Math 240 (J. Wu) Homework #11 Due Fri ,April 9

  In Kreyszig, 8.10,8.11,9.1,9.2
  To be covered next week: 9.3-9.6

  Do all core problems in 8.10,8.11,9.1,9.2

8.10 # 22,28
8.11 # 18,26
9.1  # 16,22
9.2  # 20

Spring 95 # 6,7,9
Fall   95 # 8,9,

1. [10 points] The taxi problem. A taxi company has divided a city into three regions, region I,II,and III. It has determined that typically of the passengers picked up in region I, 60% have a destination in the same region , 30% in region II and 10% in region III. Of the passengers picked up in region II, 40% have a destination in region I, 30% in region II, and 30% in region III. Of the passengers picked up in region III, 30% have a destination in region I, 30% in region II, and 40% in region III. Suppose that at the beginning of the day 80% of the taxis are in region I, 15% in region II, and 5% in region III. What is the distribution of the taxis after a long time? Does it depend on the intial distribution of taxis? This problem is a Markov chain example. You can do most of your work in Maple.
2. [10points] Let r(t)= be a curve in 3-space and suppose that the three components f,g,h are second degree polynomials in t. Show that the curve r(t) lies in some plane in 3-space. First, prove this in general. Then determine an equation of the plane in which r(t)=(t^2-2t+1,-2t^2+3t-1,-2t+1) lies.
3. [10 points] You can use Maple in this problem if you like. Consider the force field F=c*|r|^(-n)*r where r=x*i+y*j+z*k and c is a constant. (If n=3 this is the inverse square field familiar from gravity or the electric field). a) Determine for what values of n this field is conservative, and if it is, find a potential. b) If c < 0 , for what values of n can you "escape" the influence of this "gravitational" field, i.e. the line integral from some point A to infinity is finite. c) How about if F is a "central" force field, i.e. F=f(|r|)*r/|r| for some function f? Under what conditions on f it is conservative? Under what conditions of f can you "escape" the force field.