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{SECT 0 {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 226 8 "restart;" }}}
{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 202 "" 0 "" {TEXT 227 16 "Co
ntour Diagrams" }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 228 276 "
Graphs provide one way of visualizing functions of two variables. Anot
her important way of visualizing such functions is by drawing their co
ntour diagrams. With many options available,  Maple can plot magnifice
nt contour maps. Let's focus on the following interesting example." }}
}{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 229 11 "E
xample 1. " }{TEXT 228 38 " Examine the behavior of the function " }
{XPPEDIT 18 0 "g(x,y) = x^3-3*x+y^3-3*y;" "6#/-%\"gG6$%\"xG%\"yG,**$F'
\"\"$\"\"\"*&F+F,F'F,!\"\"*$F(F+F,*&F+F,F(F,F." }{TEXT 230 1 " " }
{TEXT 228 7 "  for  " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 230 1 " " }
{TEXT 228 3 " , " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 230 1 " " }
{TEXT 228 7 "  near " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 230 1 " " }
{TEXT 228 5 "= 0, " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 230 1 " " }
{TEXT 228 44 " = 0. Create a contour map for the function." }}{PARA 
201 "" 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 228 22 "We define the fun
ction" }}{PARA 201 "" 0 "" {TEXT 228 1 " " }}}{EXCHG {PARA 201 "> " 0 
"" {MPLTEXT 1 226 26 "g:=(x,y)->x^3-3*x+y^3-3*y;" }}{PARA 203 "" 1 "" 
{XPPMATH 20 "6#>I\"gG6\"f*6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F
%,**$9$\"\"$\"\"\"F/!\"$*$9%F0F1F4F2F%F%F%" }{TEXT 231 1 " " }}}
{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 228 350 "L
et's plot the function in a rectangle around x = 0 , y = 0. Finding a \+
range around x=0, y=0 so that the graph shows clearly the behavior of \+
the function around x=0, y=0 takes a bit of experimentation. One good \+
range is x, y between -2 and 2. We choose this range. Also we lock in \+
right away an orientation that gives us a good view of the surface.  "
 }}{PARA 201 "" 0 "" }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 226 64 "p
lot3d(g(x,y),x=-2..2,y=-2..2,axes=framed,orientation=[150,70]);" }}
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000000000283FFA12F684BDA14840054BDA12F684CE4010000000000008-%+AXESLABE
LSG6%Q\"xF/Q\"yF/Q!F/-%*AXESSTYLEG6#%&FRAMEG-%%FONTG6$%*HELVETICAG\"#5
-%+PROJECTIONG6#$F>!\"\"" 1 2 5 0 10 1 2 1 1 3 2 1.0 70.0 150.0 1 0 "C
urve 1" }}}}{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 201 "" 0 "" 
{TEXT 232 165 "Rotate the surface to get a good sense of its shape! As
 you will learn later, the function has a local minimum, a local maxim
um, and two saddle points in the region." }}{PARA 201 "" 0 "" }{PARA 
201 "" 0 "" {TEXT 228 73 "Maple can create a contour diagram of the fu
nction with a simple command " }{TEXT 233 11 "contourplot" }{TEXT 228 
38 ". The command, as well as the command " }{TEXT 233 7 "display" }
{TEXT 228 40 " that we use below, is contained in the " }{TEXT 233 5 "
plots" }{TEXT 228 74 " package. Hence, we load the package first and t
hen ask for a contour map." }}{PARA 201 "" 0 "" }}{EXCHG {PARA 201 "> \+
" 0 "" {MPLTEXT 1 226 12 "with(plots):" }}{PARA 205 "" 1 "" {TEXT 234 
49 "Warning, the name changecoords has been redefined" }{TEXT 234 1 "
\n" }}}{EXCHG {PARA 201 "" 0 "" }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 226 36 "contourplot(g(x,y),x=-2..2,y=-2..2);" }}{PARA 204 "" 1 "" 
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1" }}}}{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 
235 25 "Maple drew a contour map." }{TEXT 232 93 " Note the characteri
stic appearance of the diagram near local extrema and near saddle poin
ts!" }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 228 481 "As you will
 see below, we can obtain a much fancier contour map by using a few ad
ditional options. Before we do it, let's consider how the contour diag
ram is created. As you know, a contour diagram is obtained by intersec
ting the graph of a function with horizontal planes corresponding to e
qually-spaced  values of z, and projecting the resulting curves onto t
he xy-plane. Maple can illustrate very nicely how a contour diagram is
 created by plotting the graph of a function  in  " }{TEXT 233 12 "pat
chcontour" }{TEXT 228 37 " style. Let's do it for our function " }
{XPPEDIT 18 0 "g(x,y);" "6#-%\"gG6$%\"xG%\"yG" }{TEXT 230 1 " " }
{TEXT 228 2 ". " }}{PARA 201 "" 0 "" }}{EXCHG {PARA 201 "> " 0 "" 
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JECTIONG6#$F>!\"\"" 1 4 5 0 10 1 2 1 1 3 2 1.0 70.0 150.0 1 0 "Curve 1
" }}}}{EXCHG {PARA 201 "> " 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 228 
165 "Here you see the contours before they are projected onto the xy-p
lane, that is, the intersections of the graph of the function with equ
ally-spaced horizontal planes." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 ""
 {TEXT 228 109 "Maple can be used to illustrate how intersections them
selves are obtained, as well.  Let's draw the graph of " }{XPPEDIT 18 
0 "g(x,y);" "6#-%\"gG6$%\"xG%\"yG" }{TEXT 230 1 " " }{TEXT 228 73 " an
d the horizontal planes z = -2, z = 0, z =2 in one coordinate system. \+
" }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 229 5 "Note." }{TEXT 
228 109 " Whenever you want to display more than one 3d graph in one c
oordinate system, you'd be better off using the " }{TEXT 233 7 "displa
y" }{TEXT 228 25 " command, as we do below." }}{PARA 201 "" 0 "" }
{PARA 201 "" 0 "" {TEXT 228 126 "We define the four plots first. Some \+
options that are common to all plots can be specified directly under t
he display command." }}{PARA 201 "" 0 "" }}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 226 47 "p1:=plot3d(g(x,y),x=-2..2,y=-2..2,color=black):" }}
}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 226 42 "p2:=plot3d(-2,x=-2..2,y
=-2..2,color=blue):" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 226 41 "
p3:=plot3d(0,x=-2..2,y=-2..2,color=blue):" }}}{EXCHG {PARA 201 "> " 0 
"" {MPLTEXT 1 226 41 "p4:=plot3d(2,x=-2..2,y=-2..2,color=blue):" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 226 69 "display([p1,p2,p3,p4],ax
es=framed,style=hidden,orientation=[150,70]);" }}{PARA 204 "" 1 "" 
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