→	Inverse function theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2938
`depends_on`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2938
`depends_on`	→	Implicit function theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2938
`depends_on`	←	The image of a constant-rank map, locally	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3002
`depends_on`	←	The orbit map has constant rank $d$	declared	appendices/A-long-proofs.tex:26013
`depends_on`	←	Submersions have smooth local sections	declared	appendices/A-long-proofs.tex:26279
`depends_on`	←	Quotient manifold theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3082
`proves`	←	ch:11-manifolds-tensors-curvature@proof-12	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2941