definition 7.102 Differential of a function

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:3065 · p. 243

ground object -- no derivation owed

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 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 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺

Supports

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 proposition A.511 The substitution property is universal ¶
    - 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 lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶
  - 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 corollary A.553 Quasi-periodic motion ¶
      - depends_on definition A.554 Actions on the torus ¶
      - 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 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 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 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 theorem 13.67 Quotient manifold theorem ¶
      - depends_on example 13.68 Why each hypothesis is there ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
- 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 corollary 31.13 Incompressible flow ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
    - depends_on theorem 31.22 Euler's equations of motion ¶
      - depends_on lemma A.712 The force is a far-field integral ¶
      - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶ ↺
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.80 Rayleigh's circulation criterion ¶
      - depends_on proposition 52.24 The p-mode cavity ¶ ↺
      - depends_on remark 31.23 Boundary conditions, and what the ideal model omits ¶
      - depends_on theorem 31.24 Bernoulli ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶ ↺
      - depends_on theorem 31.77 Rayleigh's inflexion-point criterion ¶
- 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 theorem A.311 General Stokes theorem ¶ ↺
- 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 lemma A.716 The creeping-flow equation for $\psi$ ¶
  - 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 corollary A.778 The powder collects at the antinodes ¶
    - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on lemma A.735 Transfer of the flow ¶
    - 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 corollary 31.76 Kelvin–Helmholtz instability ¶
      - depends_on corollary 31.75 Rayleigh–Taylor instability ¶
      - depends_on theorem 31.94 Gravity–capillary waves on a free surface ¶
  - 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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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declared in the source
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Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:3082
`depends_on`	→	Partial derivative; gradient	declared	parts/02-mathematical-methods/05-real-analysis.tex:3082
`depends_on`	←	Chain rule in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105
`depends_on`	←	rem:ana-differential-scalar	declared	parts/02-mathematical-methods/05-real-analysis.tex:3090