proposition 7.105 Clairaut–Schwarz

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:3158 · p. 244

Rests on

depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
- depends_on equation 7.5 eq:ana-limit-right ¶
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 definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - 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 definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - 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 definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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-64 ¶

Supports

depends_on definition 10.4 The second-order operator ¶
- depends_on definition 10.10 Characteristic surface ¶
  - depends_on proposition 10.13 Canonical form in two variables ¶
    - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
    - depends_on theorem 10.55 d'Alembert's formula ¶
      - depends_on corollary 10.56 Domain of dependence and influence ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
  - depends_on proposition 10.11 What a characteristic surface is ¶
    - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on corollary 10.60 Huygens' principle in three space dimensions ¶
    - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on definition A.103 First-order quasilinear system with zero data ¶
      - depends_on lemma A.105 Formal reduction of a normal problem ¶
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
- depends_on definition 10.42 Green's function ¶
  - depends_on definition 10.50 Retarded and advanced Green's functions ¶
  - depends_on example 10.49 The method of images ¶
  - depends_on proposition 28.24 Impulse response of the damped oscillator ¶
  - depends_on proposition 10.47 Bilinear expansion ¶
  - depends_on proposition 10.43 Representation of the solution ¶
  - depends_on theorem 10.45 Symmetry of the Green's function ¶
- depends_on definition 10.6 Type of a second-order operator ¶
  - depends_on definition 10.79 Parabolic boundary ¶
    - depends_on theorem 10.80 Parabolic maximum principle ¶
      - depends_on corollary 10.81 Uniqueness for the heat equation ¶
      - depends_on proposition A.163 The viscous problem produces the entropy inequality ¶
  - depends_on example 10.9 ex:pde-three-types ¶
  - depends_on proposition 10.25 The backward heat problem is ill posed ¶
  - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶ ↺
  - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on lemma A.90 Affine normalization ¶
      - depends_on lemma A.91 Smallness after rescaling ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶ ↺
    - depends_on proposition 10.11 What a characteristic surface is ¶ ↺
depends_on lemma A.75 Differentiating a pullback along a flow ¶
- depends_on lemma A.77 Poincaré lemma, converse form ¶
depends_on lemma 22.20 The symplectic condition ¶
- depends_on corollary 22.24 Invariance of the phase-space volume ¶
  - depends_on remark 22.25 This is Liouville's theorem ¶
- depends_on proposition 22.27 Properties of the Lagrange bracket ¶
  - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
- depends_on proposition 22.30 Properties of the Poisson bracket ¶
  - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
    - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
- depends_on remark 22.33 What the duality is good for ¶
- depends_on remark 22.28 Where the content of the fundamental brackets lies ¶ ↺
- depends_on theorem 22.22 Poincaré ¶
  - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
  - depends_on remark 22.23 What the invariant does not depend on ¶
depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
- depends_on remark 7.124 The Laplacian of a vector field is a Cartesian notion ¶
- depends_on theorem 7.137 Helmholtz decomposition ¶
  - depends_on remark A.527 The hypotheses, and what happens without them ¶
  - depends_on remark 30.46 Why the plane-wave route, and not the potentials ¶
  - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
  - depends_on remark 10.73 What compact support is doing, and the rate that replaces it ¶
  - depends_on remark 10.74 Where the decomposition is used, and what it costs ¶
- depends_on theorem A.523 Helmholtz decomposition: existence ¶
- depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
  - depends_on remark A.527 The hypotheses, and what happens without them ¶ ↺
depends_on proposition 30.25 Twenty-one constants ¶
- depends_on definition 30.37 Voigt notation ¶
  - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶
- depends_on proposition 30.38 The Cauchy relations ¶
  - depends_on remark 30.39 The rari-constant controversy, decided by measurement ¶ ↺
- depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
- depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
depends_on proposition 22.30 Properties of the Poisson bracket ¶ ↺
depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
depends_on proposition 10.16 Cauchy's characteristic strips ¶
depends_on theorem 7.132 Stokes ¶
- depends_on definition 31.63 Circulation ¶
  - depends_on theorem 31.64 Kelvin's circulation theorem ¶
    - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶
      - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
    - depends_on remark 31.66 Only three things make circulation ¶
    - depends_on theorem 31.59 Helmholtz's vortex theorems ¶
      - depends_on corollary 31.65 Lagrange: an irrotational flow stays irrotational ¶ ↺
      - depends_on example 31.60 Vortex rings ¶
- 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 theorem 7.106 Taylor's theorem in several variables ¶
- depends_on remark 7.107 What the Hessian is for ¶
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
- depends_on definition 13.153 Contractions ¶
  - depends_on definition 44.2 Einstein tensor ¶
    - depends_on postulate 44.3 Einstein field equations ¶
      - depends_on corollary 44.24 Conservation is an integrability condition ¶
      - depends_on phenomenon 49.18 Orbital decay of a binary pulsar ¶
      - depends_on phenomenon 46.1 phen:gravitational-waves ¶
