proposition 10.19 An envelope of solutions is a solution

open in the book · parts/02-mathematical-methods/08-pdes.tex:564 · p. 345

Rests on

depends_on definition 10.18 Complete integral; envelope ¶
- depends_on equation 10.15 eq:pde-fully-nonlinear ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 7.99 Differentiability 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 definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - proves proof ch:05-real-analysis@proof-63 ¶
depends_on proposition 7.104 Chain rule in several variables ¶ ↺
proves proof ch:08-pdes@proof-7 ¶

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type	direction	node	provenance	where
`depends_on`	→	Complete integral; envelope	declared	parts/02-mathematical-methods/08-pdes.tex:568
`depends_on`	→	Chain rule in several variables	declared	parts/02-mathematical-methods/08-pdes.tex:568
`proves`	←	ch:08-pdes@proof-7	declared	parts/02-mathematical-methods/08-pdes.tex:571