equation 7.14 eq:ana-derivh

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Supports

• depends_on proposition 7.28 Derivative of a constant function ¶
• depends_on proposition 7.29 Linearity ¶
  • 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 theorem 7.43 Fundamental theorem of calculus, II ¶
      • depends_on lemma 7.67 The limit over real exponents ¶
        • depends_on proposition 7.68 The natural base ¶
      • depends_on lemma 7.75 The first quadrant ¶
        • depends_on proposition 7.76 $\pi$ as the circle constant ¶
        • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
        • depends_on proposition 125.2 Crystallographic restriction ¶
      • depends_on lemma 7.78 Chord, arc, tangent ¶
        • depends_on proposition 7.79 The polygon recursion ¶
      • 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 7.108 Euler's theorem on homogeneous functions ¶
      • 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 9.19 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 theorem A.513 Change of variables ¶
    • depends_on lemma A.73 Differentiation under the integral sign ¶
      • depends_on lemma A.77 Poincaré lemma, converse form ¶
    • depends_on lemma A.287 The Newton map contracts ¶
      • depends_on lemma A.288 The iteration converges ¶
        • depends_on corollary A.289 The zero set is a graph ¶
        • depends_on lemma A.290 $h$ is Lipschitz ¶
    • depends_on lemma A.290 $h$ is Lipschitz ¶ ↺
    • 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 corollary A.445 Displacing 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 theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    • 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 definition 10.10 Characteristic surface ¶
        • depends_on definition 10.42 Green's function ¶
        • depends_on definition 10.6 Type of a second-order operator ¶
      • 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 proposition 22.27 Properties of the Lagrange bracket ¶
        • depends_on proposition 22.30 Properties of the Poisson bracket ¶
        • 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 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 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 proposition 30.38 The Cauchy relations ¶
        • depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
      • 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 definition 31.63 Circulation ¶
        • depends_on proposition 7.136 Properties of conservative fields ¶
      • 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 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 proposition 44.44 Harmonic-gauge reduction ¶
        • depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
        • depends_on proposition 13.140 Killing's equation ¶
        • depends_on proposition 13.162 Curvature induced on the quadric ¶
        • depends_on proposition 13.154 Symmetries of the curvature ¶
        • depends_on remark 30.16 Incompatibility is curvature ¶
        • depends_on theorem 13.156 Cartan structure equations ¶
      • … 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 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 lemma A.650 Flattening the constraints ¶
        • 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 theorem 13.63 Constant rank theorem ¶
        • depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
    • … 2 more
• depends_on proposition 7.30 Leibniz 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 proposition 16.6 The isoperimetric extremal is a circle ¶
        • depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      • depends_on theorem 16.22 Euler–Lagrange ¶
        • depends_on definition 16.62 Field of extremals; slope function ¶
        • depends_on phenomenon 30.52 Euler buckling ¶
        • depends_on proposition 29.39 Steady precession ¶
        • depends_on theorem 16.35 System of Euler–Lagrange equations ¶ ↺
        • depends_on theorem 16.31 Natural boundary condition ¶
      • depends_on theorem 16.43 Euler's rule for integral constraints ¶
        • depends_on example 16.45 Two solved isoperimetric problems ¶
        • depends_on proposition 16.4 The hanging chain is a catenary ¶
        • depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺
    • depends_on lemma 9.136 Wallis integrals ¶
      • 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 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 example 30.58 A steel ruler ¶
        • depends_on proposition 30.57 Bending waves are dispersive ¶
      • depends_on theorem 30.64 The Kirchhoff plate equation ¶
        • depends_on phenomenon 30.65 Chladni figures ¶
        • depends_on proposition 30.66 Chladni scaling ¶
        • depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
  • 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 proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    • depends_on lemma 44.7 Variation of inverse and determinant ¶
      • depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
        • depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
      • depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
        • depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
        • depends_on proposition 44.30 The cosmological term ¶
    • 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 proposition 32.17 Classification of planar fixed points ¶
        • depends_on example 32.18 A linear centre that is really a stable focus ¶
        • depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
        • depends_on definition 32.22 Bifurcation ¶
        • depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
      • depends_on theorem 32.13 Linear stability ¶
        • depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
        • depends_on proposition 32.23 Saddle-node bifurcation ¶
        • depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
        • depends_on proposition 28.69 Which branch is stable, and where it ends ¶
    • depends_on lemma A.754 Exact depth expansion ¶
      • depends_on proposition A.755 The Boussinesq pair ¶
        • depends_on theorem A.756 The Korteweg–de Vries equation ¶
    • 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 definition A.141 Mollification ¶
        • depends_on lemma A.140 Properties of the 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 theorem A.490 Equicontinuity of the normalised class ¶
      • depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
        • depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
        • depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
        • depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
      • 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.212 Second-order flatness ¶
      • depends_on lemma A.213 The three minima ¶
        • depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
    • 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.213 The three minima ¶ ↺
      • depends_on lemma A.211 Root-$n$ localisation ¶
        • depends_on lemma A.213 The three minima ¶ ↺
    • 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 remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
        • depends_on remark 28.26 Causality is visible in the susceptibility ¶
        • depends_on remark 28.62 Neither velocity is a speed limit ¶
      • depends_on theorem 17.93 Fundamental strip and holomorphy ¶
        • depends_on theorem 17.94 Mellin inversion ¶
    • depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
    • … 2 more