theorem A.286 Implicit function theorem

open in the book · appendices/A-long-proofs.tex:14347 · p. 2933

Rests on

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 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- 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 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 ¶
- 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 definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - 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-19 ¶
- proves proof ch:05-real-analysis@proof-62 ¶
proves proof app:A-long-proofs@proof-185 ¶

Supports

depends_on corollary A.292 Inverse function theorem ¶
- depends_on lemma A.549 The action is transitive ¶
  - depends_on proposition A.552 The component is a torus ¶
    - depends_on corollary A.553 Quasi-periodic motion ¶
    - depends_on definition A.554 Actions on the torus ¶
      - depends_on lemma A.555 The actions are well defined ¶
      - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
    - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶ ↺
- 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 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 proposition 22.32 Duality of the two brackets ¶
  - depends_on remark 22.33 What the duality is good for ¶
- depends_on proposition 22.3 Invertibility of the Legendre map ¶
  - depends_on definition 26.3 Singular Lagrangian ¶
    - depends_on proposition 26.5 Primary constraints ¶
      - depends_on proposition 26.6 The canonical Hamiltonian is defined on the constraint surface ¶
      - depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
      - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
  - depends_on remark 22.4 When the Legendre map is singular ¶
- depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
  - depends_on theorem 13.133 Frobenius ¶
    - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
    - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on lemma A.549 The action is transitive ¶ ↺
      - depends_on lemma A.555 The actions are well defined ¶ ↺
    - depends_on theorem A.545 Liouville–Arnold ¶
  - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
- depends_on remark 22.19 Which form exists ¶
- depends_on theorem 13.63 Constant rank theorem ¶
  - 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.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 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 ¶
depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
- depends_on example A.294 The sphere, made explicit ¶
depends_on lemma A.650 Flattening the constraints ¶
- depends_on lemma A.651 Acyclicity of $\delta$ ¶
  - depends_on proposition A.655 Uniqueness up to a canonical transformation ¶
  - depends_on theorem A.653 Existence of a nilpotent BRST charge ¶
    - depends_on proposition A.655 Uniqueness up to a canonical transformation ¶ ↺
depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
depends_on remark 23.3 What Part II owes this chapter ¶
depends_on theorem 23.6 Jacobi ¶
- depends_on example 23.25 Central force in plane polar coordinates ¶
- depends_on example 23.22 Free particle ¶
  - depends_on remark 23.42 Phase speed and particle speed ¶
- depends_on example 23.23 Uniform gravity, and the parabola ¶
- depends_on proposition 23.7 The principal function is the action ¶
  - depends_on remark 23.8 The action as a function, not a functional ¶
depends_on theorem 13.63 Constant rank theorem ¶ ↺
depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
- depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiability at a point	declared	appendices/A-long-proofs.tex:14371
`depends_on`	→	Chain rule in several variables	declared	appendices/A-long-proofs.tex:14371
`depends_on`	→	$C^{1}$ implies differentiable	declared	appendices/A-long-proofs.tex:14371
`depends_on`	←	Inverse function theorem	declared	appendices/A-long-proofs.tex:14683
`depends_on`	←	Solving one scalar equation for one coordinate	declared	appendices/A-long-proofs.tex:14736
`depends_on`	←	Flattening the constraints	declared	appendices/A-long-proofs.tex:31651
`depends_on`	←	Both hypotheses are needed, and the conclusion is local	declared	appendices/A-long-proofs.tex:14813
`depends_on`	←	What Part II owes this chapter	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:127
`depends_on`	←	Jacobi	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:173
`depends_on`	←	Constant rank theorem	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2938
`depends_on`	←	Regular value theorem in codimension $k$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2879
`proves`	←	app:A-long-proofs@proof-185	declared	appendices/A-long-proofs.tex:14641