definition 7.26 Derivative of a function at a point

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:429 · p. 208

ground object -- no derivation owed

Rests on

depends_on definition 7.16 Limit ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.9 Real function ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
depends_on definition 7.9 Real function ¶ ↺

Supports

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 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 corollary 16.23 du Bois-Reymond form ¶
    - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
  - 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 theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
  - depends_on lemma A.520 A $C^{1}$ limit ¶
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.524 Rate of decay ¶
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
  - depends_on lemma A.172 Dirichlet ¶
    - depends_on lemma A.193 A cosine integral ¶
      - depends_on lemma A.194 The Fejér kernel and its transform ¶
    - depends_on theorem A.177 Fourier inversion for a distribution function ¶
      - depends_on corollary A.178 The characteristic function determines the distribution ¶
      - depends_on lemma A.196 Inversion for a difference ¶
  - 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 corollary A.723 Potential of one infinite row ¶
      - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - 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 definition 28.25 Complex susceptibility ¶
      - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
  - 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 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 ¶ ↺
  - … 8 more
- 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 lemma A.549 The action is transitive ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶ ↺
      - depends_on proposition A.535 Existence of a slice ¶
      - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on remark 22.19 Which form exists ¶
      - depends_on theorem 13.63 Constant rank 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 lemma 31.11 Transport theorem for a material volume ¶ ↺
    - 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 proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - 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 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 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
  - 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 proposition 26.24 The Faddeev–Popov determinant ¶
    - 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 postulate 26.47 Dirac quantization ¶
  - depends_on definition 26.12 First and second class ¶
    - depends_on proposition 26.13 The first-class functions close ¶
      - depends_on proposition 26.43 The hypersurface-deformation algebra ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
    - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.20 Dirac bracket ¶ ↺
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
    - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 113.3 Degrees of freedom before and after ¶
  - 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 26.16 Extended Hamiltonian ¶ ↺
      - depends_on postulate 26.47 Dirac quantization ¶ ↺
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶ ↺
  - depends_on definition 25.11 Generators of the Galilei transformations ¶ ↺
  - 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 proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
    - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶ ↺
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶ ↺
    - 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 lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on proposition 26.41 The ADM form of the Einstein–Hilbert Lagrangian ¶
    - depends_on notation A.636 Densities and smearing ¶
      - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
    - depends_on notation 26.36 Gauge fields and their SI dimensions ¶
      - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶ ↺
  - 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 ¶
      - depends_on corollary 20.3 Range, apex, time of flight ¶
      - depends_on phenomenon 20.6 Air drag ¶
      - depends_on remark 20.5 rem:projectile-idealisation ¶
    - 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 phenomenon 50.6 No material surface ¶
      - depends_on phenomenon 19.33 Mechanical energy is conserved ¶
      - depends_on theorem 21.43 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 proposition 29.14 Angular momentum and kinetic energy ¶
      - depends_on theorem 19.58 Decomposition of the angular momentum ¶
    - depends_on postulate 19.19 Newton's first law, rotational case ¶
      - depends_on phenomenon 19.21 Equal areas in equal times ¶
      - depends_on proposition 20.13 Eastward deflection of a body dropped from rest ¶
    - depends_on postulate 19.22 Newton's second law, rotational case ¶
      - depends_on theorem 29.25 Euler's equations ¶
    - 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 definition 19.30 Conservative force and potential energy ¶
      - depends_on definition 19.34 Work ¶
      - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
      - depends_on phenomenon 42.1 Universality of free fall ¶
      - depends_on phenomenon 19.10 Force, mass and acceleration ¶
      - depends_on postulate 19.22 Newton's second law, rotational case ¶ ↺
      - depends_on postulate 19.48 Newton's third law ¶
      - depends_on theorem 20.2 Parabolic trajectory ¶ ↺
      - depends_on theorem 29.51 Equation of motion in a rotating frame ¶
    - 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 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 definition 7.41 Antiderivative ¶ ↺
