equation 5.19 eq:lin-leibniz-det

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• depends_on definition A.508 Primitive map ¶
  • depends_on lemma A.510 Every diffeomorphism factorises locally ¶
    • depends_on proposition A.511 The substitution property is universal ¶
      • depends_on remark A.516 The hypotheses of the global form ¶
      • 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.509 Primitive maps have the substitution property ¶
    • depends_on proposition A.511 The substitution property is universal ¶ ↺
• depends_on definition 5.58 Special linear group ¶
• depends_on definition 5.70 Characteristic polynomial ¶
  • depends_on definition 5.73 Algebraic and geometric multiplicity ¶
    • depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
  • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      • depends_on theorem 5.98 Primary decomposition ¶
        • depends_on corollary 5.99 Semisimple and nilpotent parts ¶
      • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
        • depends_on definition 30.20 Pressure and deviatoric stress ¶
        • depends_on proposition 30.9 Principal strains ¶
        • depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
        • depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
        • depends_on theorem 28.33 Normal modes ¶
        • depends_on theorem 29.16 Principal axes ¶
    • 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 5.84 Spectral theorem for a real symmetric operator ¶ ↺
    • depends_on theorem 9.26 Structure of the solutions ¶
      • depends_on corollary 9.27 Linear stability ¶
        • depends_on corollary 9.41 Hill's equation ¶
        • depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
        • depends_on proposition 9.29 The planar classification ¶
        • depends_on theorem 9.39 Floquet ¶
• depends_on lemma A.497 The determinant is multiplicative ¶
  • depends_on lemma A.504 Transitivity ¶
• depends_on lemma A.507 Coordinate permutations ¶
• depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
  • depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
  • depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    • depends_on lemma A.742 Isotropic representation ¶
      • depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
        • depends_on lemma A.748 The second-order structure function ¶
        • depends_on lemma A.747 The third-order structure functions ¶
        • depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
      • depends_on lemma A.747 The third-order structure functions ¶ ↺
      • depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
        • depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    • depends_on lemma A.671 The three products are independent ¶
      • depends_on proposition A.673 Cubic tensors of rank four ¶
        • depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    • depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      • depends_on phenomenon 30.28 Hooke's law ¶
        • depends_on phenomenon 30.32 Poisson contraction ¶
        • depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
    • depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
    • depends_on remark 30.27 Two results this chapter borrows from Part II ¶
    • depends_on remark 13.7 What the classification is used for ¶
• depends_on lemma 5.137 The alternating top form is unique up to scale ¶
  • depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    • depends_on corollary 22.24 Invariance of the phase-space volume ¶
      • depends_on remark 22.25 This is Liouville's theorem ¶
    • depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
    • depends_on proposition 24.5 Properties of the symplectic group ¶
      • depends_on remark 24.6 Connectedness, and what it rests on ¶
      • depends_on remark 24.7 Contrast with the orthogonal group ¶
• depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
• depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶ ↺
• depends_on proposition 5.2 Jacobi's formula, column form ¶
  • depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
    • depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
  • depends_on lemma 44.7 Variation of inverse and determinant ¶
    • depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      • depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
    • depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
      • depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶ ↺
      • depends_on proposition 44.30 The cosmological term ¶
        • depends_on proposition 44.31 de Sitter geometry ¶
  • depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
• depends_on proposition 13.6 Levi-Civita identities in three dimensions ¶
  • depends_on proposition 13.96 The cross product is an axial vector ¶
  • depends_on proposition 13.95 The elementary cross-product identities ¶