definition 13.74 Length of a curve

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3633 · p. 497

ground object -- no derivation owed

Rests on

depends_on definition 13.118 Line element ¶
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
depends_on definition 13.81 Vector on a manifold ¶ ↺

Supports

depends_on definition 13.76 Arclength parametrisation ¶
- depends_on definition 13.77 Curvature vector ¶
  - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
    - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶
depends_on proposition 13.75 Invariance of the length ¶

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`	→	Line element	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3644
`depends_on`	→	Metric tensor of signature $(p,q)$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3644
`depends_on`	→	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3644
`depends_on`	←	Arclength parametrisation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3688
`depends_on`	←	Invariance of the length	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3651