      - depends_on proposition 44.29 Linearized field equations ¶
      - depends_on proposition 44.4 Trace-reversed form ¶
      - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
      - depends_on theorem 48.5 The Friedmann equations ¶
      - depends_on theorem 44.42 The constraint equations ¶
    - depends_on theorem 44.23 Contracted Bianchi identity ¶
      - depends_on corollary 44.24 Conservation is an integrability condition ¶ ↺
      - depends_on proposition 44.41 Four of the ten equations are constraints ¶
      - depends_on proposition 44.30 The cosmological term ¶
      - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶ ↺
  - depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
    - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
  - depends_on lemma A.631 The normal–normal Ricci contraction ¶
    - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶ ↺
  - depends_on lemma A.640 Variation of the integrated three-curvature ¶
    - depends_on theorem A.641 The normal–normal bracket ¶
      - depends_on corollary A.642 The constraints are first class ¶
  - depends_on proposition 15.31 Radius and cosmological constant ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
      - depends_on remark 15.33 Which branch nature is on ¶
    - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
    - depends_on remark 15.33 Which branch nature is on ¶ ↺
  - depends_on theorem A.49 thm:app-eh-equivalence ¶
  - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
    - depends_on proposition 44.31 de Sitter geometry ¶
    - depends_on proposition 15.31 Radius and cosmological constant ¶ ↺
    - depends_on theorem 48.3 The Robertson–Walker metric ¶
      - depends_on phenomenon 48.12 The relic microwave background ¶
      - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
      - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on proposition 48.7 Luminosity distance ¶
- depends_on example 21.74 Rindler coordinates ¶
- depends_on lemma A.640 Variation of the integrated three-curvature ¶ ↺
- depends_on lemma 44.8 Palatini identity ¶
  - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on proposition 44.30 The cosmological term ¶ ↺
- depends_on proposition 44.44 Harmonic-gauge reduction ¶
  - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
- depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
- depends_on proposition 13.140 Killing's equation ¶
  - depends_on example 13.144 The two Killing tensors every metric carries ¶
  - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶
  - depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
    - depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
      - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
      - depends_on example A.359 The Lorentz algebra ¶
    - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
  - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
    - depends_on phenomenon 45.52 The shadow ¶
      - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
    - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
    - depends_on proposition 45.6 Radial equation and effective potential ¶
      - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on proposition 45.8 Orbit equation ¶
      - depends_on proposition 45.21 Radial plunge: the cycloid ¶
    - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
  - depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
  - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
- depends_on proposition 13.162 Curvature induced on the quadric ¶
  - depends_on theorem 48.3 The Robertson–Walker metric ¶ ↺
- depends_on proposition 13.154 Symmetries of the curvature ¶
  - depends_on definition 13.111 Linearised curvature of a symmetric field ¶
    - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
  - depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
  - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
- depends_on remark 30.16 Incompatibility is curvature ¶
- depends_on theorem 13.156 Cartan structure equations ¶
  - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
  - depends_on proposition 13.157 Bianchi identities ¶
    - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
  - depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
    - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
    - depends_on theorem 43.3 Equivalence of the two variable sets ¶
      - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
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Edges

type	direction	node	provenance	where
`depends_on`	→	Continuity at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:3162
`depends_on`	→	Mean value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:3162
`depends_on`	←	The second-order operator	declared	parts/02-mathematical-methods/08-pdes.tex:130
`depends_on`	←	Differentiating a pullback along a flow	declared	appendices/A-long-proofs.tex:4866
`depends_on`	←	The symplectic condition	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:597
`depends_on`	←	Second-order identities of the nabla calculus	declared	parts/02-mathematical-methods/05-real-analysis.tex:4113
`depends_on`	←	Twenty-one constants	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:788
`depends_on`	←	Saint-Venant compatibility is necessary	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:402
`depends_on`	←	Properties of the Poisson bracket	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1003
`depends_on`	←	prop:mfd-torsion-tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6538
`depends_on`	←	Cauchy's characteristic strips	declared	parts/02-mathematical-methods/08-pdes.tex:480
`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`	←	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6697
`depends_on`	←	Symmetric analogue of the converse Poincaré lemma	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5306
`depends_on`	←	Liouville	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`proves`	←	ch:05-real-analysis@proof-64	declared	parts/02-mathematical-methods/05-real-analysis.tex:3165