definition 7.20 Continuity at a point

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:325 · p. 207

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 equation 7.7 eq:ana-limit-left ¶
depends_on equation 7.5 eq:ana-limit-right ¶

Supports

depends_on definition 7.98 Functions of class $C^{1}$ ¶
- 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 definition 7.119 Envelope of a family ¶
  - depends_on proposition 7.120 The envelope touches every member it meets ¶
- depends_on definition 7.126 Line and surface integrals ¶
  - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
  - depends_on definition 30.17 Traction ¶
    - depends_on proposition 30.19 The stress tensor is symmetric ¶
      - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.24 The elasticity tensor ¶
      - depends_on definition 31.1 Fluid ¶
      - depends_on proposition 30.19 The stress tensor is symmetric ¶ ↺
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on remark 30.72 Stress concentration ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
      - depends_on theorem 30.21 Cauchy's equation of motion ¶
- depends_on definition 7.127 Simple regions ¶
  - depends_on lemma A.512 The boundary strip is thin ¶
    - depends_on theorem A.513 Change of variables ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
  - depends_on remark A.516 The hypotheses of the global form ¶ ↺
  - depends_on theorem 7.133 Gauss ¶
    - depends_on lemma A.708 The disturbance flux vanishes ¶
      - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
    - depends_on lemma A.712 The force is a far-field integral ¶
    - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
      - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - 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 lemma 106.86 A shift of a divergent integral leaves a surface term ¶
      - depends_on proposition 106.88 The obstruction ¶
    - depends_on lemma 10.44 Green's identities ¶
      - depends_on proposition 10.71 Green's representation formula ¶
      - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on theorem 10.45 Symmetry of the Green's function ¶
    - depends_on lemma 10.58 Darboux's equation for spherical means ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on theorem 10.69 Mean-value property ¶
    - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on remark 23.59 What is observed ¶
    - depends_on proposition 10.57 Energy in a backward cone ¶
      - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
    - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on definition 44.6 Stress–energy tensor ¶
      - depends_on proposition 28.42 The wave equation of a stretched string ¶
    - depends_on theorem 32.6 Evolution of phase volume ¶
      - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
      - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
    - … 1 more
  - depends_on theorem 7.131 Green ¶
    - depends_on proposition 32.33 Bendixson's negative criterion ¶
    - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
    - depends_on theorem 7.132 Stokes ¶
      - depends_on definition 31.63 Circulation ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on theorem 8.12 Cauchy ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on lemma 100.27 Wick rotation ¶
      - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.94 Mellin inversion ¶
    - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
    - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
- depends_on definition A.508 Primitive map ¶
  - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
  - depends_on lemma A.509 Primitive maps have the substitution property ¶
    - depends_on proposition A.511 The substitution property is universal ¶ ↺
- depends_on definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
  - depends_on definition A.125 Ellipsoidal coordinates ¶
    - depends_on lemma A.126 Cartesian coordinates and scale factors ¶
      - depends_on proposition A.128 The ellipsoidal system separates ¶
  - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
    - depends_on theorem A.120 Stäckel and Robertson conditions ¶ ↺
  - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶ ↺
- depends_on definition A.298 Half-space; smoothness on it ¶
  - depends_on definition A.299 Manifold with boundary ¶
    - 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 lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
- depends_on definition 16.11 Admissible class; functional ¶
  - depends_on definition 16.59 Weierstrass excess function ¶
  - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on example 16.14 A weak minimum that is not strong ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.60 Weierstrass' necessary condition ¶
  - depends_on definition 16.15 Variation; the first variation ¶
    - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
    - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
      - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
      - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
    - depends_on theorem 16.43 Euler's rule for integral constraints ¶
      - depends_on example 16.45 Two solved isoperimetric problems ¶
      - depends_on proposition 16.4 The hanging chain is a catenary ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
  - depends_on proposition 16.68 Weierstrass' counterexample ¶
- depends_on definition 10.1 Partial differential equation; order ¶
  - 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.104 The recursion ¶
      - depends_on proposition A.110 Explicit solution of the majorant problem ¶
    - depends_on lemma A.105 Formal reduction of a normal problem ¶
  - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
    - depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
    - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on example 10.97 Burgers shock ¶ ↺
