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

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

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

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

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

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

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

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

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

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

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

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

13. 12.1 Vector-Valued Functions and Space Curves (12) Derivatives and Motion on Smooth Curves Suppose that r(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 of t. Then the difference between the particle’s positions at time t+  t and the time t is r(t)r(t) r(t+  t) rt)rt) P Q y x z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

35. 12.3 Curvature, Torsion, and the TNB Frame (1) Curvature, Torsion, and TNB Frame x y O P0P0 T P

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

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

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

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

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

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

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

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

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

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

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

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

48. 12.3 Curvature, Torsion, and the TNB Frame (14) Torsion and the Binormal Vector T N B 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. 12.3 Curvature, Torsion, and the TNB Frame (15) Torsion and the Binormal Vector T N B

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

51. 12.3 Curvature, Torsion, and the TNB Frame (17) Torsion and the Binormal Vector

52. 12.3 Curvature, Torsion, and the TNB Frame (18) Torsion and the Binormal Vector

53. 12.3 Curvature, Torsion, and the TNB Frame (19) Tangential and Normal Components of Acceleration T N a s

54. 12.3 Curvature, Torsion, and the TNB Frame (20) Tangential and Normal Components of Acceleration

55. 12.3 Curvature, Torsion, and the TNB Frame (21) Tangential and Normal Components of Acceleration C a P

56. 12.3 Curvature, Torsion, and the TNB Frame (22) Tangential and Normal Components of Acceleration (Ex. 6-1)

57. 12.3 Curvature, Torsion, and the TNB Frame (23) Tangential and Normal Components of Acceleration (Ex. 6-2) t

58. 12.3 Curvature, Torsion, and the TNB Frame (24) Formulas for Computing Curvature and Torsion

61. 12.4 Planetary Motion and Satellites omitted