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256 34 "Why a saddle is called a \"saddle\"?" }}{PARA 0 "" 0 "" {TEXT 
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{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
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on of the " }{TEXT 257 15 "gradient vector" }{TEXT -1 15 " of a functi
on " }{XPPEDIT 18 0 "f(x,y);" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 13 " at
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0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 32 "The following command plots the " }{TEXT 259 12 "level
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\"yG" }{TEXT -1 27 ", that is the solutions of " }{XPPEDIT 18 0 "f(x,y
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35.000000 29.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 138 "Notice that the gradient vector field \+
is orthogonal to the level curves. To see this better, in the next pic
ture we plot the two together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "plot1:=gradplot(f(x,y),x=-3
..3,y=-3..3,arrows=thick):plot2:=contourplot(f(x,y),x=-3..3,y=-3..3,co
ntours=15):display(plot1,plot2,scaling=constrained);" }}{PARA 13 "" 1 
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ONSTRAINEDG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "
Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "
Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve \+
14" "Curve 15" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 260 12 "Excercise 1." }{TEXT -1 1 " " }{TEXT 261 96 "Write \+
down a system of differential equations in the plane defined by the gr
adient vector field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "It is pretty o
bvious from the picture that the system from Excercise 1 will have a s
addle point at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 93 "What does this have to do with the word \"saddle
\"? Let us visualize the graph of the function " }{XPPEDIT 18 0 "f(x,y
);" "6#-%\"fG6$%\"xG%\"yG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot3d(f(x,y),x=-3..3,y=-3..3,style
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LABELSG6%%\"xG%\"yGQ!6\"-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 1 1 
1 2 1.000000 64.000000 66.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Rotate the image aro
und to see that it indeed is shaped like a saddle." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 12 "Excercise 2." }{TEXT -1 
1 " " }{TEXT 263 103 "Verify that the type of the equilibrium point of
 the system from Excercise 1 at the origin is a saddle." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 12 "Excercise 3." }
{TEXT -1 2 "  " }{TEXT 265 48 "Verify that the direction of the gradie
nt vector" }{TEXT -1 6 " Grad " }{XPPEDIT 18 0 "f(x[0], y[0]);" "6#-%
\"fG6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 2 " i" }{TEXT 266 47 "s the d
irection of steepest ascent of the graph" }{TEXT -1 1 " " }{TEXT 267 
2 "of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y);" "6#-%\"fG6$%\"xG%\"yG
" }{TEXT -1 2 "  " }{TEXT 268 13 "at the point " }{TEXT -1 1 "(" }
{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 2 ") \+
" }{TEXT 269 52 "while the magnitude is the slope in that direction. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 14 "Exce
rcise 4.**" }{TEXT 270 94 " Finally, here is the physical interpretati
on of the gradient. Imagine an object sliding along" }}{PARA 0 "" 0 "
" {TEXT 271 12 "the graph of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y);
" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 272 76 "in the above p
icture. Then the force is (up to a constant multiple) equal to" }}
{PARA 0 "" 0 "" {TEXT -1 55 "                                         \+
        -Grad " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }}
{PARA 0 "" 0 "" {TEXT 273 12 "Verify this." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "13 7 1" 88 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }