definition 7.9 Real function

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:152 · p. 204

ground object -- no derivation owed

Rests on

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 7.15 Absolute-value function ¶
depends_on definition 7.13 Composite function ¶
- depends_on definition 7.14 Inverse function ¶
  - 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 corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on definition 7.62 Real powers ¶
      - 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 9.141 First properties ¶ ↺
    - depends_on proposition 9.19 Separation of variables ¶
- depends_on proposition 7.31 Chain rule ¶
  - depends_on corollary 7.44 Substitution and integration by parts ¶
    - depends_on definition 8.8 Contour integral ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶
    - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
    - depends_on proposition 16.50 The second variation ¶
    - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
    - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 30.56 The beam equation ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
  - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on proposition 16.26 Covariance under change of dependent variable ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
    - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
  - depends_on theorem 16.29 Beltrami identity ¶
    - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
    - depends_on proposition 16.39 Ostrogradsky instability ¶
  - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on lemma A.90 Affine normalization ¶
      - depends_on lemma A.91 Smallness after rescaling ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert's formula ¶
    - depends_on proposition 10.11 What a characteristic surface is ¶
      - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - 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 corollary 7.110 Variable limits of integration ¶
  - depends_on corollary A.292 Inverse function theorem ¶
    - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on definition A.309 The induced orientation of the boundary ¶
    - depends_on proposition A.535 Existence of a slice ¶
      - depends_on lemma A.536 A slice is a chart domain downstairs ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
    - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on remark 22.33 What the duality is good for ¶
    - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on definition 26.3 Singular Lagrangian ¶
      - depends_on remark 22.4 When the Legendre map is singular ¶
    - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.133 Frobenius ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on remark 22.19 Which form exists ¶
    - 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 definition 30.4 Material and spatial descriptions ¶
  - depends_on definition 31.4 Material derivative ¶
    - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on proposition 31.7 The three coincide in steady flow ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
  - depends_on definition 22.26 Lagrange bracket ¶
  - depends_on definition 10.18 Complete integral; envelope ¶
    - depends_on proposition 10.19 An envelope of solutions is a solution ¶
  - 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 lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
  - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on definition A.715 Stokes stream function ¶
    - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on example 31.29 Flow past a sphere ¶
      - depends_on lemma A.777 The tangential field above a bending plate ¶
      - depends_on lemma A.733 The circle with incidence and circulation ¶
      - depends_on proposition A.797 The outer limit loses a boundary condition ¶
      - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
      - depends_on theorem A.707 D'Alembert's paradox ¶
      - depends_on theorem 31.74 The interface dispersion relation ¶
    - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
  - depends_on proposition 7.120 The envelope touches every member it meets ¶
  - … 9 more
depends_on definition 7.10 Constant function ¶
- depends_on proposition 7.28 Derivative of a constant function ¶
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 proposition A.521 Every derivative passes onto $g$ ¶
    - 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 remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - 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 corollary 7.113 Inverse function theorem ¶ ↺
    - 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 definition 7.127 Simple regions ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 7.131 Green ¶
    - 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 definition A.116 Orthogonal curvilinear coordinates; scale factors ¶
      - depends_on definition A.125 Ellipsoidal coordinates ¶
      - depends_on definition A.118 Simple separation of the Helmholtz equation ¶
      - 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 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 16.15 Variation; the first variation ¶
      - 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 10.2 Linear, semilinear, quasilinear ¶
      - depends_on definition 10.24 Well-posed problem ¶
    - depends_on definition 10.66 Harmonic function ¶
      - depends_on theorem A.135 Converse of the mean-value property ¶
      - depends_on theorem A.491 Dirichlet problem for the disc; quoted ¶
      - depends_on theorem 10.69 Mean-value property ¶
      - depends_on theorem 10.75 Weak maximum principle ¶
    - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
    - … 8 more
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    - 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.77 Poincaré lemma, converse form ¶
    - 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 lemma A.77 Poincaré lemma, converse form ¶ ↺
- 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 16.22 Euler–Lagrange ¶ ↺
    - depends_on theorem 16.51 Legendre's necessary condition ¶
      - depends_on corollary 16.65 Sufficiency for a weak minimum ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on lemma A.306 The standard smooth bump ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
      - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶
      - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on lemma A.226 The level factor ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - … 2 more
    - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      - depends_on 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 16.104 Snell's law of refraction ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on proposition 7.31 Chain rule ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.61 The logarithm ¶ ↺
  - 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 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 theorem 9.23 Solution of a constant-coefficient system ¶
- 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.463 Poincaré inequality; $B_{K}$ is an inner product ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - depends_on lemma A.452 Young and Hölder ¶ ↺
  - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
  - depends_on lemma A.440 Grönwall's inequality ¶ ↺
  - depends_on lemma 16.19 du Bois-Reymond ¶
    - depends_on lemma 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 11.27 Probability integral transform ¶
  - 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 9.88 The model problem, and the error of the composite ¶ ↺
  - depends_on proposition 9.19 Separation of variables ¶ ↺
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
  - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on corollary 10.53 Duhamel for the wave equation ¶
depends_on definition 7.16 Limit ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶ ↺
  - 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 9.22 The exponential and its derivative ¶ ↺
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on corollary 32.5 Trajectories do not cross ¶
      - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on proposition 32.4 The flow is a one-parameter group ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶ ↺
      - depends_on proposition 20.7 Terminal speed and the approach to it ¶
      - depends_on proposition 9.11 Continuous dependence on the initial data ¶ ↺
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - depends_on theorem 9.34 Poincaré–Bendixson; quoted ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶ ↺
  - depends_on proposition 7.145 Delta as a limit; elementary properties ¶ ↺
  - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
  - 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 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 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
      - depends_on lemma A.499 What zero content buys ¶
      - depends_on lemma A.512 The boundary strip is thin ¶ ↺
      - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
      - depends_on lemma A.176 Helly–Bray ¶
      - depends_on lemma A.488 Small chords cut off small arcs ¶
      - 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 ¶
      - … 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 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 7.99 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 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 theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶ ↺
- 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 ¶ ↺
depends_on definition 7.12 Product of functions ¶
- depends_on proposition 7.30 Leibniz rule ¶ ↺
depends_on definition 7.11 Sum of functions ¶
- depends_on proposition 7.29 Linearity ¶
  - depends_on theorem 7.35 Mean value theorem ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Map	declared	parts/02-mathematical-methods/05-real-analysis.tex:156
`depends_on`	←	Absolute-value function	declared	parts/02-mathematical-methods/05-real-analysis.tex:211
`depends_on`	←	Composite function	declared	parts/02-mathematical-methods/05-real-analysis.tex:195
`depends_on`	←	Constant function	declared	parts/02-mathematical-methods/05-real-analysis.tex:165
`depends_on`	←	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:443
`depends_on`	←	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:229
`depends_on`	←	Product of functions	declared	parts/02-mathematical-methods/05-real-analysis.tex:185
`depends_on`	←	Sum of functions	declared	parts/02-mathematical-methods/05-real-analysis.tex:175