definition 13.11 Arc length

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:666 · p. 457

ground object -- no derivation owed

Rests on

depends_on definition 13.8 Curve ¶
depends_on definition 13.9 Tangent vector to a curve ¶
- depends_on definition 13.8 Curve ¶ ↺

Supports

depends_on definition 13.21 First fundamental form ¶
- depends_on definition 13.22 First fundamental coefficients ¶
  - depends_on definition A.476 Disc-type surfaces and the two functionals ¶
    - 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 lemma A.485 Conformal automorphisms of the disc ¶
    - 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 ¶
  - depends_on definition 13.29 Gaussian curvature ¶
    - depends_on theorem 13.30 Theorema Egregium; Gauss, 1827 ¶
      - depends_on corollary 13.31 Bending invariance ¶
  - depends_on definition 13.23 Inverse metric tensor ¶
    - depends_on definition 13.26 Christoffel symbols ¶
      - depends_on definition 13.37 Geodesic lines ¶
      - depends_on definition 13.27 Riemann symbols ¶
- depends_on proposition 13.40 Parallel transport is an isometry of the tangent plane ¶
  - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
    - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
depends_on definition 13.12 Unit tangent vector ¶
- depends_on definition 13.15 Unit binormal vector ¶
- depends_on definition 13.13 Normal curvature vector of a curve ¶
  - depends_on definition 13.36 Geodesic curvature vector ¶
    - depends_on definition 13.37 Geodesic lines ¶ ↺
  - depends_on definition 13.28 Normal curvature vector of a surface ¶
    - depends_on definition 13.29 Gaussian curvature ¶ ↺
    - depends_on definition 13.36 Geodesic curvature vector ¶ ↺
  - depends_on definition 13.14 Principal unit normal vector ¶
    - depends_on definition 13.15 Unit binormal vector ¶ ↺
    - depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶
      - depends_on remark 18.34 The two ways an acceleration can be zero ¶
- depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶ ↺
depends_on proposition 7.76 $\pi$ as the circle constant ¶

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`	→	Curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:671
`depends_on`	→	Tangent vector to a curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:671
`depends_on`	←	First fundamental form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:983
`depends_on`	←	Unit tangent vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:696
`depends_on`	←	$\pi$ as the circle constant	declared	parts/02-mathematical-methods/05-real-analysis.tex:2007