proposition 16.105 The optical invariant in a stratified medium

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:3333 · p. 668

Rests on

depends_on equation 16.89 eq:calcvar-optical-length ¶
depends_on theorem 16.29 Beltrami identity ¶
- depends_on equation 16.20 eq:calcvar-euler-lagrange ¶
- depends_on proposition 7.31 Chain rule ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - 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-15 ¶
- proves proof ch:14-calculus-of-variations@proof-15 ¶
proves proof ch:14-calculus-of-variations@proof-50 ¶

Supports

Nothing declares a dependency on this node yet.

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:calcvar-optical-length	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3341
`depends_on`	→	Beltrami identity	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3341
`proves`	←	ch:14-calculus-of-variations@proof-50	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3344