Explanation Calculation: If a smooth curve in space is indicated by the position vector r ( t ) , and if s is the arc length parameter of the curve, then the unit tangent vector T is determined as follows: T = d r d s = v | v | (1) Then, the curvature in space is defined as, κ = | d T d s | = 1 | v | | d T d t | (2) Since the vector d T d s is perpendicular to T , then normal vector is defined as, N = 1 κ d T d s = d T d t | d T d t | (3) For example: Calculate the principal unit normal ( N ) for the helix r ( t ) = ( a cos t ) i + ( a sin t ) j + b t k , a , b ≥ 0 , a 2 + b 2 ≠ 0 which is shown in Figure 12.22 in the textbook and describe how the vector is pointing. The general expression to calculate the velocity. v = d r d t (4) Here, d r d t is the rate of change in displacement with respect to time. Substitute ( a cos t ) i + ( a sin t ) j + b t k for r ( t ) in equation (4). v = d d t [ ( a cos t ) i + ( a sin t ) j + b t k ] = − ( a sin t ) i + ( a cos t ) j + b k The magnitude of v is calculated as follows, | v | = ( − a sin t ) 2 + ( a cos t ) 2 + b 2 = a 2 sin 2 t + a 2 cos 2 t + b 2 = a 2 ( sin 2 t + cos 2 t ) + b 2 = a 2 + b 2 { ∵ sin 2 t + cos 2 t = 1 } Substitute − ( a sin t ) i + ( a cos t ) j + b k for v and a 2 + b 2 for | v | in equation (1)

Thomas' Calculus: Early Transcendentals plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (14th Edition)

14th Edition

ISBN: 9780134768496

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13, Problem 10GYR

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How to define the principal unit normal (N) and the curvature (κ) for curves in space? Also find the relationship between N and κ. Give some examples.

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Chapter 13, Problem 9GYR

Chapter 13, Problem 11GYR

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Thomas' Calculus: Early Transcendentals plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (14th Edition)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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How to define the principal unit normal ( N ) and the curvature ( κ ) for curves in space? Also find the relationship between N and κ . Give some examples.

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How to define the principal unit normal (N) and the curvature (κ) for curves in space? Also find the relationship between N and κ. Give some examples.

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Chapter 13 Solutions