lemma A.777 The tangential field above a bending plate

open in the book · appendices/A-long-proofs.tex:38153 · p. 3177

Rests on

depends_on equation A.1292 eq:app-acoustic-streaming-delta ¶
depends_on equation 31.25 eq:fluid-laplace ¶
depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
- depends_on corollary 31.13 Incompressible flow ¶
  - depends_on theorem 31.12 Conservation of mass ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on definition 31.4 Material derivative ¶
      - depends_on theorem 7.133 Gauss ¶
      - proves proof ch:14-fluid-dynamics@proof-4 ¶
    - proves proof ch:14-fluid-dynamics@proof-5 ¶
  - proves proof ch:14-fluid-dynamics@proof-6 ¶
- depends_on equation 10.24 eq:pde-poisson-equation ¶
- depends_on proposition 7.136 Properties of conservative fields ¶
  - depends_on definition 7.135 Conservative field ¶
    - depends_on equation 7.124 eq:ana-nabla-gradient ¶
  - 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 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - 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 ¶
    - proves proof ch:05-real-analysis@proof-63 ¶
  - depends_on theorem 7.132 Stokes ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-64 ¶
    - depends_on theorem 7.131 Green ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-77 ¶
    - proves proof ch:05-real-analysis@proof-78 ¶
  - proves proof ch:05-real-analysis@proof-80 ¶
- proves proof ch:14-fluid-dynamics@proof-14 ¶
proves proof app:A-long-proofs@proof-452 ¶

Supports

depends_on corollary A.778 The powder collects at the antinodes ¶

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:app-acoustic-streaming-delta	declared	appendices/A-long-proofs.tex:38173
`depends_on`	→	eq:fluid-laplace	declared	appendices/A-long-proofs.tex:38173
`depends_on`	→	Potential flow reduces to Laplace's equation	declared	appendices/A-long-proofs.tex:38173
`depends_on`	←	The powder collects at the antinodes	declared	appendices/A-long-proofs.tex:38222
`proves`	←	app:A-long-proofs@proof-452	declared	appendices/A-long-proofs.tex:38177