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on lemma 7.75 The first quadrant ¶
      - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
      - depends_on proposition 7.61 The logarithm ¶
      - depends_on proposition 9.19 Separation of variables ¶
      - depends_on theorem 7.115 Constant rank ¶ ↺
    - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
    - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on lemma A.512 The boundary strip is thin ¶ ↺
    - depends_on lemma A.73 Differentiation under the integral sign ¶ ↺
    - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.288 The iteration converges ¶
    - depends_on lemma A.290 $h$ is Lipschitz ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
    - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on lemma A.226 The level factor ¶
    - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.470 $T$ is compact ¶
    - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶ ↺
      - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on theorem 7.132 Stokes ¶ ↺
      - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - … 2 more
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
      - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - … 2 more
  - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
    - depends_on definition 32.12 Linearization ¶
      - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
      - depends_on theorem 32.13 Linear stability ¶
    - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on proposition A.755 The Boussinesq pair ¶
    - depends_on lemma A.187 Exponentiation ¶
    - depends_on lemma A.138 The flat exponential ¶
      - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.213 The three minima ¶
    - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.211 Root-$n$ localisation ¶
    - depends_on phenomenon 28.13 Universality of small oscillations ¶
    - depends_on proposition 16.50 The second variation ¶ ↺
    - depends_on proposition 17.67 Holomorphy ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - … 2 more
- depends_on theorem 16.104 Snell's law of refraction ¶
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 lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
  - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.91 The golden ratio ¶
    - depends_on lemma 7.67 The limit over real exponents ¶ ↺
    - depends_on proposition 7.68 The natural base ¶
      - depends_on remark 7.70 What the condition says ¶
    - depends_on proposition 7.91 The golden ratio ¶ ↺
    - depends_on proposition 7.89 Irrationality of square roots ¶
      - depends_on proposition 7.91 The golden ratio ¶ ↺
  - depends_on proposition 16.26 Covariance under change of dependent variable ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶ ↺
  - depends_on theorem 16.29 Beltrami identity ¶
    - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
    - depends_on proposition 16.39 Ostrogradsky instability ¶
  - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on lemma A.90 Affine normalization ¶
      - depends_on lemma A.91 Smallness after rescaling ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert's formula ¶
    - depends_on proposition 10.11 What a characteristic surface is ¶
      - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶ ↺
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶ ↺
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.61 The logarithm ¶ ↺
- depends_on proposition 7.30 Leibniz rule ¶
  - depends_on corollary 7.44 Substitution and integration by parts ¶ ↺
  - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶ ↺
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
  - depends_on proposition 7.122 Product identities of the nabla calculus ¶
  - depends_on proposition 5.2 Jacobi's formula, column form ¶
    - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶ ↺
    - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
  - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
    - depends_on 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 corollary 7.81 The decimal expansion of $\pi$ ¶
      - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.82 What Archimedes proved ¶
      - depends_on remark 7.80 Why these are the polygon perimeters ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
      - depends_on proposition 7.83 Viète's product ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
    - 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 proposition 9.143 A cubic quadrature between two turning points ¶ ↺
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
  - depends_on lemma 7.75 The first quadrant ¶ ↺
  - depends_on proposition 9.22 The exponential and its derivative ¶
    - depends_on 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 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 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 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 9.19 Separation of variables ¶ ↺
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
- depends_on theorem 10.52 Duhamel's principle ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:443
`depends_on`	→	Real function	declared	parts/02-mathematical-methods/05-real-analysis.tex:443
`depends_on`	←	Antiderivative	declared	parts/02-mathematical-methods/05-real-analysis.tex:823
`depends_on`	←	Partial derivative; gradient	declared	parts/02-mathematical-methods/05-real-analysis.tex:2915
`depends_on`	←	Acceleration	declared	parts/03-classical-mechanics/01-kinematics.tex:283
`depends_on`	←	Velocity	declared	parts/03-classical-mechanics/01-kinematics.tex:261
`depends_on`	←	Fermat: interior extremum	declared	parts/02-mathematical-methods/05-real-analysis.tex:647
`depends_on`	←	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:458
`depends_on`	←	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/05-real-analysis.tex:834