definition 13.8 Curve

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Supports

• 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 13.19 Parameter curves ¶
  • depends_on definition 13.22 First fundamental coefficients ¶ ↺
  • depends_on definition 13.20 Unit normal vector of a surface ¶
    • depends_on definition 13.28 Normal curvature vector of a surface ¶ ↺
    • depends_on definition 13.25 Second fundamental coefficients ¶
      • depends_on definition 13.29 Gaussian curvature ¶ ↺
      • depends_on definition 13.27 Riemann symbols ¶
        • depends_on theorem 13.30 Theorema Egregium; Gauss, 1827 ¶ ↺
      • depends_on proposition A.493 A harmonic conformal map is a minimal surface ¶
    • depends_on definition 13.24 Second fundamental form ¶
      • depends_on definition 13.25 Second fundamental coefficients ¶ ↺
• depends_on definition 13.9 Tangent vector to a curve ¶
  • 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 phenomenon 18.31 Uniform circular motion is accelerated motion ¶
        • 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 ¶
      • depends_on proposition 18.33 Intrinsic decomposition of the acceleration ¶ ↺
      • 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 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 postulate 19.19 Newton's first law, rotational case ¶
        • depends_on postulate 19.22 Newton's second law, rotational case ¶
        • 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 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 13.12 Unit tangent vector ¶ ↺