      - depends_on proposition 10.94 Gradient catastrophe ¶
    - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on definition 10.10 Characteristic surface ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.83 Weak solution ¶
    - depends_on lemma 10.28 The separation constant ¶
      - depends_on definition A.118 Simple separation of the Helmholtz equation ¶ ↺
      - depends_on example 10.31 Cartesian separation ¶
      - depends_on example 10.32 Cylindrical separation ¶
      - depends_on example 10.33 Spherical separation ¶
      - depends_on proposition 23.9 Separation of the time ¶
      - depends_on proposition 10.29 Time separation reduces the canonical equations to Helmholtz ¶
      - depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
    - depends_on proposition 28.44 Superposition ¶
    - depends_on proposition 10.15 Method of characteristics, first order ¶
      - depends_on definition 10.93 Scalar conservation law ¶ ↺
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
      - depends_on remark 31.90 Why a compression wave must steepen ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
    - depends_on proposition 10.43 Representation of the solution ¶
    - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
    - depends_on theorem 10.52 Duhamel's principle ¶
      - depends_on corollary 10.53 Duhamel for the wave equation ¶
  - depends_on definition 10.24 Well-posed problem ¶
    - depends_on corollary A.152 Uniqueness ¶
    - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
    - depends_on corollary 10.81 Uniqueness for the heat equation ¶
    - depends_on definition 30.22 Boundary conditions of elastostatics ¶
      - depends_on proposition 30.50 Love waves need a slow surface layer ¶
      - depends_on proposition 30.48 Rayleigh's secular equation ¶
      - depends_on proposition 30.23 Virtual work; the weak form ¶
    - depends_on proposition 10.25 The backward heat problem is ill posed ¶
    - 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 definition 10.66 Harmonic function ¶
  - depends_on theorem A.135 Converse of the mean-value property ¶
    - depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
    - depends_on corollary A.145 Harmonic functions are smooth ¶
  - depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
  - depends_on theorem 10.69 Mean-value property ¶ ↺
  - depends_on theorem 10.75 Weak maximum principle ¶
    - depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶ ↺
    - depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
      - depends_on remark A.527 The hypotheses, and what happens without them ¶
    - 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 definition 9.30 Hyperbolic equilibrium ¶
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
- depends_on lemma A.520 A $C^{1}$ limit ¶
  - depends_on proposition A.521 Every derivative passes onto $g$ ¶
    - depends_on proposition A.524 Rate of decay ¶ ↺
    - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
      - depends_on theorem A.523 Helmholtz decomposition: existence ¶
- … 8 more
depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
- depends_on proposition 9.139 Series for the complete integral of the first kind ¶
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
- depends_on proposition 9.22 The exponential and its derivative ¶
  - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶ ↺
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶ ↺
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on corollary 32.5 Trajectories do not cross ¶
    - depends_on definition 32.29 Limit cycle ¶
      - depends_on phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on definition 32.91 The circle map ¶
      - depends_on example 32.70 The Hénon map ¶
      - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
      - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
      - depends_on corollary 32.32 No chaos in a planar flow ¶
  - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
    - depends_on definition 9.37 Fundamental matrix ¶
      - depends_on proposition 9.38 Liouville's formula ¶
      - depends_on theorem 9.39 Floquet ¶
    - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
      - depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
      - depends_on remark 20.11 What has no closed form, and what to do about it ¶
    - depends_on proposition 9.103 The positive zeros and their spacing ¶
    - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
    - depends_on theorem A.74 Flow of a time-dependent vector field ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
    - depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
    - depends_on theorem 9.13 Dimension of the solution space ¶
      - depends_on definition 9.37 Fundamental matrix ¶ ↺
      - depends_on proposition 28.18 The three regimes ¶
      - depends_on proposition 28.4 General solution and isochronism ¶
      - depends_on proposition 9.106 The modified series and what it does not do ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      - depends_on proposition 9.16 Wronskian test for independence ¶
  - depends_on proposition 32.4 The flow is a one-parameter group ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶ ↺
  - depends_on proposition 10.16 Cauchy's characteristic strips ¶ ↺
  - depends_on proposition 20.7 Terminal speed and the approach to it ¶
    - depends_on example 20.8 A raindrop ¶
  - depends_on proposition 9.11 Continuous dependence on the initial data ¶
  - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
  - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
  - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶ ↺
  - depends_on theorem 9.34 Poincaré–Bendixson; quoted ¶
    - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶ ↺
depends_on proposition 7.105 Clairaut–Schwarz ¶
