proposition 7.104 Chain rule in several variables

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:3093 · p. 243

Rests on

depends_on definition 7.13 Composite function ¶
- 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 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 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 7.102 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
proves proof ch:05-real-analysis@proof-63 ¶

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 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 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 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 corollary A.743 Two scalar functions instead of a tensor field ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on remark 31.14 When incompressibility is legitimate ¶
    - depends_on phenomenon 31.88 Sound travels at the adiabatic speed ¶
      - depends_on remark 31.89 The Mach number and the regimes it separates ¶
      - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on proposition 52.24 The p-mode cavity ¶
      - depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
    - depends_on theorem 31.91 Rankine–Hugoniot jump conditions ¶
      - depends_on phenomenon 31.93 The Mach cone ¶
      - depends_on remark 31.92 Entropy selects the physical branch ¶
  - 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 remark 31.81 Where the threshold comes from, and what it costs ¶
    - 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 corollary A.799 The impressed pressure gradient ¶
      - depends_on lemma 31.67 Blasius' force formula ¶
      - depends_on phenomenon 31.25 Pressure falls where the flow speeds up ¶
      - depends_on theorem A.707 D'Alembert's paradox ¶
      - depends_on theorem 31.74 The interface dispersion relation ¶
    - depends_on theorem 31.64 Kelvin's circulation theorem ¶
      - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
      - depends_on remark 31.66 Only three things make circulation ¶
      - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
    - depends_on theorem 31.37 The Navier–Stokes equations ¶
      - depends_on definition A.741 The statistical setting ¶
      - depends_on definition A.760 The convecting layer ¶
      - depends_on definition 31.43 Creeping flow ¶
      - depends_on phenomenon 31.78 Convective onset at a computed threshold ¶
      - depends_on phenomenon 31.41 The fourth-power law of pipe flow ¶
      - depends_on phenomenon 31.49 Dynamical similarity ¶
      - depends_on proposition A.761 The Boussinesq equations ¶
      - depends_on proposition 31.40 Circular Couette flow ¶
      - depends_on proposition 31.39 Plane Couette flow ¶
      - depends_on remark 31.48 What is actually proved ¶
      - depends_on remark 31.38 Whose equations, and the Stokes hypothesis ¶
      - depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
      - … 3 more
    - 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 proposition 31.28 Potential flow reduces to Laplace's equation ¶ ↺
- 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

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Composite function	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105
`depends_on`	→	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105
`depends_on`	→	Differential of a function	declared	parts/02-mathematical-methods/05-real-analysis.tex:3105
`depends_on`	←	Inverse function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3545
`depends_on`	←	Variable limits of integration	declared	parts/02-mathematical-methods/05-real-analysis.tex:3384
`depends_on`	←	Inverse function theorem	declared	appendices/A-long-proofs.tex:14683
`depends_on`	←	Material and spatial descriptions	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:133
`depends_on`	←	Material derivative	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:114
`depends_on`	←	Lagrange bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:857
`depends_on`	←	Complete integral; envelope	declared	parts/02-mathematical-methods/08-pdes.tex:561
`depends_on`	←	Every diffeomorphism factorises locally	declared	appendices/A-long-proofs.tex:24909
`depends_on`	←	Transitivity	declared	appendices/A-long-proofs.tex:24683
`depends_on`	←	The standard smooth bump	declared	appendices/A-long-proofs.tex:15103
`depends_on`	←	Properties of conservative fields	declared	parts/02-mathematical-methods/05-real-analysis.tex:4614
`depends_on`	←	The envelope touches every member it meets	declared	parts/02-mathematical-methods/05-real-analysis.tex:3840
`depends_on`	←	Euler's theorem on homogeneous functions	declared	parts/02-mathematical-methods/05-real-analysis.tex:3274
`depends_on`	←	Lagrange multipliers in finitely many variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3752
`depends_on`	←	An envelope of solutions is a solution	declared	parts/02-mathematical-methods/08-pdes.tex:568
`depends_on`	←	Euler's theorem on homogeneous functions is owed	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:358
`depends_on`	←	Constant rank	declared	parts/02-mathematical-methods/05-real-analysis.tex:3588
`depends_on`	←	Implicit function theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3523
`depends_on`	←	Stokes	declared	parts/02-mathematical-methods/05-real-analysis.tex:4409
`depends_on`	←	Taylor's theorem in several variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3206
`depends_on`	←	Implicit function theorem	declared	appendices/A-long-proofs.tex:14371
`proves`	←	ch:05-real-analysis@proof-63	declared	parts/02-mathematical-methods/05-real-analysis.tex:3109