proposition 44.17 Stress–energy of the electromagnetic field

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:746 · p. 1270

Rests on

depends_on definition 44.6 Stress–energy tensor ¶
- depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
  - depends_on equation 16.31 eq:calcvar-field-functional ¶
  - depends_on lemma 16.18 Fundamental lemma ¶
    - depends_on definition 16.15 Variation; the first variation ¶
      - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on equation 7.177 eq:ana-functional-derivative ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - proves proof ch:14-calculus-of-variations@proof-8 ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof ch:14-calculus-of-variations@proof-20 ¶
depends_on lemma 44.7 Variation of inverse and determinant ¶
- 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 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 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - 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 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-1 ¶
- proves proof ch:03-einstein-field-equations@proof-2 ¶
proves proof ch:03-einstein-field-equations@proof-8 ¶

Supports

depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶

Neighborhood

Every logical edge within two steps of this node.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Stress–energy tensor	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:766
`depends_on`	→	Variation of inverse and determinant	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:766
`depends_on`	←	The electromagnetic field satisfies NEC, WEC, SEC and DEC	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1569
`proves`	←	ch:03-einstein-field-equations@proof-8	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:769