definition A.476 Disc-type surfaces and the two functionals

open in the book · appendices/A-long-proofs.tex:23464 · p. 3027

ground object -- no derivation owed

Rests on

depends_on definition 13.22 First fundamental coefficients ¶
- depends_on definition 13.21 First fundamental form ¶
  - depends_on definition 13.11 Arc length ¶
    - depends_on definition 13.8 Curve ¶
    - depends_on definition 13.9 Tangent vector to a curve ¶
      - depends_on definition 13.8 Curve ¶ ↺
  - depends_on definition 13.18 Surface ¶
- depends_on definition 13.19 Parameter curves ¶
  - depends_on definition 13.8 Curve ¶ ↺
  - depends_on definition 13.18 Surface ¶ ↺
depends_on equation 16.60 eq:calcvar-dirichlet-integral ¶

Supports

depends_on definition A.481 Douglas' boundary functional ¶
- depends_on definition A.486 The normalised admissible class ¶
- depends_on lemma A.484 Lower semicontinuity of the Douglas functional ¶
- depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
- depends_on theorem A.482 Douglas' identity ¶
depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
- depends_on lemma A.487 The normalisation costs nothing ¶
  - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶ ↺
- depends_on lemma A.485 Conformal automorphisms of the disc ¶
  - depends_on definition A.486 The normalised admissible class ¶ ↺
  - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
depends_on lemma A.489 Courant–Lebesgue ¶
- depends_on theorem A.490 Equicontinuity of the normalised class ¶
depends_on proposition A.478 $D\ge A$, with equality exactly for conformal maps ¶

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`	→	First fundamental coefficients	declared	appendices/A-long-proofs.tex:23485
`depends_on`	→	eq:calcvar-dirichlet-integral	declared	appendices/A-long-proofs.tex:23485
`depends_on`	←	Douglas' boundary functional	declared	appendices/A-long-proofs.tex:23617
`depends_on`	←	Conformal invariance of the Dirichlet integral	declared	appendices/A-long-proofs.tex:23548
`depends_on`	←	Courant–Lebesgue	declared	appendices/A-long-proofs.tex:23982
`depends_on`	←	$D\ge A$, with equality exactly for conformal maps	declared	appendices/A-long-proofs.tex:23516