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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 39 "MAPLE PROJECT: PATH OF ST
EEPEST DESCENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "by " }{TEXT 257 15 "Steve Pederson " }{TEXT -1 21 "(versio
n of 10-13-02)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "A path of steepest descent is a curve on a surface which \+
goes \"downhill\"" }}{PARA 0 "" 0 "" {TEXT -1 75 "as rapidly as possib
le. In this project, you will use a program written for" }}{PARA 0 "" 
0 "" {TEXT -1 74 "Maple to approximate the \"path of steepest descent
\" given a starting point" }}{PARA 0 "" 0 "" {TEXT -1 70 "on a surface
.  Each path of steepest descent will be approximated by a" }}{PARA 0 
"" 0 "" {TEXT -1 24 "finite number of points." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Maple contains a programm
ing language which is useful for writing short" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "programs. An algorithm to generate the points has been in
corporated into" }}{PARA 0 "" 0 "" {TEXT -1 71 "the Maple program belo
w. For the program you enter the following items:" }}{PARA 0 "" 0 "" 
{TEXT -1 54 "1. f(x,y), where z = f(x,y) is the surface's equation," }
}{PARA 0 "" 0 "" {TEXT -1 58 "2. n = the number of points desired for \+
the approximation," }}{PARA 0 "" 0 "" {TEXT -1 76 "3. (x0,y0), where (
x0,y0,f(x0,y0)) is the starting point on the surface, and" }}{PARA 0 "
" 0 "" {TEXT -1 36 "4. d = the step size between points." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Any program line b
eginning with \"#\" is a comment line and is not read by" }}{PARA 0 "
" 0 "" {TEXT -1 26 "Maple as the program runs." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 20 "TEXT OF THE PROGRAM:" }}
{PARA 0 "" 0 "" {TEXT 266 40 "#enter your information on the next line
" }}{PARA 0 "" 0 "" {TEXT 267 49 "f:=(x,y)-> x^2-y^2; n:=100; x0:=4; y
0:=0; d:=1/n:" }}{PARA 0 "" 0 "" {TEXT 268 43 "pts:=array(0..n): pts[0
]:=[x0,y0,f(x0,y0)]:" }}{PARA 0 "" 0 "" {TEXT 269 17 "for i from 1 to \+
n" }}{PARA 0 "" 0 "" {TEXT 270 2 "do" }}{PARA 0 "" 0 "" {TEXT 271 34 "
newx0:=evalf(x0-d*D[1](f)(x0,y0));" }}{PARA 0 "" 0 "" {TEXT 272 34 "ne
wy0:=evalf(y0-d*D[2](f)(x0,y0));" }}{PARA 0 "" 0 "" {TEXT 273 47 "x0:=
newx0; y0:=newy0; pts[i]:=[x0,y0,f(x0,y0)];" }}{PARA 0 "" 0 "" {TEXT 
274 3 "od:" }}{PARA 0 "" 0 "" {TEXT 275 12 "#print(pts):" }}{PARA 0 "
" 0 "" {TEXT 276 78 "h1:=plots[pointplot3d](convert(pts,list),color=re
d,connect=true, thickness=4):" }}{PARA 0 "" 0 "" {TEXT 277 25 "h2:=plo
t3d(f,0..4,-1..1):" }}{PARA 0 "" 0 "" {TEXT 278 51 "plots[display](\{h
1,h2\},style=WIREFRAME,axes=BOXED);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 72 "Next let's try out the program. In this \+
example, the surface is a saddle" }}{PARA 0 "" 0 "" {TEXT -1 59 "f(x,y
)= x^2-y^2. Starting at the point on the surface where" }}{PARA 0 "" 
0 "" {TEXT -1 71 "x=4 and y=0, the program will generate 100 points an
d then plot them on" }}{PARA 0 "" 0 "" {TEXT -1 82 "the surface. To ru
n the program, put the cursor in the first line and press enter." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 447 "#e
nter your information on the next line\nf:=(x,y)-> x^2-y^2; n:=100; x0
:=4; y0:=0; d:=1/n:\npts:=array(0..n): pts[0]:=[x0,y0,f(x0,y0)]:\nfor \+
i from 1 to n\ndo\nnewx0:=evalf(x0-d*D[1](f)(x0,y0));\nnewy0:=evalf(y0
-d*D[2](f)(x0,y0));\nx0:=newx0; y0:=newy0; pts[i]:=[x0,y0,f(x0,y0)];\n
od:\n#print(pts):\nh1:=plots[pointplot3d](convert(pts,list),color=red,
connect=true, thickness=4):\nh2:=plot3d(f,0..4,-1..1):\nplots[display]
(\{h1,h2\},style=WIREFRAME,axes=BOXED);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "To rotate the plot, put the arrow cursor \+
on the plot. Hold down the mouse" }}{PARA 0 "" 0 "" {TEXT -1 71 "butto
n and a box will appear. Continuing to hold the mouse button down," }}
{PARA 0 "" 0 "" {TEXT -1 70 "move the cursor to rotate the box, then r
elease the mouse button. Move" }}{PARA 0 "" 0 "" {TEXT -1 73 "the curs
or to \"edit\" and choose \"redraw\" (on some versions of Maple, just
" }}{PARA 0 "" 0 "" {TEXT -1 74 "press enter). Obtain a view of the pl
ot which best illustrates the surface" }}{PARA 0 "" 0 "" {TEXT -1 56 "
