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

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

**
CORE PROBLEMS:**
Do all core problems in 8.10,8.11,9.1,9.2

**
ADDITIONAL PROBLEMS: **
8.10 # 22,28
8.11 # 18,26
9.1 # 16,22
9.2 # 20

**
PAST FINAL EXAMS:**
Spring 95 # 6,7,9
Fall 95 # 8,9,

**
EXTRA CREDIT PROBLEMS:**
- [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.
- [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.
- [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.