Homework #3
Assigned: Jan. 25th, 2011
Due: Feb. 1st, 2011
Points: 60
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 15 points
   Imagine a one-dimensional body (say, a wire) that has zero bending stiffness but infinite axial stiffness. So, this wire can bend easily but cannot stretch or contract. Let its length be L. Let it be located along the x-axis of with one end (call it the tail) at the origin and the head (call it the head) at (L,0). Let us the head is pulled along the vertical line passing through (L,0) to (L,L). Your task is to determine the shape of the entire wire as the head is pulled as mentioned. Formulate it as a calculus of variations problem. What is your function space for this problem? What is your norm? Does it satisfy the four properties that the norm should satisfy. What can you say about the nature of the solution?
2. 15 points
   What if you have a flexible cloth that can bend easily but cannot stretch or contract? How do you pose this problem by extending your solution to the preceding problem? You need not work all the details but give an outline. If you do work out all the details, you will get bonus points!
3. 15 points
   Pose a calculus of variations problem that uses the concept of the inner product rather than just the norm. Give enough background to the problem.
4. 15 points
   Pose a structural optimization problem in the framework of calculus of variations. Comment on the function space and the norm (or inner product) that you need to define your functional(s).