definition 7.16 Limit

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:216 · p. 205

ground object -- no derivation owed

Rests on

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 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶

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 definition 7.26 Derivative of a function at a point ¶
- depends_on definition 7.41 Antiderivative ¶
  - 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 corollary 16.33 Weierstrass–Erdmann corner conditions ¶
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶ ↺
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
    - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
    - depends_on lemma A.172 Dirichlet ¶
      - depends_on lemma A.193 A cosine integral ¶
      - depends_on theorem A.177 Fourier inversion for a distribution function ¶
    - 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 theorem 8.13 Goursat ¶
      - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 17.41 The identities physics uses ¶
      - depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
      - depends_on proposition 17.52 Linear translation-invariant systems ¶
    - depends_on proposition 17.73 Initial- and final-value theorems ¶
    - … 4 more
- depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on corollary 7.113 Inverse function theorem ¶ ↺
      - depends_on corollary 7.110 Variable limits of integration ¶ ↺
      - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on definition 30.4 Material and spatial descriptions ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 22.26 Lagrange bracket ¶
      - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
      - depends_on lemma A.504 Transitivity ¶ ↺
      - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
      - … 9 more
    - depends_on remark 7.103 rem:ana-differential-scalar ¶
  - depends_on definition 7.119 Envelope of a family ¶ ↺
  - depends_on definition A.70 Pullback ¶
    - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.43 Momentum map ¶
      - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
    - depends_on lemma A.75 Differentiating a pullback along a flow ¶ ↺
    - depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
  - depends_on definition 32.19 Lyapunov function ¶
    - depends_on theorem 32.20 Lyapunov's direct method ¶
      - depends_on example 32.21 The damped pendulum ¶
  - depends_on definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 30.4 Material and spatial descriptions ¶ ↺
    - depends_on definition 30.17 Traction ¶ ↺
    - depends_on example 30.3 How small a piece of copper is still a continuum ¶
    - depends_on remark 30.5 Where the hypothesis fails, and how one can tell ¶
  - depends_on definition 22.29 Poisson bracket ¶
    - depends_on definition 26.20 Dirac bracket ¶
      - depends_on definition 26.23 Gauge conditions ¶
      - depends_on example 26.22 Holonomic constraints ¶
      - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on definition 26.12 First and second class ¶
      - depends_on proposition 26.13 The first-class functions close ¶
      - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on proposition 26.34 Proca constraints are second class ¶
    - depends_on definition 26.8 Total Hamiltonian ¶
      - depends_on definition 26.16 Extended Hamiltonian ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
      - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
    - depends_on definition 25.11 Generators of the Galilei transformations ¶
      - depends_on proposition 25.12 The algebra of generators closes ¶
    - depends_on definition 25.6 Infinitesimal canonical transformation ¶ ↺
    - depends_on definition 25.2 Poisson algebra ¶
    - depends_on definition 24.36 Poisson manifold ¶
      - depends_on definition 25.2 Poisson algebra ¶ ↺
      - depends_on definition 24.37 Casimir function ¶
      - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
    - depends_on notation 26.28 Field systems ¶
      - depends_on definition 26.40 The $3+1$ decomposition ¶
      - depends_on notation A.636 Densities and smearing ¶
      - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
    - depends_on notation 26.1 Weak and strong equality ¶
      - depends_on notation 26.28 Field systems ¶ ↺
    - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
      - depends_on definition 24.37 Casimir function ¶ ↺
      - depends_on theorem 24.45 Noether, symplectic form ¶
  - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
- depends_on definition 18.11 Acceleration ¶
  - depends_on definition 18.27 Angular position, velocity and acceleration ¶
    - depends_on definition 18.29 Angular velocity vector ¶
      - depends_on remark 18.30 The angular velocity is an element of $\mathfrak{so}(3)$ ¶
    - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶
      - depends_on proposition 18.32 Tangential and centripetal components ¶
    - depends_on proposition 18.32 Tangential and centripetal components ¶ ↺
    - depends_on remark 18.28 The radian carries no dimension ¶
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
  - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶ ↺
  - depends_on phenomenon 18.12 Uniformly accelerated motion ¶
    - depends_on phenomenon 20.1 phen:parabolic-trajectory ¶
