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 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 defines 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, ✅ 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))_0 at s = 0 in terms of (t) and V(t). (4) Similarly, find the derivative (t)||-o at s = 0 in terms of ÿ(t) and V(t). (5) Let L(7) denote the length of y. WriteL(%) as an integral involving Ÿ(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? If y: [0, 1] → R³ is the circle y(t) = (cost, sint, 0), what V(t) makes L(s) 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 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 defines 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, ✅ 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))_0 at s = 0 in terms of (t) and V(t). (4) Similarly, find the derivative (t)||-o at s = 0 in terms of ÿ(t) and V(t). (5) Let L(7) denote the length of y. WriteL(%) as an integral involving Ÿ(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? If y: [0, 1] → R³ is the circle y(t) = (cost, sint, 0), what V(t) makes L(s) 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 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 ys 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, ✅ 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))_0
at s = 0 in terms of (t) and V(t).
(4) Similarly, find the derivative (t)||-o at s = 0 in terms of ÿ(t) and V(t).
(5) Let L(7) denote the length of y. WriteL(%) as an integral involving
Ÿ(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? 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%2Fdb765cc0-57ef-4934-8932-f38f84ef7ba0%2Fswlwjyr_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 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 ys 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, ✅ 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))_0
at s = 0 in terms of (t) and V(t).
(4) Similarly, find the derivative (t)||-o at s = 0 in terms of ÿ(t) and V(t).
(5) Let L(7) denote the length of y. WriteL(%) as an integral involving
Ÿ(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? 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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