- depends_on definition 10.4 The second-order operator ¶ ↺
- 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 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 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 theorem 44.23 Contracted Bianchi identity ¶
      - 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 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 15.33 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 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 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 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 proposition 48.4 Peculiar momentum decays with the expansion ¶
      - depends_on proposition 45.6 Radial equation and effective potential ¶
      - 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 ¶
- … 2 more
depends_on proposition 7.145 Delta as a limit; elementary properties ¶
depends_on proposition 7.27 Differentiable implies continuous ¶
- depends_on proposition 7.31 Chain rule ¶
  - depends_on corollary 7.44 Substitution and integration by parts ¶
    - depends_on definition 8.8 Contour integral ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶ ↺
    - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶ ↺
    - depends_on proposition 7.145 Delta as a limit; elementary properties ¶ ↺
    - depends_on proposition 16.50 The second variation ¶
    - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
    - depends_on theorem 16.38 Euler–Poisson equation ¶ ↺
  - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
  - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.91 The golden ratio ¶
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
    - depends_on proposition 7.68 The natural base ¶ ↺
    - depends_on proposition 7.91 The golden ratio ¶ ↺
    - depends_on proposition 7.89 Irrationality of square roots ¶
      - depends_on proposition 7.91 The golden ratio ¶ ↺
  - depends_on proposition 16.26 Covariance under change of dependent variable ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶ ↺
  - depends_on theorem 16.29 Beltrami identity ¶
    - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
    - depends_on proposition 16.39 Ostrogradsky instability ¶
  - 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 theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert'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 definition A.102 Normal Cauchy problem of order $k$ ¶ ↺
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶ ↺
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.61 The logarithm ¶
  - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
    - depends_on remark 7.70 What the condition says ¶
  - depends_on definition 7.62 Real powers ¶
    - depends_on definition 7.90 The golden ratio ¶ ↺
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
    - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
  - depends_on lemma 7.67 The limit over real exponents ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on proposition 7.30 Leibniz rule ¶
  - depends_on corollary 7.44 Substitution and integration by parts ¶ ↺
  - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
  - depends_on proposition 7.122 Product identities of the nabla calculus ¶
  - depends_on proposition 5.2 Jacobi's formula, column form ¶
    - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶ ↺
    - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶ ↺
    - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
  - depends_on proposition 9.28 The scalar constant-coefficient equation ¶ ↺
  - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
    - depends_on definition 32.12 Linearization ¶
      - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
      - depends_on theorem 32.13 Linear stability ¶
    - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on proposition A.755 The Boussinesq pair ¶
    - depends_on lemma A.187 Exponentiation ¶
    - depends_on lemma A.138 The flat exponential ¶
      - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.213 The three minima ¶
    - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.211 Root-$n$ localisation ¶
    - depends_on phenomenon 28.13 Universality of small oscillations ¶
    - depends_on proposition 16.50 The second variation ¶ ↺
    - depends_on proposition 17.67 Holomorphy ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶ ↺
      - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
    - … 2 more
- depends_on theorem 7.51 Termwise differentiation ¶
  - depends_on definition 7.74 $\pi$ ¶
    - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
    - depends_on proposition 7.79 The polygon recursion ¶
      - depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
      - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.82 What Archimedes proved ¶
      - depends_on remark 7.80 Why these are the polygon perimeters ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
  - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    - depends_on lemma 7.75 The first quadrant ¶ ↺
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on lemma 9.136 Wallis integrals ¶ ↺
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on proposition 9.141 First properties ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶ ↺
    - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
  - depends_on lemma 7.75 The first quadrant ¶ ↺
  - depends_on proposition 9.22 The exponential and its derivative ¶ ↺
depends_on proposition 7.22 Sequential characterization ¶
- depends_on theorem 7.24 Extreme value theorem ¶
  - depends_on lemma A.221 Counting identity ¶
  - depends_on lemma 6.19 Continuous argument along a path ¶
    - depends_on definition 6.20 Winding number ¶
      - depends_on example 6.21 ex:top-winding-values ¶
      - depends_on lemma 6.22 Nearby loops wind alike ¶
