definition 7.99 Differentiability at a point

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2925 · p. 242

ground object -- no derivation owed

Rests on

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 3.23 Predicate ¶
    - 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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - 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 ¶

Supports

depends_on definition 7.102 Differential of a function ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on corollary 7.113 Inverse function theorem ¶
    - depends_on definition A.503 The substitution property ¶
      - depends_on lemma A.506 Locality ¶
      - 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.519 The convolution exists ¶
    - depends_on theorem 7.115 Constant rank ¶
  - depends_on corollary 7.110 Variable limits of integration ¶
  - 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 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 proposition A.535 Existence of a slice ¶
      - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - 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 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 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 lemma A.538 Submersions have smooth local sections ¶
      - depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on definition 30.4 Material and spatial descriptions ¶
  - depends_on definition 31.4 Material derivative ¶
    - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on proposition 31.7 The three coincide in steady flow ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
  - depends_on definition 22.26 Lagrange bracket ¶
  - depends_on definition 10.18 Complete integral; envelope ¶
    - depends_on proposition 10.19 An envelope of solutions is a solution ¶
  - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
  - depends_on lemma A.504 Transitivity ¶ ↺
  - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
  - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on definition A.715 Stokes stream function ¶
    - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on example 31.29 Flow past a sphere ¶
      - depends_on lemma A.777 The tangential field above a bending plate ¶
      - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
      - depends_on theorem A.707 D'Alembert's paradox ¶
      - depends_on theorem 31.74 The interface dispersion relation ¶
    - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on proposition 7.120 The envelope touches every member it meets ¶
  - … 9 more
- depends_on remark 7.103 rem:ana-differential-scalar ¶
depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
depends_on proposition 7.104 Chain rule in several variables ¶ ↺
depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
- depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
- depends_on remark 7.101 Partial derivatives alone do not suffice ¶
- depends_on theorem 7.112 Implicit function theorem ¶
  - depends_on corollary 7.113 Inverse function theorem ¶ ↺
  - 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 ¶ ↺
- depends_on theorem A.286 Implicit function theorem ¶
  - depends_on corollary A.292 Inverse function theorem ¶ ↺
  - 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 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 ¶ ↺
depends_on theorem 7.112 Implicit function theorem ¶ ↺
depends_on theorem A.286 Implicit function theorem ¶ ↺

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`	→	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:2953
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/05-real-analysis.tex:2953
`depends_on`	→	eq:top-euclidean-metric	declared	parts/02-mathematical-methods/05-real-analysis.tex:2953
`depends_on`	←	Differential of a function	declared	parts/02-mathematical-methods/05-real-analysis.tex:3082
`depends_on`	←	$h$ is differentiable, with the stated derivative	declared	appendices/A-long-proofs.tex:14587
`depends_on`	←	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105
`depends_on`	←	$C^{1}$ implies differentiable	declared	parts/02-mathematical-methods/05-real-analysis.tex:2967
`depends_on`	←	Implicit function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	←	Implicit function theorem	declared	appendices/A-long-proofs.tex:14371