Prove that the length of this curve is decreasing

  • #1
chipotleaway
174
0

Homework Statement


The problem asks us to prove the length of a curve decreases as one of its parameters, t, increases. Here is the full statement

http://img94.imageshack.us/img94/6560/yf6j.jpg [Broken]

Homework Equations


[tex]L_t=\int_0^1 ||\partial_s x(t,s)||ds[/tex] is the length of the curve as a function of t, for all t greater than 0.

[tex]\partial_t L_t=\int_0^1 \partial_t ||\partial_s x(t,s)||ds[/tex] is the derivative of the length of the curve with respect to t.


The Attempt at a Solution


To show that the length L(t) decreases as t increases, I think we have to show that the derivative of the length L(t) with respect to t is negative, that is, [tex]\partial_t L_t=\int_0^1 \partial_t ||\partial_s x(t,s)||ds < 0[/tex] is negative.

The only strategy I have for the moment is the mean-value theorem which I've seen used to prove inequalities before (e.g. show that sin(x)<x for all x) but so far, I'm still a bit unclear on how to apply it to this situation. The other attempt I've tried was to take the hint, do some substitutions and hope something will pop out but for a 3 mark question, there's no way this method alone will work. I'm also not yet able to see how the information at the start is relevant to doing this problem.

Any helpful pointers will be greatly appreciated, thanks!
 
Last edited by a moderator:

Answers and Replies

  • #2
UltrafastPED
Science Advisor
Gold Member
1,912
216
You have to use all of the given assumptions ... see the "givens" at the bottom of the problem statement!
 
  • #3
chipotleaway
174
0
I've used both the assumption and the hint in my attempt to manipulate it into something I can apply the MVT to...but I'm not sure if this is going down the right path or not, and also I don't have a clear idea of how to get a the desired inequality out of it.
 
  • #4
chipotleaway
174
0
Any more suggestions, anyone?
 

Suggested for: Prove that the length of this curve is decreasing

Replies
4
Views
4K
Replies
15
Views
2K
Replies
4
Views
838
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
12
Views
3K
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
7
Views
664
  • Last Post
Replies
18
Views
2K
  • Last Post
Replies
6
Views
2K
Top