lemma 44.7 Variation of inverse and determinant

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:251 · p. 1266

Rests on

depends_on proposition 5.2 Jacobi's formula, column form ¶
- 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 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 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 proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.6 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 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - 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 5.14 Linear independence ¶
    - 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 ¶
proves proof ch:03-einstein-field-equations@proof-2 ¶

Supports

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 ¶

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`	→	Jacobi's formula, column form	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:263
`depends_on`	←	Stress–energy of the electromagnetic field	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:766
`depends_on`	←	Variation of the Einstein–Hilbert action	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:403
`proves`	←	ch:03-einstein-field-equations@proof-2	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:266