definition 10.18 Complete integral; envelope

open in the book · parts/02-mathematical-methods/08-pdes.tex:541 · p. 345

ground object -- no derivation owed

Rests on

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 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - 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:05-real-analysis@proof-63 ¶

Supports

depends_on proposition 10.19 An envelope of solutions is a solution ¶

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:pde-fully-nonlinear	declared	parts/02-mathematical-methods/08-pdes.tex:561
`depends_on`	→	Chain rule in several variables	declared	parts/02-mathematical-methods/08-pdes.tex:561
`depends_on`	←	An envelope of solutions is a solution	declared	parts/02-mathematical-methods/08-pdes.tex:568