{VERSION 2 3 "APPLE_PPC_MAC" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1
{CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0
0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Dash Item
" 0 16 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1
3 3 0 0 0 0 0 0 16 3 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0
257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 1 12 0 0
0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4
259 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0
0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 26
"Three Dimensional Graphics" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 259 "" 0 "" {TEXT -1
21 "School of Mathematics" }}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math
.gatehc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 569 " Maple can be used to give good intuition for the \+
shape of surfaces, for the calculus on multidimensional functions, and
for the computation of algorithms. If any computer program is opened \+
for the first time on the occasion that there is a hard problem to be \+
done, the tasks can be formidable. Rather, we begin with some simple p
roblems that are introductory in nature. Some tools are suggested in t
his assignment. The tools are put in a context, but it is the introduc
tion to using Maple to draw surfaces that is of primary importance in
this assignment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 418 " Diffusion is a common phenomena in all of t
he sciences. Think of it in connection with population studies and in \+
relation to migrations of populations. For example, the movement of Th
e Plague across Europe, the migration of armadillos northward through \+
the United States, and the introduction and spread of rabbits in Austr
alia -- these are all examples of the diffusion of a population throu
gh a region." }}{PARA 0 "" 0 "" {TEXT -1 543 " Diffusion is
modeled as a multidimensional function, u(t, x) or u(t,x,y), dependin
g on time and position. How the population changes in time is modeled \+
by stating what the rate of change is. The rate of change in time is c
onnected with the shape of the curve. Where the second derivative is \+
negative and the curve is concave down, the model for diffusion predic
ates that the function decreases. In that case, the derivative in time
should be negative, also. The graph of the distribution being concave
up has the opposite effect. " }}{PARA 0 "" 0 "" {TEXT -1 42 " \+
These ideas might be modeled as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 " \+
" }{XPPEDIT 18 0 "diff(u,t) = diff(u,x,x) " "/-%%diffG6$%\"uG%\"t
G-F$6%F&%\"xGF*" }{TEXT -1 14 ", or " }{XPPEDIT 18 0 "diff(u,
t) = diff(u,x,x) + diff(u,y,y)" "/-%%diffG6$%\"uG%\"tG,&-F$6%F&%\"xGF+
\"\"\"-F$6%F&%\"yGF/F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 59 "in one space dimension, x, or in two, \{x,y\}, resp
ectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT
-1 86 "We draw the function u(0,x), where u is as given below, on the \+
interval 0 < x < 2 ¹ ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u
:=(t,x)->exp(-t)*sin(x);\nplot(u(0,x),x=0..2*Pi);" }}}{PARA 16 "" 0 "
" {TEXT -1 67 "We show that this function satisfies the above model fo
r diffusion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "diff(u(t,x),
t)-diff(u(t,x),x,x);" }}}{PARA 16 "" 0 "" {TEXT -1 62 "We draw an anim
ation to actually see the diffusion take place." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "with(plots):\nanimate(u(t,x),x=0..2*Pi,t=0..2)
;" }}}{PARA 0 "" 0 "" {TEXT -1 105 "(Suggestion: One way to illustrate
an animation on a static document is to draw a series of \"snapshots.
\")" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(\{u(0,x),u(2/3,x
),u(5/3,x),u(2,x)\},x=0..2*Pi);" }}}{PARA 16 "" 0 "" {TEXT -1 131 " W \+
provide an alternate way to view a one-dimensional graph changing in t
ime by drawing a surface, where one axis is the time axis." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot3d(u(t,x),x=0..2*Pi,t=0..2,axes
=NORMAL,orientation=[-135,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 147 "Fina
lly, if the surface at been defined in polar coordinates, we could hav
e drawn the surface in that form -- even we could animate such a surfa
ce." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=(t,r,theta)->t*(1-
r^2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 119 "animate3d([r,theta,u(t,r,theta)],r=0..1,theta=0..2*Pi,t=-1..1,
\n coords=cylindrical,axes=NORMAL,orientation=[60,80]);" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercis
e for the student" }}{PARA 0 "" 0 "" {TEXT -1 27 "Draw the graph of u(
x,y) = " }{XPPEDIT 18 0 "x^2-y^2" ",&*$%\"xG\"\"#\"\"\"*$%\"yGF%!\"\"
" }{TEXT -1 38 " over the rectangle [-1, 1] x [-1, 1]." }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}}{MARK "1 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 }