(a) Let 7 : [a, b] → R" be a curve with 7' (t) # 0 for all t e [a, b]. Let s = s(t) be the arc-length function. Show that the reparametrization by arc-length traverses the curve with unit speed. That is, show ||7" (s)|| = 1 for all s.
(a) Let 7 : [a, b] → R" be a curve with 7' (t) # 0 for all t e [a, b]. Let s = s(t) be the arc-length function. Show that the reparametrization by arc-length traverses the curve with unit speed. That is, show ||7" (s)|| = 1 for all s.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you solve a)?
![(a) Let 7 : [a, b| → R" be a curve with 7 (t) 7 0 for all t E [a, b|]. Let s = s(t) be the arc-length function. Show
that the reparametrization by arc-length traverses the curve with unit speed. That is, show |T (s)||
= 1 for all s.
(b) Consider the curve 7 (t) = (- sin t, cos t, 1). Reparametrize this curve by arc-length starting from the point
(-1,0, 1).
(c) Use your answer from part (b) to find the point on the curve ř(t) = (- sint, cos t, 1) that is T/4 units away
from (–1,0, 1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f7898e2-76d3-4472-b0b2-48c9c34ae41b%2F42c491f7-be6c-4ee3-91d1-81a55a43e990%2Fjhf67bo_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let 7 : [a, b| → R" be a curve with 7 (t) 7 0 for all t E [a, b|]. Let s = s(t) be the arc-length function. Show
that the reparametrization by arc-length traverses the curve with unit speed. That is, show |T (s)||
= 1 for all s.
(b) Consider the curve 7 (t) = (- sin t, cos t, 1). Reparametrize this curve by arc-length starting from the point
(-1,0, 1).
(c) Use your answer from part (b) to find the point on the curve ř(t) = (- sint, cos t, 1) that is T/4 units away
from (–1,0, 1).
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