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

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

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

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

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

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

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

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

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

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

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

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

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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) rt)rt) P Q y x z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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

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

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

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

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

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

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

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

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

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

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12.4 Planetary Motion and Satellites omitted

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