      - assumes experiment Absolute Ballistic Gravimetry (1995–present) ¶
      - depends_on theorem 20.2 Parabolic trajectory ¶
      - tests experiment Galileo's Inclined Planes ¶
  - depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶
    - depends_on remark 18.34 The two ways an acceleration can be zero ¶
  - depends_on proposition 18.23 Transformation of velocity and acceleration ¶
    - depends_on phenomenon 18.26 Composition of velocities, and its limit ¶
- depends_on definition 18.10 Velocity ¶
  - depends_on definition 18.11 Acceleration ¶ ↺
  - depends_on definition 18.27 Angular position, velocity and acceleration ¶ ↺
  - depends_on definition 18.29 Angular velocity vector ¶ ↺
  - depends_on definition 19.29 Kinetic energy ¶
    - depends_on definition 19.31 Energy ¶
      - depends_on proposition 27.20 Conservation of energy ¶
      - depends_on theorem 19.32 Conservation of energy ¶
    - depends_on theorem 19.38 Work–energy theorem ¶
  - depends_on definition 19.5 Linear momentum ¶
    - depends_on definition 27.2 Translational equilibrium ¶
    - depends_on definition 19.18 Angular momentum ¶
      - depends_on definition 27.3 Rotational equilibrium ¶
      - depends_on definition 19.55 Angular momentum of the centre of mass ¶
      - depends_on postulate 19.19 Newton's first law, rotational case ¶
      - depends_on postulate 19.22 Newton's second law, rotational case ¶
      - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶
    - depends_on definition 19.51 Linear momentum of the centre of mass ¶
      - depends_on definition 19.55 Angular momentum of the centre of mass ¶ ↺
    - depends_on postulate 19.6 Newton's first law ¶
      - depends_on phenomenon 19.7 Persistence of uniform motion ¶
      - depends_on postulate 19.19 Newton's first law, rotational case ¶ ↺
      - depends_on postulate 19.8 Newton's second law ¶
      - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶ ↺
    - depends_on postulate 19.8 Newton's second law ¶ ↺
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
  - depends_on proposition 18.23 Transformation of velocity and acceleration ¶ ↺
- 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 ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
depends_on definition 7.99 Differentiability at a point ¶
- depends_on definition 7.102 Differential of a function ¶ ↺
- depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
- depends_on proposition 7.104 Chain rule in several variables ¶ ↺
- 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 corollary 7.113 Inverse function theorem ¶ ↺
    - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
    - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶ ↺
    - depends_on remark 7.114 Where the proof is, and why ¶ ↺
  - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on corollary A.292 Inverse function theorem ¶ ↺
    - depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
      - depends_on example A.294 The sphere, made explicit ¶
    - depends_on lemma A.650 Flattening the constraints ¶
      - depends_on lemma A.651 Acyclicity of $\delta$ ¶
    - depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
    - depends_on theorem 23.6 Jacobi ¶
      - depends_on example 23.25 Central force in plane polar coordinates ¶
      - depends_on example 23.22 Free particle ¶
      - depends_on example 23.23 Uniform gravity, and the parabola ¶
      - depends_on proposition 23.7 The principal function is the action ¶
    - depends_on theorem 13.63 Constant rank theorem ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
      - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on theorem 13.67 Quotient manifold theorem ¶
    - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺
- depends_on theorem 7.112 Implicit function theorem ¶ ↺
- depends_on theorem A.286 Implicit function theorem ¶ ↺
depends_on definition 7.19 Infinite limits ¶
- depends_on lemma 7.67 The limit over real exponents ¶ ↺
depends_on definition 9.86 Outer and inner expansions; matching ¶
depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
- depends_on lemma 7.33 Fermat: interior extremum ¶ ↺
depends_on remark 7.21 rem:ana-discontinuities ¶ ↺
depends_on remark 7.17 rem:ana-limit-punctured ¶
depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	Absolute value	declared	parts/02-mathematical-methods/05-real-analysis.tex:229
`depends_on`	→	Real function	declared	parts/02-mathematical-methods/05-real-analysis.tex:229
`depends_on`	←	Continuity at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:336
`depends_on`	←	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:443
`depends_on`	←	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:2953
`depends_on`	←	Infinite limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:315
`depends_on`	←	Outer and inner expansions; matching	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3296
`depends_on`	←	Two-sided limit from one-sided limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:294
`depends_on`	←	rem:ana-discontinuities	declared	parts/02-mathematical-methods/05-real-analysis.tex:347
`depends_on`	←	rem:ana-limit-punctured	declared	parts/02-mathematical-methods/05-real-analysis.tex:239
`depends_on`	←	Cauchy mean value theorem and l'Hôpital's rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:715