axiom 7.1 Completeness of $\R$

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• depends_on definition 7.39 Darboux sums and the definite integral ¶
  • depends_on definition 7.125 Multiple integral ¶
    • 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 theorem 30.18 Cauchy: the stress tensor exists ¶
    • depends_on definition 7.138 Set of measure zero ¶
      • depends_on definition 7.140 Hausdorff measure ¶
        • depends_on definition 7.142 Box-counting dimension ¶
        • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
        • depends_on remark 7.144 What is developed here, and what is not ¶
      • depends_on lemma 7.139 A countable union of null sets is null ¶
        • depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
        • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
        • depends_on remark 7.94 Three numbers, two kinds ¶
      • depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
      • depends_on remark 7.144 What is developed here, and what is not ¶ ↺
    • depends_on definition A.498 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 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 definition A.304 Integral over a chart ¶
      • depends_on lemma A.305 The chart integral is well defined ¶
        • depends_on definition A.308 Integral over the manifold ¶
    • depends_on definition 29.13 Inertia tensor, continuum form ¶
      • depends_on definition 29.19 Inertia ellipsoid ¶
        • depends_on theorem 29.29 Poinsot's construction ¶
      • depends_on example 29.23 The polar moment of the Earth, and what it reveals ¶
        • depends_on phenomenon 29.34 Free nutation of the Earth ¶
      • depends_on proposition 49.17 Magnetic-dipole spin-down ¶
      • depends_on proposition 29.14 Angular momentum and kinetic energy ¶
      • depends_on proposition 29.15 Positivity ¶
        • depends_on definition 29.19 Inertia ellipsoid ¶ ↺
        • depends_on theorem 29.16 Principal axes ¶
      • depends_on proposition 29.24 MacCullagh's formula ¶
        • depends_on phenomenon 29.49 Precession of the equinoxes ¶
      • depends_on proposition 29.22 Perpendicular-axis theorem ¶
      • depends_on theorem 29.20 Parallel-axis theorem ¶
        • depends_on corollary 29.21 The centre of mass minimizes the moment ¶
      • depends_on theorem 29.16 Principal axes ¶ ↺
    • 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 9.143 A cubic quadrature between two turning points ¶
        • depends_on proposition 9.141 First properties ¶
        • depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
      • 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 lemma A.501 Iterated integration over a box ¶
      • depends_on corollary A.502 All iterated orders agree ¶
        • depends_on lemma A.507 Coordinate permutations ¶
      • depends_on lemma A.509 Primitive maps have the substitution property ¶
        • depends_on proposition A.511 The substitution property is universal ¶ ↺
    • depends_on lemma A.499 What zero content buys ¶ ↺
    • depends_on lemma A.518 The excised ball ¶
      • 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 proposition A.519 The convolution exists ¶
        • depends_on proposition A.524 Rate of decay ¶ ↺
    • 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 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 7.133 Gauss ¶
        • depends_on lemma A.708 The disturbance flux vanishes ¶
        • depends_on lemma A.712 The force is a far-field integral ¶
        • depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
        • depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
        • depends_on lemma 31.11 Transport theorem for a material volume ¶
        • depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
        • depends_on lemma 10.44 Green's identities ¶
        • depends_on lemma 10.58 Darboux's equation for spherical means ¶
        • depends_on proposition 23.54 The geometrical amplitude diverges ¶
        • depends_on proposition 10.57 Energy in a backward cone ¶
        • depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
        • depends_on theorem 32.6 Evolution of phase volume ¶
        • … 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 theorem 8.12 Cauchy ¶
        • depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
        • depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      • depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
        • depends_on lemma A.305 The chart integral is well defined ¶ ↺
    • depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
    • … 3 more
  • depends_on definition 11.20 Density ¶
    • depends_on definition 11.21 Distribution function ¶
      • depends_on definition A.174 Tightness ¶
        • depends_on lemma A.175 Tightness makes the Helly limit proper ¶
        • depends_on lemma A.180 Tightness ¶
      • depends_on lemma 11.52 Slutsky ¶
        • depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶
        • depends_on theorem 11.67 Asymptotics of the MLE ¶
        • depends_on theorem 11.88 Wilks ¶
      • depends_on proposition 11.22 Properties of the distribution function ¶
        • depends_on proposition 11.27 Probability integral transform ¶
      • depends_on theorem A.168 Lévy's continuity theorem ¶
        • depends_on corollary A.181 The converse ¶
      • depends_on theorem A.173 Helly's selection theorem ¶
        • depends_on lemma A.176 Helly–Bray ¶
        • depends_on lemma A.175 Tightness makes the Helly limit proper ¶ ↺
      • depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶ ↺
    • depends_on definition 11.23 Expectation, moments and quantiles in the continuous case ¶
