Homework #2
Assigned: Aug. 11th, 2005
Due: Aug. 18th, 2005
Points: 20

1. 5 points
   A discrete metric is defined for real number domain R, as follows:
   d(x,y) = 0 if x is equal to y
   d(x,y) = 1 if x is not equal to y
   Verify that it satisfies the four properties that a metric should satisfy.
2. 5 points
   If d1(x,y) is a metric on a real number domain, then show that d2(x,y) defined as follows is also a valid metric.
   d2(x,y) = { d1(x,y) } / { 1 + d1(x,y) }
   
3. 10 points
   If J is a functional of y(x) and its derivatives, then its Gateaux variation can be obtained using the fact that it is equal to the derivarive of J(x+epsilon*h) with respect to epsilon evaluated at epsilon = 0, where h(x) is an arbitray function that satisfies the end conditions and differentiability requirements. Use this and intergration by parts to reduce the Gateaux variation of the following two functionals to int( f(x) h(x) dx ) with integration limits of x_a and x₂. Comment on the end conditions at the limits.
   • J(y,y') = int( y * sqrt{ (1+y'^2) } )
     with integration limits x₁ and x₂.
   • J(y,y',y") = int( D*y''^2 - w(x)*y )
     where D is a constant with integration limits x₁ and x₂.