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{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 256 42 "Space Graphs Occur in Mathematical Biology" }}{PARA 0 "
" 0 "" {TEXT -1 418 " Diffusion is a common phenomena in a
ll of the sciences. Think of it in connection with population studies \+
and in relation to migrations of populations. For example, the movemen
t of The Plague across Europe, the migration of armadillos northward t
hrough the United States, and the introduction and spread of rabbits i
n Australia -- these are all examples of the diffusion of a populatio
n through a region." }}{PARA 0 "" 0 "" {TEXT -1 543 " Diffu
sion is modeled as a multidimensional function, u(t, x) or u(t,x,y), d
epending on time and position. How the population changes in time is m
odeled by stating what the rate of change is. The rate of change in ti
me is connected with the shape of the curve. Where the second derivat
ive is negative and the curve is concave down, the model for diffusion
predicates 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,`$`(x,2));" "6#/-%%di
ffG6$%\"uG%\"tG-F%6$F'-%\"$G6$%\"xG\"\"#" }{TEXT -1 14 ", or \+
" }{XPPEDIT 18 0 "diff(u,t) = diff(u,`$`(x,2))+diff(u,`$`(y,2));" "6#/
-%%diffG6$%\"uG%\"tG,&-F%6$F'-%\"$G6$%\"xG\"\"#\"\"\"-F%6$F'-F-6$%\"yG
F0F1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "i
n one space dimension, x, or in two, \{x,y\}, respectively." }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT -1 85 "We draw the fu
nction u(0,x), where u is as given below, on the interval 0 < x < 2 \+
." }}{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 for 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 animation to actually see t
he 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 sta
tic 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 w
ay to view a one-dimensional graph changing in time by drawing a surfa
ce, 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,orientatio
n=[-135,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 147 "Finally, if the surfac
e at been defined in polar coordinates, we could have drawn the surfac
e in that form -- even we could animate such a surface." }}{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 "animat
e3d([r,theta,u(t,r,theta)],r=0..1,theta=0..2*Pi,t=-1..1,\n coords=c
ylindrical,axes=NORMAL,orientation=[60,80]);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24
"Exercise for the student" }}{PARA 0 "" 0 "" {TEXT -1 27 "Draw the gra
ph of u(x,y) = " }{XPPEDIT 18 0 "x^2-y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"
yGF&!\"\"" }{TEXT -1 38 " over the rectangle [-1, 1] x [-1, 1]." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1
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