theorem 7.34 Rolle

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:658 · p. 214

Rests on

depends_on lemma 7.33 Fermat: interior extremum ¶
- depends_on definition 7.26 Derivative of a function at a point ¶
  - 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 7.9 Real function ¶ ↺
- depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
  - depends_on definition 7.16 Limit ¶ ↺
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
  - proves proof ch:05-real-analysis@proof-6 ¶
- proves proof ch:05-real-analysis@proof-17 ¶
depends_on theorem 7.24 Extreme value theorem ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on proposition 7.22 Sequential characterization ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
    - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-7 ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶ ↺
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- proves proof ch:05-real-analysis@proof-9 ¶
proves proof ch:05-real-analysis@proof-18 ¶

Supports

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 lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on definition A.308 Integral over the manifold ¶
      - depends_on theorem A.311 General Stokes theorem ¶
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 corollary 7.44 Substitution and integration by parts ¶
      - depends_on corollary 16.23 du Bois-Reymond form ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on lemma A.520 A $C^{1}$ limit ¶
      - depends_on lemma A.172 Dirichlet ¶
      - 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 proposition 17.41 The identities physics uses ¶
      - depends_on proposition 17.73 Initial- and final-value theorems ¶
      - … 4 more
  - 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 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 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 7.78 Chord, arc, tangent ¶ ↺
  - 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 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 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 corollary A.514 Degeneracy on a negligible set ¶ ↺
      - depends_on remark A.516 The hypotheses of the global form ¶
- 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.291 $h$ is differentiable, with the stated derivative ¶
- 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 A.472 The pairing identity ¶
- 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 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 remark A.527 The hypotheses, and what happens without them ¶ ↺
  - 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 definition 31.63 Circulation ¶
      - depends_on theorem 31.64 Kelvin's circulation theorem ¶
    - 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 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 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 definition A.503 The substitution property ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶ ↺
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - 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 theorem 7.115 Constant rank ¶ ↺
    - 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 7.118 The multiplier rule and its functional counterpart ¶
    - 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 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 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 ¶
- … 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 example 32.18 A linear centre that is really a stable focus ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶ ↺
    - 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 proposition A.143 Reproduction identity ¶
    - depends_on lemma A.140 Properties of the mollifier ¶
      - depends_on proposition A.143 Reproduction identity ¶ ↺
      - depends_on proposition A.142 The mollification is smooth ¶
  - 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 definition A.467 Green function of the shifted problem ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - 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 theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
- 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 17.96 Poles give asymptotics ¶
      - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
- depends_on proposition 9.88 The model problem, and the error of the composite ¶
- … 2 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Fermat: interior extremum	declared	parts/02-mathematical-methods/05-real-analysis.tex:661
`depends_on`	→	Extreme value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:661
`depends_on`	←	Cauchy mean value theorem and l'Hôpital's rule	declared	parts/02-mathematical-methods/05-real-analysis.tex:715
`depends_on`	←	Mean value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:678
`depends_on`	←	Taylor's theorem with Lagrange remainder	declared	parts/02-mathematical-methods/05-real-analysis.tex:742
`proves`	←	ch:05-real-analysis@proof-18	declared	parts/02-mathematical-methods/05-real-analysis.tex:664