lemma 7.33 Fermat: interior extremum

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:644 · p. 214

Rests on

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 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 ¶

Supports

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 definition 16.62 Field of extremals; slope function ¶
    - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.53 A caustic point is a conjugate point ¶
    - depends_on remark 23.8 The action as a function, not a functional ¶
    - 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 16.64 Weierstrass' sufficient condition ¶ ↺
  - depends_on phenomenon 30.52 Euler buckling ¶
    - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
    - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
    - depends_on remark 30.55 Beyond the critical load: the elastica ¶
    - depends_on remark 30.68 Shells carry load in a different way ¶
  - depends_on proposition 29.39 Steady precession ¶
  - 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 proposition 21.56 The two prescriptions and their difference ¶
  - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
    - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on remark A.692 The corner force is real, and you can feel it ¶
    - depends_on theorem 16.32 Transversality ¶
      - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
- 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 proposition 23.53 A caustic point is a conjugate point ¶ ↺
    - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶ ↺
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 lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
- 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 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
depends_on theorem 16.104 Snell's law of refraction ¶

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`	→	Derivative of a function at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:647
`depends_on`	→	Two-sided limit from one-sided limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:647
`depends_on`	←	Lagrange multipliers in finitely many variables	declared	parts/02-mathematical-methods/05-real-analysis.tex:3752
`depends_on`	←	Stationarity is necessary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	←	Rolle	declared	parts/02-mathematical-methods/05-real-analysis.tex:661
`depends_on`	←	Snell's law of refraction	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3299
`proves`	←	ch:05-real-analysis@proof-17	declared	parts/02-mathematical-methods/05-real-analysis.tex:650