equation 7.7 eq:ana-limit-left

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Supports

• depends_on definition 7.20 Continuity at a point ¶
  • 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 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.519 The convolution exists ¶
      • 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 theorem 30.18 Cauchy: the stress tensor exists ¶
    • 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 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 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 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 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 lemma A.307 Partition of unity on a compact 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 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 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.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 10.3 Principal part and principal symbol ¶
        • depends_on lemma 10.28 The separation constant ¶
        • depends_on proposition 28.44 Superposition ¶
        • depends_on proposition 10.15 Method of characteristics, first order ¶
        • 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 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 10.25 The backward heat problem is ill posed ¶
        • depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; 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 corollary A.145 Harmonic functions are smooth ¶ ↺
        • depends_on theorem A.135 Converse of the mean-value property ¶ ↺
        • depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
      • 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 theorem 10.80 Parabolic maximum principle ¶
    • 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 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 ¶
    • … 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 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 9.8 Picard–Lindelöf ¶
      • depends_on corollary 32.5 Trajectories do not cross ¶
        • depends_on definition 32.29 Limit cycle ¶
        • depends_on definition 32.38 Poincaré section and return map ¶
        • depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
      • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on definition 9.37 Fundamental matrix ¶
        • depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
        • depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
        • depends_on proposition 9.103 The positive zeros and their spacing ¶
        • depends_on proposition 9.18 The linear equation: integrating factor ¶
        • depends_on theorem A.74 Flow of a time-dependent vector field ¶
        • depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
        • depends_on theorem 9.13 Dimension of the solution space ¶
      • 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 lemma A.548 The joint flow is a translation action on the level set ¶
        • depends_on lemma A.567 The differential of the momentum map along the orbit ¶
        • depends_on proposition 13.131 Commuting fields have commuting flows ¶
        • depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
        • depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
      • 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 definition 10.10 Characteristic surface ¶
        • depends_on proposition 10.13 Canonical form in two variables ¶
        • depends_on proposition 10.11 What a characteristic surface is ¶
      • 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 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.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 22.33 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 lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
        • depends_on lemma A.631 The normal–normal Ricci contraction ¶
        • depends_on lemma A.640 Variation of the integrated three-curvature ¶
        • depends_on proposition 15.31 Radius and cosmological constant ¶
        • depends_on theorem A.49 thm:app-eh-equivalence ¶
        • depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
      • 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 44.44 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 proposition 13.141 The invariant of a Killing vector along a geodesic ¶
        • 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 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 13.158 The connection determined by vielbein and torsion ¶
    • … 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 lemma 9.136 Wallis integrals ¶
        • 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 lemma A.485 Conformal automorphisms of the disc ¶
      • 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 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 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 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 7.111 Jacobi's formula, cofactor form ¶ ↺
      • depends_on proposition 9.28 The scalar constant-coefficient equation ¶
        • depends_on proposition 9.88 The model problem, and the error of the composite ¶
      • 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 lemma A.754 Exact depth expansion ¶
        • depends_on lemma A.187 Exponentiation ¶
        • depends_on lemma A.138 The flat exponential ¶
        • depends_on lemma A.452 Young and Hölder ¶
        • depends_on lemma A.212 Second-order flatness ¶
        • depends_on lemma A.209 Uniform third-order remainder ¶
        • 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 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 7.79 The polygon recursion ¶
        • 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 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 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 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 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 definition 16.11 Admissible class; functional ¶ ↺
        • 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 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 theorem 7.35 Mean value theorem ¶
        • 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 lemma A.500 Graphs and $C^{1}$ images have zero content ¶
        • depends_on corollary A.514 Degeneracy on a negligible set ¶
        • depends_on lemma A.512 The boundary strip is thin ¶ ↺
      • 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.77 Poincaré lemma, converse form ¶ ↺
      • 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 lemma A.599 One variable, complex coefficient ¶
        • 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 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 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 proposition 7.18 Two-sided limit from one-sided limits ¶
  • depends_on lemma 7.33 Fermat: interior extremum ¶
    • depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
      • depends_on remark 7.118 The multiplier rule and its functional counterpart ¶
    • depends_on proposition 16.16 Stationarity is necessary ¶ ↺
    • depends_on theorem 7.34 Rolle ¶ ↺
    • depends_on theorem 16.104 Snell's law of refraction ¶