(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
icon
Related questions
Question

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).
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).
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,