Explanation Given: Let r ( s ) be an arc length parameterization of a closed curve C of length L. Here, C is an oval if d θ d s > 0 . The curve C 1 is defined as: r 1 ( s ) = r ( s ) − k N … ( 1 ) Proof: (a) Consider that r 1 ( s ) = r ( s ) − k N We know that κ ( s ) = | d θ d s | And T = 〈 cos θ , sin θ 〉 N = 〈 − sin θ , cos θ 〉 Now, differentiating equation (1), we get r 1 ' ( s ) = r ' ( s ) − k d N d s … ( 2 ) Now, d N d s = d N d θ ⋅ d θ d s = ( 〈 − cos θ , − sin θ 〉 ) d θ d s = − T ( s ) κ ( s ) Substitute this value in equation (2), we get r 1 ' ( s ) = r ' ( s ) + k κ ( s ) T ( s ) Since r ( s ) be an arc length parameterization, ‖ r ' ( s ) ‖ = 1 , T ( s ) = r ' ( s ) Then r 1 ' ( s ) = r ' ( s ) + k κ ( s ) r ' ( s ) = r ' ( s ) ( 1 + k κ ( s ) ) ‖ r 1 ' ( s ) ‖ = ‖ r ' ( s )...

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Author: Rogawski

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Chapter 14.4, Problem 84E

To determine

To prove:

(a) $‖ r_{1}' (s) ‖ = ‖ r' (s) ‖ + k κ (s)$

(b) $\int_{0}^{L} κ (s) d s = 2 π$ as P moves around the oval counterclockwise, $θ$ increases by $2 π$ .

(c) That $C_{1}$ has length $L + 2 π k$ .

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Chapter 14.4, Problem 83E

Chapter 14.4, Problem 85E

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Chapter 14 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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To prove: (a) ‖ r 1 ' ( s ) ‖ = ‖ r ' ( s ) ‖ + k κ ( s ) (b) ∫ 0 L κ ( s ) d s = 2 π as P moves around the oval counterclockwise, θ increases by 2 π . (c) That C 1 has length L + 2 π k .

Solution Summary: The author explains that r ( s ) is an arc length parameterization of a closed curve C of length L.

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To prove:

(a) $‖ r_{1}' (s) ‖ = ‖ r' (s) ‖ + k κ (s)$

(b) $\int_{0}^{L} κ (s) d s = 2 π$ as P moves around the oval counterclockwise, $θ$ increases by $2 π$ .

(c) That $C_{1}$ has length $L + 2 π k$ .

Want to see the full answer?

Chapter 14 Solutions

To prove: (a) ‖ r 1 ' ( s ) ‖ = ‖ r ' ( s ) ‖ + k κ ( s ) (b) ∫ 0 L κ ( s ) d s = 2 π as P moves around the oval counterclockwise, θ increases by 2 π . (c) That C 1 has length L + 2 π k .

Solution Summary: The author explains that r ( s ) is an arc length parameterization of a closed curve C of length L.

Videos

To prove: (a) ‖r1'(s)‖=‖r'(s)‖+kκ(s) (b) ∫0Lκ(s)ds=2π as P moves around the oval counterclockwise,θ increases by 2π. (c) That C1 has length L+2πk.

Want to see the full answer?

Chapter 14 Solutions

To prove:

(a) $‖ r_{1}' (s) ‖ = ‖ r' (s) ‖ + k κ (s)$

(b) $\int_{0}^{L} κ (s) d s = 2 π$ as P moves around the oval counterclockwise, $θ$ increases by $2 π$ .

(c) That $C_{1}$ has length $L + 2 π k$ .