    - depends_on lemma 6.22 Nearby loops wind alike ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on corollary A.502 All iterated orders agree ¶
      - depends_on lemma A.507 Coordinate permutations ¶
    - depends_on definition 16.11 Admissible class; functional ¶ ↺
    - depends_on definition 9.137 Elliptic integrals of the three kinds ¶
      - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶ ↺
    - depends_on lemma A.501 Iterated integration over a box ¶
      - depends_on corollary A.502 All iterated orders agree ¶ ↺
      - depends_on lemma A.509 Primitive maps have the substitution property ¶ ↺
    - depends_on lemma 16.18 Fundamental lemma ¶ ↺
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on lemma A.312 The computation in one chart ¶
      - depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
      - depends_on theorem 7.133 Gauss ¶ ↺
      - depends_on theorem 7.131 Green ¶ ↺
      - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on corollary 7.110 Variable limits of integration ¶
      - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
      - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on lemma 16.19 du Bois-Reymond ¶
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 11.22 Properties of the distribution function ¶
      - depends_on proposition 9.18 The linear equation: integrating factor ¶ ↺
      - depends_on proposition 9.19 Separation of variables ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on theorem 10.52 Duhamel's principle ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
  - depends_on theorem 7.34 Rolle ¶
    - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶ ↺
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶
      - depends_on lemma A.440 Grönwall's inequality ¶ ↺
      - depends_on lemma A.226 The level factor ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - … 2 more
    - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶ ↺
- depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
  - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
    - depends_on lemma A.196 Inversion for a difference ¶
      - depends_on theorem A.198 Esseen's smoothing inequality ¶
  - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
  - depends_on lemma A.499 What zero content buys ¶
    - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
    - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
    - depends_on theorem A.513 Change of variables ¶ ↺
  - depends_on lemma A.512 The boundary strip is thin ¶ ↺
  - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
  - depends_on lemma A.176 Helly–Bray ¶
    - depends_on corollary A.181 The converse ¶
  - depends_on lemma A.488 Small chords cut off small arcs ¶
    - depends_on theorem A.490 Equicontinuity of the normalised class ¶
  - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
  - depends_on lemma 6.19 Continuous argument along a path ¶ ↺
  - depends_on remark 7.128 What the derivations below take as given ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
  - depends_on theorem 7.109 Leibniz integral rule ¶
    - depends_on corollary 7.110 Variable limits of integration ¶ ↺
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶ ↺
    - depends_on lemma A.607 A zero-free entire function is an exponential ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on lemma A.599 One variable, complex coefficient ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    - depends_on proposition A.524 Rate of decay ¶ ↺
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
    - depends_on theorem 7.137 Helmholtz decomposition ¶ ↺
  - … 1 more
depends_on remark 7.21 rem:ana-discontinuities ¶
depends_on remark 7.101 Partial derivatives alone do not suffice ¶
depends_on theorem 7.23 Intermediate value theorem ¶
- depends_on definition 7.74 $\pi$ ¶ ↺
- depends_on lemma 7.75 The first quadrant ¶ ↺
- depends_on lemma 6.19 Continuous argument along a path ¶ ↺
- depends_on proposition 7.32 Derivative of the inverse function ¶
  - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶ ↺
  - depends_on proposition 7.61 The logarithm ¶ ↺
  - depends_on proposition 9.141 First properties ¶ ↺
  - depends_on proposition 9.19 Separation of variables ¶ ↺
- depends_on proposition 7.61 The logarithm ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on theorem A.443 Sturm separation theorem ¶
- depends_on theorem 11.101 Davies' bound ¶
  - depends_on corollary 11.105 Trials factor at high significance ¶
  - depends_on example 11.107 Degrading a local five sigma ¶
- depends_on theorem 29.37 Reduction to one degree of freedom ¶
  - depends_on phenomenon 29.38 Nutation of a released top ¶
  - depends_on proposition 29.40 The sleeping top ¶
  - depends_on remark 29.42 The closed form, and the mathematics it needs ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:336
`depends_on`	→	eq:ana-limit-left	declared	parts/02-mathematical-methods/05-real-analysis.tex:336
`depends_on`	→	eq:ana-limit-right	declared	parts/02-mathematical-methods/05-real-analysis.tex:336
`depends_on`	←	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2922
`depends_on`	←	Weierstrass $M$-test; uniform limits are continuous	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:207
`depends_on`	←	Clairaut–Schwarz	declared	parts/02-mathematical-methods/05-real-analysis.tex:3162
`depends_on`	←	Delta as a limit; elementary properties	declared	parts/02-mathematical-methods/05-real-analysis.tex:4984
`depends_on`	←	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	←	Sequential characterization	declared	parts/02-mathematical-methods/05-real-analysis.tex:353
`depends_on`	←	rem:ana-discontinuities	declared	parts/02-mathematical-methods/05-real-analysis.tex:347
`depends_on`	←	Partial derivatives alone do not suffice	declared	parts/02-mathematical-methods/05-real-analysis.tex:3062
`depends_on`	←	Intermediate value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:374