proposition 125.2 Crystallographic restriction

open in the book · parts/12-statistical-mechanics/08-phonons-lattices.tex:113 · p. 2660

Rests on

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 theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.4 Convergence ¶
    - 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 definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.9 Real function ¶
    - 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 ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - 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 definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - 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 corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - 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 definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - 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 corollary 7.36 cor:ana-mvt-consequences ¶ ↺
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on equation 7.42 eq:ana-exp-derivative ¶
      - depends_on lemma 7.60 Functional equation of the exponential ¶
      - depends_on proposition 7.32 Derivative of the inverse function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-39 ¶
  - 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 definition 7.13 Composite function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-15 ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on lemma 7.64 Well-ordering of $\N$ ¶
      - depends_on axiom 3.33 Peano axioms ¶ ↺
      - depends_on axiom 3.34 Principle of induction ¶
      - proves proof ch:05-real-analysis@proof-41 ¶
    - 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 ¶
      - depends_on definition 18.4 Body ¶
      - depends_on definition 19.3 Mass ¶
  - 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 ¶

Supports

depends_on remark 125.3 What the theorem does and does not forbid ¶

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`	→	$\pi$	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	Derivatives; the Pythagorean identity	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	The first quadrant	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	Addition theorems	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	The golden ratio	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	Special values, periodicity, and the kernel	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	→	Euler's rotation theorem	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	←	What the theorem does and does not forbid	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:212
`proves`	←	ch:08-phonons-lattices@proof-1	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:132