Question 1. In this question we consider what happens to the length of a curve when we deform it along a vector field. We answer the question: "Which way should you push a curve to shorten it quickest?” Throughout this question: 7: [a, b] → R³ is a Cº-smooth unit speed curve; for each t € [a, b], V(t) is a vector perpendicular to (t), varying smoothly with t; and for any real s, we define 7s by Ys: [a, b] → R³, Ys(t) = y(t) + sV (t). (1) Show that the curves {s}SER form a deformation of %, along V in the sense that (a) for each real number s, 7, is a smooth curve [a, b] →→→→ R³; (b) % = y; and (c) Ys(t) = V(t). (2) Show that when s is sufficiently close to zero, s is a regular curve. (Hint: you might want to take s such that s < 1/M, where M maxte[a,b] ||V (t)| is the maximum length of the vectors V(t). You can assume this maximum exists. A version of the triangle inequality |a - b] ≥ |a|-|b| might also be useful.) (3) Show that y(t).V (t) = −ï(t).V (t), and hence find the derivative (†s(t) · √s(t))|,- at s = 0 in terms of (t) and V(t). = (4) Similarly, find the derivative (t)||_oat s = 0 in terms of ÿ(t) and V(t). (5) Let L(y) denote the length of y. WriteL(s) as an integral involving (t) and V(t). (Don't worry about any convergence issues if you want to pass a derivative through an integral.) (6) Suppose now that |V(t)| = 1 for all t. Given y(t), which vector field V(t) makes L(7) decrease as quickly as possible in s, at s = 0? If y: [0, 1] →→ R³ is the circle y(t) = (cost, sint, 0), what V(t) makes L(7) decrease as fast as possible? -
Question 1. In this question we consider what happens to the length of a curve when we deform it along a vector field. We answer the question: "Which way should you push a curve to shorten it quickest?” Throughout this question: 7: [a, b] → R³ is a Cº-smooth unit speed curve; for each t € [a, b], V(t) is a vector perpendicular to (t), varying smoothly with t; and for any real s, we define 7s by Ys: [a, b] → R³, Ys(t) = y(t) + sV (t). (1) Show that the curves {s}SER form a deformation of %, along V in the sense that (a) for each real number s, 7, is a smooth curve [a, b] →→→→ R³; (b) % = y; and (c) Ys(t) = V(t). (2) Show that when s is sufficiently close to zero, s is a regular curve. (Hint: you might want to take s such that s < 1/M, where M maxte[a,b] ||V (t)| is the maximum length of the vectors V(t). You can assume this maximum exists. A version of the triangle inequality |a - b] ≥ |a|-|b| might also be useful.) (3) Show that y(t).V (t) = −ï(t).V (t), and hence find the derivative (†s(t) · √s(t))|,- at s = 0 in terms of (t) and V(t). = (4) Similarly, find the derivative (t)||_oat s = 0 in terms of ÿ(t) and V(t). (5) Let L(y) denote the length of y. WriteL(s) as an integral involving (t) and V(t). (Don't worry about any convergence issues if you want to pass a derivative through an integral.) (6) Suppose now that |V(t)| = 1 for all t. Given y(t), which vector field V(t) makes L(7) decrease as quickly as possible in s, at s = 0? If y: [0, 1] →→ R³ is the circle y(t) = (cost, sint, 0), what V(t) makes L(7) decrease as fast as possible? -
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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Need help with part (1). Please explain each step and neatly type up. Thank you :)
![Question 1. In this question we consider what happens to the length of a curve
when we deform it along a vector field. We answer the question: "Which way should
you push a curve to shorten it quickest?”
Throughout this question: 7: [a, b] → R³ is a Cº-smooth unit speed curve; for
each t € [a, b], V(t) is a vector perpendicular to (t), varying smoothly with t; and
for any real s, we define 7s by
Ys: [a, b] → R³, Ys(t) = y(t) + sV (t).
(1) Show that the curves {s}SER form a deformation of %, along V in the sense
that
(a) for each real number s, %, is a smooth curve [a, b] → R³;
(b) % = y; and
(c)(t) = V(t).
(2) Show that when s is sufficiently close to zero, s is a regular curve.
(Hint: you might want to take s such that |s| < 1/M, where M
maxte[a,b] ||V (t)| is the maximum length of the vectors V(t). You can assume
this maximum exists. A version of the triangle inequality |ab| ≥ |a|-|b|
might also be useful.)
(3) Show that y(t).V (t) = −ÿ(t).V (t), and hence find the derivative (†(t) · Ÿs(t))|-0
at s = 0 in terms of (t) and V(t).
=
(4) Similarly, find the derivative (t)||_oat s = 0 in terms of ÿ(t) and V(t).
(5) Let L(y) denote the length of y. WriteL(s) as an integral involving
(t) and V(t).
(Don't worry about any convergence issues if you want to pass a derivative
through an integral.)
(6) Suppose now that |V(t) = 1 for all t. Given y(t), which vector field V (t)
makes L(7) decrease as quickly as possible in s, at s = 0? If y: [0, 1] →→ R³
is the circle y(t) = (cost, sint, 0), what V(t) makes L(7) decrease as fast as
possible?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe1fc35-672a-49fd-831b-9642c77888ed%2F7c498f6c-4bb7-4d83-9be3-651a6d2adcbb%2Fhcgd1as_processed.png&w=3840&q=75)
Transcribed Image Text:Question 1. In this question we consider what happens to the length of a curve
when we deform it along a vector field. We answer the question: "Which way should
you push a curve to shorten it quickest?”
Throughout this question: 7: [a, b] → R³ is a Cº-smooth unit speed curve; for
each t € [a, b], V(t) is a vector perpendicular to (t), varying smoothly with t; and
for any real s, we define 7s by
Ys: [a, b] → R³, Ys(t) = y(t) + sV (t).
(1) Show that the curves {s}SER form a deformation of %, along V in the sense
that
(a) for each real number s, %, is a smooth curve [a, b] → R³;
(b) % = y; and
(c)(t) = V(t).
(2) Show that when s is sufficiently close to zero, s is a regular curve.
(Hint: you might want to take s such that |s| < 1/M, where M
maxte[a,b] ||V (t)| is the maximum length of the vectors V(t). You can assume
this maximum exists. A version of the triangle inequality |ab| ≥ |a|-|b|
might also be useful.)
(3) Show that y(t).V (t) = −ÿ(t).V (t), and hence find the derivative (†(t) · Ÿs(t))|-0
at s = 0 in terms of (t) and V(t).
=
(4) Similarly, find the derivative (t)||_oat s = 0 in terms of ÿ(t) and V(t).
(5) Let L(y) denote the length of y. WriteL(s) as an integral involving
(t) and V(t).
(Don't worry about any convergence issues if you want to pass a derivative
through an integral.)
(6) Suppose now that |V(t) = 1 for all t. Given y(t), which vector field V (t)
makes L(7) decrease as quickly as possible in s, at s = 0? If y: [0, 1] →→ R³
is the circle y(t) = (cost, sint, 0), what V(t) makes L(7) decrease as fast as
possible?
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