      • depends_on example 11.30 Exponential lifetimes ¶
      • depends_on proposition 11.29 Gaussian moments ¶
      • depends_on proposition 11.27 Probability integral transform ¶ ↺
      • depends_on proposition 11.24 The algebra of the discrete case transfers ¶
        • depends_on example 11.30 Exponential lifetimes ¶ ↺
        • depends_on proposition 11.26 Product rule ¶
    • depends_on definition 11.28 Gaussian distribution ¶
      • depends_on definition A.218 The hypotheses ¶
        • depends_on lemma A.220 At each point, field and derivative are independent Gaussians ¶
        • depends_on theorem A.219 Rice's formula ¶
      • depends_on lemma A.226 The level factor ¶
      • depends_on lemma A.225 The mean positive part ¶
      • depends_on proposition 11.29 Gaussian moments ¶ ↺
      • depends_on theorem A.192 Berry–Esseen ¶
      • depends_on theorem A.183 Lindeberg–Feller ¶
        • depends_on corollary A.188 Lindeberg–Lévy is the special case ¶
        • depends_on corollary A.189 Lyapunov's condition ¶
      • depends_on theorem 11.102 Rice's formula ¶
        • depends_on theorem 11.103 Level dependence of the mean upcrossing count ¶
    • depends_on definition 11.25 Joint density and independence ¶
      • depends_on proposition 11.26 Product rule ¶ ↺
  • depends_on lemma A.501 Iterated integration over a box ¶ ↺
  • depends_on lemma 17.7 Riemann–Lebesgue ¶
    • depends_on corollary 17.8 Localisation ¶
    • depends_on proposition 17.10 Smoothness and coefficient decay ¶
      • depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶
    • depends_on proposition 17.30 Elementary properties ¶
      • depends_on example 17.31 The three standard pairs ¶
      • depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
        • depends_on definition 17.40 Transform and derivative of a distribution ¶
      • depends_on theorem 17.43 Bandwidth theorem ¶
        • depends_on corollary 17.44 Duration and bandwidth in SI ¶
        • depends_on remark 17.46 The quantum reading, in SI ¶
    • depends_on theorem 17.9 Dirichlet ¶
      • depends_on corollary 17.11 Uniform convergence for $C^{1}$ functions ¶ ↺
      • depends_on theorem 17.47 Poisson summation ¶
        • depends_on corollary 17.81 Sampling periodises the spectrum; aliasing ¶
        • depends_on corollary 17.49 Lattice sums and the reciprocal lattice ¶
  • depends_on theorem 7.40 Continuous functions are integrable ¶
    • depends_on corollary A.502 All iterated orders agree ¶ ↺
    • 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 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 proposition 16.16 Stationarity is necessary ¶
        • depends_on theorem 16.43 Euler's rule for integral constraints ¶
      • depends_on proposition 16.68 Weierstrass' counterexample ¶
    • depends_on definition 9.137 Elliptic integrals of the three kinds ¶ ↺
    • depends_on lemma A.501 Iterated integration over a box ¶ ↺
    • depends_on lemma 16.18 Fundamental lemma ¶ ↺
    • depends_on remark 7.128 What the derivations below take as given ¶ ↺
    • 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 definition A.462 The weighted space and the energy space ¶
        • depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
        • depends_on lemma A.452 Young and Hölder ¶
      • depends_on lemma A.520 A $C^{1}$ limit ¶
        • depends_on proposition A.521 Every derivative passes onto $g$ ¶ ↺
      • depends_on lemma A.440 Grönwall's inequality ¶
        • depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
      • depends_on lemma 16.19 du Bois-Reymond ¶
        • depends_on lemma 16.20 Mixed form ¶
      • depends_on lemma 9.10 Grönwall's inequality ¶
        • depends_on proposition 9.11 Continuous dependence on the initial data ¶
        • depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      • depends_on proposition 11.22 Properties of the distribution function ¶ ↺
      • 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 proposition 9.19 Separation of variables ¶
      • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • depends_on corollary 7.44 Substitution and integration by parts ¶
        • depends_on corollary 16.23 du Bois-Reymond form ¶
        • depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
        • depends_on lemma A.72 Iterated integral inequality ¶
        • depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
        • depends_on lemma A.172 Dirichlet ¶
        • depends_on lemma A.221 Counting identity ¶
        • depends_on lemma A.312 The computation in one chart ¶ ↺
        • depends_on proposition 7.85 Irrationality of $\pi$ ¶
        • depends_on proposition 8.10 Fundamental theorem for contours ¶
        • depends_on proposition 17.41 The identities physics uses ¶
        • depends_on proposition 17.73 Initial- and final-value theorems ¶
        • … 4 more
      • depends_on theorem 10.52 Duhamel's principle ¶
        • depends_on corollary 10.53 Duhamel for the wave equation ¶
• depends_on definition 7.140 Hausdorff measure ¶ ↺
• depends_on definition 7.125 Multiple integral ¶ ↺
• 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 lemma 7.78 Chord, arc, tangent ¶
        • depends_on proposition 7.79 The polygon recursion ¶
      • depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      • depends_on proposition 7.79 The polygon recursion ¶ ↺
      • depends_on proposition 7.83 Viète's product ¶
        • depends_on remark 7.84 One construction, two faces ¶
      • depends_on proposition 125.2 Crystallographic restriction ¶
        • depends_on remark 125.3 What the theorem does and does not forbid ¶
      • depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
    • 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 proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
  • depends_on proposition 125.2 Crystallographic restriction ¶ ↺
• depends_on proposition 12.10 $\ell^{2}$ is complete ¶
  • depends_on proposition 12.85 The direct sum is a Hilbert space ¶
    • depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
      • depends_on lemma A.250 Decomposition into cyclic subspaces ¶
        • depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  • depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
• 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 proposition 6.23 The punctured plane is not simply connected ¶
    • depends_on lemma 6.22 Nearby loops wind alike ¶ ↺
  • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
  • 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 lemma A.307 Partition of unity on a compact manifold ¶
    • depends_on theorem 7.35 Mean value theorem ¶
      • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on definition 7.41 Antiderivative ¶
        • depends_on lemma 7.67 The limit over real exponents ¶
        • depends_on lemma 7.75 The first quadrant ¶ ↺
        • depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
        • depends_on lemma 9.10 Grönwall's inequality ¶ ↺
        • depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
        • depends_on proposition 7.61 The logarithm ¶
        • depends_on proposition 9.19 Separation of variables ¶ ↺
        • depends_on theorem 7.115 Constant rank ¶
      • 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.77 Poincaré lemma, converse form ¶
      • depends_on lemma A.287 The Newton map contracts ¶
        • depends_on lemma A.288 The iteration converges ¶
      • depends_on lemma A.290 $h$ is Lipschitz ¶
        • depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
      • 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 theorem A.470 $T$ is compact ¶
      • depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
      • 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 22.20 The symplectic condition ¶
        • depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
        • depends_on proposition 30.25 Twenty-one constants ¶
        • depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
        • 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 theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
        • … 2 more
      • 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 theorem A.286 Implicit function theorem ¶
      • … 2 more
    • 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.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 7.23 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 ¶
• depends_on theorem 7.50 Power series; radius of convergence ¶
  • depends_on definition 7.59 The real exponential ¶
    • depends_on lemma 7.60 Functional equation of the exponential ¶
      • depends_on proposition 7.68 The natural base ¶
        • depends_on remark 7.70 What the condition says ¶
      • depends_on proposition 7.61 The logarithm ¶ ↺
      • depends_on proposition 7.63 Laws of real powers ¶
        • depends_on definition 7.90 The golden ratio ¶
        • depends_on lemma 7.67 The limit over real exponents ¶ ↺
        • depends_on proposition 7.68 The natural base ¶ ↺
        • depends_on proposition 7.91 The golden ratio ¶
        • depends_on proposition 7.89 Irrationality of square roots ¶
  • depends_on definition 8.3 Exponential, sine, cosine ¶
    • depends_on lemma A.786 From the half-strip to the half plane ¶
      • depends_on corollary A.788 The two parameters are the same parameter ¶
        • depends_on theorem A.789 Kirchhoff's contraction coefficient ¶
      • depends_on lemma A.787 From the potential strip to the half plane ¶
        • depends_on corollary A.788 The two parameters are the same parameter ¶ ↺
    • depends_on lemma A.787 From the potential strip to the half plane ¶ ↺
    • depends_on lemma A.785 The hodograph image ¶
      • depends_on lemma A.786 From the half-strip to the half plane ¶ ↺
  • depends_on theorem 7.51 Termwise differentiation ¶
    • depends_on definition 7.74 $\pi$ ¶ ↺
    • 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 lemma 9.136 Wallis integrals ¶
        • depends_on proposition 9.139 Series for the complete integral of the first kind ¶ ↺
      • depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
      • depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      • depends_on proposition 125.2 Crystallographic restriction ¶ ↺
      • depends_on proposition 9.141 First properties ¶ ↺
      • 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 theorem 9.23 Solution of a constant-coefficient system ¶
        • depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
        • depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
        • 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 theorem 6.11 Heine–Borel on $\R$ ¶
  • depends_on lemma A.365 Lebesgue number ¶
    • depends_on lemma A.367 Homotopy lifting ¶
    • depends_on lemma A.366 Unique path lifting ¶
      • depends_on lemma A.367 Homotopy lifting ¶ ↺
  • depends_on lemma A.136 Differentiation under the integral sign ¶
    • depends_on proposition A.142 The mollification is smooth ¶
  • depends_on proposition 6.23 The punctured plane is not simply connected ¶ ↺
  • 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 theorem 6.12 Heine–Borel in $\R^{N}$ ¶
• depends_on theorem 6.14 Intervals are connected ¶
  • depends_on proposition 6.16 prop:top-path-implies-connected ¶