and points. Notice that you may copy and paste the plot." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "EXERCISE 1" }}
{PARA 0 "" 0 "" {TEXT -1 71 "Place your figure from above at the arrow
head prompt below. Be sure the" }}{PARA 0 "" 0 "" {TEXT -1 19 "points \+
are in view." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "EXERCISE 2" }}
{PARA 0 "" 0 "" {TEXT -1 69 "Copy and paste the program to the arrowhe
ad prompt below.  Do not run" }}{PARA 0 "" 0 "" {TEXT -1 71 "it yet. F
irst change the starting point to x0=4, y0=0.1. Also, near the" }}
{PARA 0 "" 0 "" {TEXT -1 70 "bottom of the program are lines that begi
n with \"h1\" and \"h2\". h1 is a" }}{PARA 0 "" 0 "" {TEXT -1 75 "supr
essed plot of the points and h2 is a supressed plot of the surface. Th
e" }}{PARA 0 "" 0 "" {TEXT -1 72 "line that begins with \"plots[displa
y]\" shows h1 and h2 in the same plot." }}{PARA 0 "" 0 "" {TEXT -1 77 
"In the bottom three or four lines of the program, change h1 to \"h3\"
 and h2 to" }}{PARA 0 "" 0 "" {TEXT -1 70 "\"h4\". This so that h1 and
 h2 from exercise 1 are not lost. Now run the" }}{PARA 0 "" 0 "" 
{TEXT -1 73 "program. Rotate your plot to obtain a suitable view and i
nclude it below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 10 "EXERCISE
 3" }}{PARA 0 "" 0 "" {TEXT -1 74 "Next we'll combine the points gener
ated in exercises 1 and 2 into a single" }}{PARA 0 "" 0 "" {TEXT -1 
73 "plot (we'll plot h1 and h3 together with the surface). In the line
 below," }}{PARA 0 "" 0 "" {TEXT -1 74 "press enter. Then rotate the p
lot to obtain a suitable view and include it" }}{PARA 0 "" 0 "" {TEXT 
-1 6 "below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plots[display](\{h1
,h3,h4\},style=WIREFRAME,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 10 "EXERCISE 4" }}{PARA 0 "
" 0 "" {TEXT -1 72 "For the saddle f(x,y) = x^2-y^2, choose 3 new star
ting points (different" }}{PARA 0 "" 0 "" {TEXT -1 71 "from those in t
he exercises above). Generate their approximate paths of" }}{PARA 0 "
" 0 "" {TEXT -1 69 "steepest descent and show them simultaneously on t
he surface. Be sure" }}{PARA 0 "" 0 "" {TEXT -1 69 "that the points fr
om all three approximations are within your plot of" }}{PARA 0 "" 0 "
" {TEXT -1 29 "the surface (see note below)." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Note: Sometimes it is necessary
 to plot a larger portion of the surface. For" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "example, if in your surface plot you want -2 <= x <= 6 an
d -3 <= y <= 8," }}{PARA 0 "" 0 "" {TEXT -1 61 "then change part of th
e next to last line of the program from" }}{PARA 0 "" 0 "" {TEXT -1 
79 "\"plot3d(f,0..4,-1..1)\" to \"plot3d(f,-2..6,-3..8)\". To help cho
ose the domain of" }}{PARA 0 "" 0 "" {TEXT -1 75 "the surface plot, yo
u may want to see the x and y coordinates of the points" }}{PARA 0 "" 
0 "" {TEXT -1 72 "generated. To do this, remove \"#\" from the line \"
#print(pts):\" and rerun" }}{PARA 0 "" 0 "" {TEXT -1 72 "the program. \+
You will get the x, y, and z coordinates of each point. Use" }}{PARA 
0 "" 0 "" {TEXT -1 58 "this information to adjust the domain of the su
rface plot." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 10 "EXERCISE 5" }}
{PARA 0 "" 0 "" {TEXT -1 46 "Repeat problem 4 with the surface f(x,y) \+
= xy." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 10 "EXERCISE 6" }}{PARA 0 "
" 0 "" {TEXT -1 76 "Let f(x,y) = sin(xy). Generate the approximate pat
hs of steepest descent for" }}{PARA 0 "" 0 "" {TEXT -1 71 "the startin
g points x0 = 0.7, y0 = 2 and x0 = 0.8, y0 = 2 and show them" }}{PARA 
0 "" 0 "" {TEXT -1 73 "simultaneously on the surface. Rotate the plot \+
to obtain a suitable view." }}{PARA 0 "" 0 "" {TEXT -1 60 "Be sure you
r points are all within your plot of the surface." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 20 "EXTRA CREDIT PROBLEM" }}
{PARA 0 "" 0 "" {TEXT -1 74 "Change the program to approximate paths o
f steepest ascent. For the saddle" }}{PARA 0 "" 0 "" {TEXT -1 73 "f(x,
y) = x^2-y^2, start at x0 = 4, y0 = 0.1 and go \"uphill\" as rapidly a
s" }}{PARA 0 "" 0 "" {TEXT -1 9 "possible." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "14 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }