Dr.-Ing. Erwin SitompulPresident University

Lecture 3

Multivariable Calculus

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Homework Aid: Cycloid MotionChapter 13 13.2 Modeling Projectile Motion

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The Vector and Parametric Equations for Ideal Projectile MotionChapter 13 13.2 Modeling Projectile Motion

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The Vector and Parametric Equations for Ideal Projectile MotionChapter 13 13.2 Modeling Projectile Motion

Example

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Arc Length Along a Space CurveChapter 13 13.3 Arc Length and the Unit Tangent Vector T

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Arc Length Along a Space CurveChapter 13 13.3 Arc Length and the Unit Tangent Vector T

Example

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Arc Length Along a Space CurveChapter 13 13.3 Arc Length and the Unit Tangent Vector T

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Speed on a Smooth Curve, Unit Tangent Vector TChapter 13 13.3 Arc Length and the Unit Tangent Vector T

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Speed on a Smooth Curve, Unit Tangent Vector TChapter 13 13.3 Arc Length and the Unit Tangent Vector T

Example

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Curvature of a Plane CurveChapter 13 13.4 Curvature and the Unit Normal Vector N

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Curvature of a Plane CurveChapter 13 13.4 Curvature and the Unit Normal Vector N

Example

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Curvature of a Plane CurveChapter 13 13.4 Curvature and the Unit Normal Vector N

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Curvature of a Plane CurveChapter 13 13.4 Curvature and the Unit Normal Vector N

Example

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Curvature and Normal Vectors for Space CurvesChapter 13 13.4 Curvature and the Unit Normal Vector N

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Curvature and Normal Vectors for Space CurvesChapter 13 13.4 Curvature and the Unit Normal Vector N

Example

Effects of increasing a or b?

Effects on reducing a or b to zero?

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Curvature and Normal Vectors for Space CurvesChapter 13 13.4 Curvature and the Unit Normal Vector N

Example

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TorsionChapter 13 13.5 Torsion and the Unit Binormal Vector B

As we are traveling along a space curve, the Cartesian i, j, and k coordinate system which are used to represent the vectors of the motion are not truly relevant.

Instead, it is more meaningful to know the vectors representative of our forward direction (unit tangent vector T), the direction in which our path is turning (the unit normal vector N), and the tendency of our motion to twist out of the plane created by these vectors in a perpendicular direction of the plane (defined as unit binormal vectorB = T N).

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TorsionChapter 13 13.5 Torsion and the Unit Binormal Vector B

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TorsionChapter 13 13.5 Torsion and the Unit Binormal Vector B

dds

B T,d

ds

B Bdds

B N

1 dds

TN

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TorsionChapter 13 13.5 Torsion and the Unit Binormal Vector B

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Tangential and Normal Components of AccelerationChapter 13 13.5 Torsion and the Unit Binormal Vector B

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Tangential and Normal Components of AccelerationChapter 13 13.5 Torsion and the Unit Binormal Vector B

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Tangential and Normal Components of AccelerationChapter 13 13.5 Torsion and the Unit Binormal Vector B

Example

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Tangential and Normal Components of AccelerationChapter 13 13.5 Torsion and the Unit Binormal Vector B

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Homework 3Chapter 13

Exercise 13.2, No. 7. Exercise 13.3, No. 5. Exercise 13.3, No. 12. Exercise 13.4, No. 3. Exercise 13.4, No. 11. Exercise 13.5, No. 12. Exercise 13.5, No. 24.

Due: Next week, at 17.15.

13.5 Torsion and the Unit Binormal Vector B