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Vector-Valued Functions and Motion in Space

Vector-Valued Functions and Motion in Space. Dr. Ching I Chen. z. r ( t ). y. O. x. curve. 12.1 Vector-Valued Functions and Space Curves (1) Space Curve. 12.1 Vector-Valued Functions and Space Curves (2) Space Curve (Example 1).

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Vector-Valued Functions and Motion in Space

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  1. Vector-Valued Functions and Motion in Space Dr. Ching I Chen

  2. z r(t) y O x curve 12.1Vector-Valued Functions and Space Curves(1)Space Curve

  3. 12.1Vector-Valued Functions and Space Curves(2)Space Curve (Example 1)

  4. 12.1Vector-Valued Functions and Space Curves(3)Space Curve

  5. 12.1Vector-Valued Functions and Space Curves(4)Space Curve (Exploration 1-1~4)

  6. 12.1Vector-Valued Functions and Space Curves(5)Space Curve (Exploration 1-5~7)

  7. 12.1Vector-Valued Functions and Space Curves(6)Space Curve (Exploration 1-8~10)

  8. 12.1Vector-Valued Functions and Space Curves(7)Limit and Continuity

  9. 12.1Vector-Valued Functions and Space Curves(8)Limit and Continuity (Example 2)

  10. 12.1Vector-Valued Functions and Space Curves(9)Limit and Continuity

  11. 12.1Vector-Valued Functions and Space Curves(10)Limit and Continuity

  12. 12.1Vector-Valued Functions and Space Curves(11)Limit and Continuity (Example 3)

  13. z y Q r(t+Dt) r(t) Dr(t) x P 12.1Vector-Valued Functions and Space Curves(12)Derivatives and Motion on Smooth Curves Suppose thatr(t) = f(t) i + g(t) j + h(t) k is the position of a particle moving along a curve in the plane and that f(t), g(t) and h(t) are differentiable functions oft. Then the difference between the particle’s positions at time t+Dtand the time t is

  14. 12.1Vector-Valued Functions and Space Curves(13)Derivatives and Motion on Smooth Curves

  15. 12.1Vector-Valued Functions and Space Curves(14)Derivatives and Motion on Smooth Curves

  16. 12.1Vector-Valued Functions and Space Curves(15)Derivatives and Motion on Smooth Curves

  17. 12.1Vector-Valued Functions and Space Curves(16)Derivatives and Motion on Smooth Curves (Example 4)

  18. 12.1Vector-Valued Functions and Space Curves(17)Derivatives and Motion on Smooth Curves

  19. 12.1Vector-Valued Functions and Space Curves(18)Differentiation Rules

  20. 12.1Vector-Valued Functions and Space Curves(19)Differentiation Rules

  21. 12.1Vector-Valued Functions and Space Curves(20)Vector Functions of Constant Length

  22. 12.1Vector-Valued Functions and Space Curves(21)Vector Functions of Constant Length(Example 5)

  23. 12.1Vector-Valued Functions and Space Curves(22)Integrals of Vector Functions

  24. 12.1Vector-Valued Functions and Space Curves (23)Integrals of Vector Functions (Example 6)

  25. 12.1Vector-Valued Functions and Space Curves(20)Integrals of Vector Functions

  26. 12.1Vector-Valued Functions and Space Curves(24)Integrals of Vector Functions (Example 7)

  27. 12.1Vector-Valued Functions and Space Curves(25)Integrals of Vector Functions (Example 8)

  28. 12.2Arc Length and the Unit Tangent Vector T(1)Arc length

  29. 12.2Arc Length and the Unit Tangent Vector T(2)Arc length (Example 1)

  30. 12.2Arc Length and the Unit Tangent Vector T(3)Arc length

  31. 12.2Arc Length and the Unit Tangent Vector T(4)Arc length (Example 2)

  32. 12.2Arc Length and the Unit Tangent Vector T(5)The Unit Tangent Vector T

  33. 12.2Arc Length and the Unit Tangent Vector T(6)The Unit Tangent Vector T (Example 4)

  34. y P(x,y) r t x O 12.2Arc Length and the Unit Tangent Vector T(7)The Unit Tangent Vector T (Example 5)

  35. y T P P0 x O 12.3Curvature, Torsion, and the TNB Frame(1)Curvature, Torsion, and TNB Frame

  36. 12.3Curvature, Torsion, and the TNB Frame(2)Curvature, Torsion, and TNB Frame

  37. T 12.3Curvature, Torsion, and the TNB Frame(3)Curvature, Torsion, and TNB Frame (Example 1)

  38. 12.3Curvature, Torsion, and the TNB Frame(4)Curvature, Torsion, and TNB Frame (Example 2)

  39. 12.3Curvature, Torsion, and the TNB Frame(5)The Principal Unit Normal Vector for Plane Curves

  40. 12.3Curvature, Torsion, and the TNB Frame(6)The Principal Unit Normal Vector for Plane Curves

  41. 12.3Curvature, Torsion, and the TNB Frame(7)The Principal Unit Normal Vector for Plane Curves

  42. 12.3Curvature, Torsion, and the TNB Frame(8)The Principal Unit Normal Vector for Plane Curves (EX.3)

  43. 12.3Curvature, Torsion, and the TNB Frame(9)Circle of Curvature and Radius of Curvature

  44. 12.3Curvature, Torsion, and the TNB Frame(10)Circle of Curvature and Radius of Curvature

  45. 12.3Curvature, Torsion, and the TNB Frame(11)Curvature and Normal Vectors for Space Curves (Ex. 4-1)

  46. 12.3Curvature, Torsion, and the TNB Frame(12)Curvature and Normal Vectors for Space Curves (Ex. 4-2)

  47. 12.3Curvature, Torsion, and the TNB Frame(13)Curvature and Normal Vectors for Space Curves (Example 5)

  48. B N T 12.3Curvature, Torsion, and the TNB Frame(14)Torsion and the Binormal Vector The binormal vector of a curve in space is B = TN, a unit vector orthogonal to both T and N. Together define a moving right-handed vector frame that always travel with a body moving along a curve in space. It is the Frenet (“fre-nay”) frame, or the TNB frame. This vector frame plays a significant role in calculating the flight paths of space vehicles.

  49. B N T 12.3Curvature, Torsion, and the TNB Frame(15)Torsion and the Binormal Vector

  50. B N T 12.3Curvature, Torsion, and the TNB Frame(16)Torsion and the Binormal Vector

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