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{SECT 0 {EXCHG {PARA 200 "" 0 "" }{PARA 200 "" 0 "" {TEXT 205 29 "Grad
ient systems - an example" }}{PARA 200 "" 0 "" }}{EXCHG {PARA 200 "> "
 0 "" {MPLTEXT 1 206 26 "with(plots):with(DEtools):" }}}{EXCHG {PARA 
200 "" 0 "" {TEXT 207 29 "Recall the definition of the " }{TEXT 208 
15 "gradient vector" }{TEXT 207 15 " of a function " }{XPPEDIT 18 0 "f
(x,y);" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 207 1 " " }{TEXT 207 13 " at a p
oint (" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yGF%" }{TEXT 
207 1 " " }{TEXT 207 30 "). This vector originates at (" }{XPPEDIT 18 
0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yGF%" }{TEXT 207 1 " " }{TEXT 207 
21 ") and has coordinates" }}{PARA 200 "" 0 "" {TEXT 207 47 "         \+
                                 Grad " }{XPPEDIT 18 0 "f(x[0],y[0]);"
 "6#-%\"fG6$&%\"xG6#\"\"!&%\"yGF(" }{TEXT 207 1 " " }{TEXT 207 6 "  = \+
( " }{XPPEDIT 18 0 "diff(f(x[0],y[0]),x);" "6#-%%diffG6$-%\"fG6$&%\"xG
6#\"\"!&%\"yGF+F*" }{TEXT 207 1 " " }{TEXT 207 1 "," }{XPPEDIT 18 0 "d
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{TEXT 207 1 " " }{TEXT 207 2 ") " }}{PARA 200 "" 0 "" }{PARA 200 "" 0 
"" {TEXT 207 20 "Maple has a command " }{TEXT 209 8 "gradplot" }{TEXT 
207 78 " which calculates and plots gradient vectors at a grid of poin
ts in the plane." }}{PARA 200 "" 0 "" }{PARA 200 "" 0 "" {TEXT 207 38 
"Let us consider the following example:" }}{PARA 200 "" 0 "" }}{PARA 
200 "" 0 "" }{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 206 33 "G:=(x,y)->(
-5*x^2+5*x^4/2+y^2)/2;" }}{PARA 201 "" 1 "" {XPPMATH 20 "6#>I\"GG6\"f*
6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"##!\"&F0*$F/\"
\"%#\"\"&F4*$9%F0#\"\"\"F0F%F%F%" }{TEXT 210 1 " " }}}{EXCHG {PARA 
200 "" 0 "" }{PARA 200 "" 0 "" {TEXT 207 32 "The following command plo
ts the " }{TEXT 208 12 "level curves" }{TEXT 207 4 " of " }{XPPEDIT 2 
0 "G(x,y)" "6#-I\"GG6\"6$I\"xGF%I\"yGF%" }{TEXT 207 1 " " }{TEXT 207 
27 ", that is the solutions of " }{XPPEDIT 2 0 "G(x,y)=C" "6#/-I\"GG6
\"6$I\"xGF&I\"yGF&I\"CGF&" }{TEXT 207 1 " " }}{PARA 200 "" 0 "" }}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 206 75 "contourplot(G(x,y),x=-2.
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