definition 13.9 Tangent vector to a curve

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:648 · p. 457

ground object -- no derivation owed

Rests on

depends_on definition 13.8 Curve ¶

Supports

depends_on definition 18.10 Velocity ¶
- depends_on definition 18.11 Acceleration ¶
  - depends_on definition 18.27 Angular position, velocity and acceleration ¶
    - depends_on definition 18.29 Angular velocity vector ¶
      - depends_on remark 18.30 The angular velocity is an element of $\mathfrak{so}(3)$ ¶
    - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶
      - depends_on proposition 18.32 Tangential and centripetal components ¶
    - depends_on proposition 18.32 Tangential and centripetal components ¶ ↺
    - depends_on remark 18.28 The radian carries no dimension ¶
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
  - depends_on phenomenon 18.31 Uniform circular motion is accelerated motion ¶ ↺
  - depends_on phenomenon 18.12 Uniformly accelerated motion ¶
    - depends_on phenomenon 20.1 phen:parabolic-trajectory ¶
      - assumes experiment Absolute Ballistic Gravimetry (1995–present) ¶
      - depends_on theorem 20.2 Parabolic trajectory ¶
      - tests experiment Galileo's Inclined Planes ¶
  - 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 18.23 Transformation of velocity and acceleration ¶
    - depends_on phenomenon 18.26 Composition of velocities, and its limit ¶
- depends_on definition 18.27 Angular position, velocity and acceleration ¶ ↺
- depends_on definition 18.29 Angular velocity vector ¶ ↺
- depends_on definition 19.29 Kinetic energy ¶
  - depends_on definition 19.31 Energy ¶
    - depends_on proposition 27.20 Conservation of energy ¶
    - depends_on theorem 19.32 Conservation of energy ¶
      - depends_on phenomenon 50.6 No material surface ¶
      - depends_on phenomenon 19.33 Mechanical energy is conserved ¶
      - depends_on theorem 21.43 Conservation of energy ¶
  - depends_on theorem 19.38 Work–energy theorem ¶
- depends_on definition 19.5 Linear momentum ¶
  - depends_on definition 27.2 Translational equilibrium ¶
  - depends_on definition 19.18 Angular momentum ¶
    - depends_on definition 27.3 Rotational equilibrium ¶
    - depends_on definition 19.55 Angular momentum of the centre of mass ¶
      - depends_on proposition 29.14 Angular momentum and kinetic energy ¶
      - depends_on theorem 19.58 Decomposition of the angular momentum ¶
    - depends_on postulate 19.19 Newton's first law, rotational case ¶
      - depends_on phenomenon 19.21 Equal areas in equal times ¶
      - depends_on proposition 20.13 Eastward deflection of a body dropped from rest ¶
    - depends_on postulate 19.22 Newton's second law, rotational case ¶
      - depends_on theorem 29.25 Euler's equations ¶
    - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶
  - depends_on definition 19.51 Linear momentum of the centre of mass ¶
    - depends_on definition 19.55 Angular momentum of the centre of mass ¶ ↺
  - depends_on postulate 19.6 Newton's first law ¶
    - depends_on phenomenon 19.7 Persistence of uniform motion ¶
    - depends_on postulate 19.19 Newton's first law, rotational case ¶ ↺
    - depends_on postulate 19.8 Newton's second law ¶
      - depends_on definition 19.30 Conservative force and potential energy ¶
      - depends_on definition 19.34 Work ¶
      - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
      - depends_on phenomenon 42.1 Universality of free fall ¶
      - depends_on phenomenon 19.10 Force, mass and acceleration ¶
      - depends_on postulate 19.22 Newton's second law, rotational case ¶ ↺
      - depends_on postulate 19.48 Newton's third law ¶
      - depends_on theorem 20.2 Parabolic trajectory ¶ ↺
      - depends_on theorem 29.51 Equation of motion in a rotating frame ¶
    - depends_on proposition 19.20 For one particle the rotational first law is not independent ¶ ↺
  - depends_on postulate 19.8 Newton's second law ¶ ↺
- depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶ ↺
- depends_on proposition 18.23 Transformation of velocity and acceleration ¶ ↺
depends_on definition 13.11 Arc length ¶
- 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 lemma A.479 Conformal invariance of the Dirichlet integral ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - 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 definition 13.23 Inverse metric tensor ¶
      - depends_on definition 13.26 Christoffel 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 proposition 18.33 Intrinsic decomposition of the acceleration ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶
depends_on definition 13.12 Unit tangent vector ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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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:653
`depends_on`	←	Velocity	declared	parts/03-classical-mechanics/01-kinematics.tex:261
`depends_on`	←	Arc length	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:671
`depends_on`	←	Unit tangent vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:696