proposition 5.2 Jacobi's formula, column form

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:416 · p. 101

Rests on

depends_on equation 5.19 eq:lin-leibniz-det ¶
depends_on proposition 7.30 Leibniz rule ¶
- depends_on definition 7.12 Product of functions ¶
  - 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 equation 7.14 eq:ana-derivh ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - 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 ¶
  - 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 definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-14 ¶
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 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶ ↺
  - 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 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on lemma 5.38 Exchange and completion ¶
    - depends_on definition 5.15 Basis ¶ ↺
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
    - depends_on definition 5.14 Linear independence ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-11 ¶
  - 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-1 ¶

Supports

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 ¶

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`	→	eq:lin-leibniz-det	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:430
`depends_on`	→	Leibniz rule	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:430
`depends_on`	→	Injective, surjective, invertible	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:430
`depends_on`	←	Divergence theorem on $(M,g)$	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:467
`depends_on`	←	Variation of inverse and determinant	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:263
`depends_on`	←	Jacobi's formula, cofactor form	declared	parts/02-mathematical-methods/05-real-analysis.tex:3440
`proves`	←	ch:03-linear-algebra-representations@proof-1	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:433