remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric

Rests on

• depends_on remark 5.134 What the theorem buys, and what it does not ¶
• depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  • depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    • depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      • depends_on definition 4.48 Symmetric group ¶
        • depends_on definition 3.47 Bijective map ¶
      • depends_on definition 5.18 Inner product ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.25 Unit vector ¶
    • depends_on equation 5.19 eq:lin-leibniz-det ¶
    • proves proof ch:03-linear-algebra-representations@proof-8 ¶
  • depends_on lemma 5.132 Parity constraint ¶
    • depends_on definition 5.131 Isotropic Cartesian tensor ¶
      • depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
      • depends_on equation 5.162 eq:rep-special-orthogonal ¶
    • depends_on equation 5.162 eq:rep-special-orthogonal ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-57 ¶
  • depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
    • depends_on definition 5.18 Inner product ¶ ↺
    • depends_on equation 5.44 eq:lin-norm-assoc ¶
    • depends_on proposition 5.49 Injective, surjective, invertible ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
      • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on definition 5.47 Inverse of a linear transformation ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-17 ¶
      • depends_on theorem 5.40 Rank–nullity ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on lemma 5.38 Exchange and completion ¶
        • proves proof ch:03-linear-algebra-representations@proof-13 ¶
      • proves proof ch:03-linear-algebra-representations@proof-18 ¶
    • proves proof ch:03-linear-algebra-representations@proof-56 ¶
  • proves proof ch:03-linear-algebra-representations@proof-58 ¶

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