lemma A.479 Conformal invariance of the Dirichlet integral

open in the book · appendices/A-long-proofs.tex:23540 · p. 3028

Rests on

depends_on definition A.476 Disc-type surfaces and the two functionals ¶
- 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 13.18 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 ¶
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 definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- 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 definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - 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 app:A-long-proofs@proof-284 ¶

Supports

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 ¶ ↺

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`	→	Disc-type surfaces and the two functionals	declared	appendices/A-long-proofs.tex:23548
`depends_on`	→	Chain rule	declared	appendices/A-long-proofs.tex:23548
`depends_on`	←	The normalisation costs nothing	declared	appendices/A-long-proofs.tex:23882
`depends_on`	←	Conformal automorphisms of the disc	declared	appendices/A-long-proofs.tex:23796
`proves`	←	app:A-long-proofs@proof-284	declared	appendices/A-long-proofs.tex:23551