remark 125.3 What the theorem does and does not forbid

open in the book · parts/12-statistical-mechanics/08-phonons-lattices.tex:199 · p. 2661

remark -- no derivation owed by its kind

Rests on

depends_on proposition 125.2 Crystallographic restriction ¶
- depends_on definition 7.74 $\pi$ ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on lemma 7.73 $\cos 2 < 0$ ¶
    - depends_on equation 7.53 eq:ana-cos-sin-series ¶
    - depends_on proposition 7.52 Alternating series test ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - proves proof ch:05-real-analysis@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-48 ¶
  - depends_on theorem 7.23 Intermediate value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - 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 ¶
    - proves proof ch:05-real-analysis@proof-8 ¶
  - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶ ↺
      - proves proof ch:05-real-analysis@proof-29 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - proves proof ch:05-real-analysis@proof-31 ¶
    - proves proof ch:05-real-analysis@proof-32 ¶
- depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
  - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
  - depends_on equation 5.3 eq:lin-binomial ¶
  - depends_on proposition 7.49 Cauchy product ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-30 ¶
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
  - proves proof ch:05-real-analysis@proof-46 ¶
- depends_on lemma 7.75 The first quadrant ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.74 $\pi$ ¶ ↺
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
  - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
  - proves proof ch:05-real-analysis@proof-49 ¶
- depends_on lemma 7.72 Addition theorems ¶
  - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
  - depends_on equation 5.3 eq:lin-binomial ¶ ↺
  - depends_on proposition 7.49 Cauchy product ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-47 ¶
- depends_on proposition 7.91 The golden ratio ¶
  - depends_on definition 7.90 The golden ratio ¶
    - depends_on definition 7.62 Real powers ¶
      - depends_on proposition 7.61 The logarithm ¶
    - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on definition 7.62 Real powers ¶ ↺
      - depends_on equation 7.42 eq:ana-exp-derivative ¶
      - depends_on lemma 7.60 Functional equation of the exponential ¶
      - depends_on proposition 7.31 Chain rule ¶
      - depends_on proposition 7.61 The logarithm ¶ ↺
      - proves proof ch:05-real-analysis@proof-40 ¶
  - depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - proves proof ch:05-real-analysis@proof-57 ¶
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on definition 7.62 Real powers ¶ ↺
    - depends_on lemma 7.65 The integer part ¶
      - depends_on axiom 3.33 Peano axioms ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on lemma 7.64 Well-ordering of $\N$ ¶
      - proves proof ch:05-real-analysis@proof-42 ¶
    - depends_on lemma 7.64 Well-ordering of $\N$ ¶ ↺
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
    - proves proof ch:05-real-analysis@proof-58 ¶
  - proves proof ch:05-real-analysis@proof-59 ¶
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
  - depends_on definition 7.74 $\pi$ ¶ ↺
  - depends_on equation 7.58 eq:ana-half-angle ¶
  - depends_on lemma 7.65 The integer part ¶ ↺
  - depends_on lemma 7.75 The first quadrant ¶ ↺
  - depends_on lemma 7.72 Addition theorems ¶ ↺
  - proves proof ch:05-real-analysis@proof-51 ¶
- depends_on theorem 29.3 Euler's rotation theorem ¶
  - depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
    - depends_on equation 14.28 eq:lie-expso2 ¶
    - depends_on equation 14.25 eq:lie-so2-matrix ¶
    - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
      - depends_on definition 14.26 The rotation group ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
  - depends_on proposition 29.2 Six degrees of freedom ¶
    - depends_on definition 21.9 Degrees of freedom ¶
    - depends_on definition 29.1 Rigid body ¶
      - depends_on definition 21.2 Holonomic constraint ¶
      - depends_on definition 19.39 Body ¶
    - proves proof ch:12-rigid-body-rotating-frames@proof-1 ¶
  - proves proof ch:12-rigid-body-rotating-frames@proof-2 ¶
- proves proof ch:08-phonons-lattices@proof-1 ¶

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type	direction	node	provenance	where
`cites`	→	Metallic Phase with Long-Range Orientational Order and No Translational Symmetry	derived	parts/12-statistical-mechanics/08-phonons-lattices.tex:208
`depends_on`	→	Crystallographic restriction	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:212
`owes`	→	ch:08-phonons-lattices@pending-2	derived	parts/12-statistical-mechanics/08-phonons-lattices.tex:215