theorem 7.112 Implicit function theorem

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:3493 · p. 248

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 ch:05-real-analysis@prooflink-1 ¶

Supports

depends_on corollary 7.113 Inverse function theorem ¶
- depends_on definition A.503 The substitution property ¶
  - depends_on lemma A.506 Locality ¶
    - depends_on proposition A.511 The substitution property is universal ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶
  - depends_on lemma A.504 Transitivity ¶
- depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
- depends_on lemma A.510 Every diffeomorphism factorises locally ¶
  - depends_on proposition A.511 The substitution property is universal ¶ ↺
- depends_on remark 7.114 Where the proof is, and why ¶
- depends_on theorem 7.129 Change of variables in a multiple integral ¶
  - depends_on example 7.130 The two Jacobians this treatise uses ¶
    - depends_on lemma A.518 The excised ball ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.519 The convolution exists ¶
  - depends_on lemma A.518 The excised ball ¶ ↺
  - depends_on proposition A.519 The convolution exists ¶ ↺
- depends_on theorem 7.115 Constant rank ¶
depends_on proposition 7.120 The envelope touches every member it meets ¶
depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
- depends_on remark 7.118 The multiplier rule and its functional counterpart ¶
depends_on remark 7.114 Where the proof is, and why ¶ ↺

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`	→	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	→	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	→	$C^{1}$ implies differentiable	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	←	Inverse function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3545
`depends_on`	←	The envelope touches every member it meets	declared	parts/02-mathematical-methods/05-real-analysis.tex:3840
`depends_on`	←	Lagrange multipliers in finitely many variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3752
`depends_on`	←	Where the proof is, and why	declared	parts/02-mathematical-methods/05-real-analysis.tex:3567
`proves`	←	ch:05-real-analysis@prooflink-1	declared	parts/02-mathematical-methods/05-real-analysis.tex:3527