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2 9 1 4 2 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" "Curve 16" 
"Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT -1 101 "We see that for some choices of (xB,yB) there i
s no solution, while for others there are two of them!" }}{PARA 0 "" 
0 "" {TEXT -1 67 "\nFor a specific example, let's look at the case whe
re xB=1, yB=0.9." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "yat1:=su
bs(x=1,y);\nplot([0.9,yat1],b=0.01..1,0..2,numpoints=300,title=\"y at \+
x=1\");\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yat1G*&%\"bG\"\"\"-%%c
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`o7$Fhbn$\"3R&Glw7T\\M\"F[`o7$F[cn$\"3mkqw*Q,eO\"F[`o7$F^cn$\"3JSv')Q&
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rV_\"[j!3V:F[`o-F[dn6&F]dnFadnF^dnFadn-%&TITLEG6#Q)y~at~x=16\"-%+AXESL
ABELSG6$Q\"bFgarQ!Fgar-%%VIEWG6$;$Fhcn!\"#Fgcn;Fadn$\"\"#Ficn" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 169 "We will use \"fsolve
\" to numerically find the two b values for which y(1)=0.9.\nNote that
 the starting guesses given below to \"fsolve\" were obtained from the
 plot above by" }}{PARA 0 "" 0 "" {TEXT -1 72 "clicking on the points \+
where the green curve crosses the red line y=0.9." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 69 "Digits:=20;\nb1:=fsolve(yat1=0.9,b=0.25);\nb2
:=fsolve(yat1=0.9,b=0.77);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Digit
sG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G$\"549#[pIrCuZ#!#?" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G$\"5Pj\\FV_Zu+y!#?" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT -1 84 "and here are the corresponding curv
es, and the surfaces they generate by revolution:" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 363 "y1:=subs(b=b1,y);\ny2:=subs(b=b2,y);\nplot([
y1,y2],x=0..1,view=[0..1,0..1],color=[red,blue]);\np1:=plot3d([x,y1*co
s(t),y1*sin(t)],x=0..1,t=0..2*Pi,\n                    grid=[20,40],co
lor=red):\np2:=plot3d([x,y2*cos(t),y2*sin(t)],x=0..1,t=0..2*Pi,\n     \+
               grid=[20,40],color=blue):\nplots[display](p1,p2,axes=bo
xed,labels=['x','y','z'],\nlightmodel=light2);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y1G,$*&$\"549#[pIrCuZ#!#?\"\"\"-%%coshG6#,&*&$\"5\\*
o-*=.'\\k.%!#>F*%\"xGF*!\"\"-%(arccoshG6#$\"5\\*o-*=.'\\k.%F2F*F*F*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*&$\"5Pj\\FV_Zu+y!#?\"\"\"-%%
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17%F]hy$\"5%pH'yHA,$[3\"F1$!5*Q8\\#)o')zV$*)F17%F]hy$\"5z?H![_<dR]#F1$
!5gdjBo/ImW')F17%F]hy$\"5$QWWFE0L#eQF1$!5v(4Y$\\6R0J\")F17%F]hy$\"5;jh
.e?Fe7^F1$!5&ob$H/$zaoS(F17%F]hy$\"5X)yN'e\"=>XB'F1$!5F)fO%4E?#3\\'F17
%F]hy$\"5D#*y9jq[)\\>(F1$!572_=9y.o1aF17%F]hy$\"5y0k)y3B/\"pzF1$!5EbkR
R[&3D=%F17%F]hy$\"5JA#HCv(z#o`)F1$!5a%\\I8U%>,]GF17%F]hy$\"5Km17ujBX$)
))F1$!5>A)*)>x_,PW\"F17%F]hyFa^il$!5\\5.**3eF()GeF[x-Fcdz6&FedzF)F)Ffd
z-%+AXESLABELSG6%%\"xG%\"yG%\"zG-%*AXESSTYLEG6#%$BOXG-%+LIGHTMODELG6#%
(LIGHT_2G" 1 2 0 1 10 0 2 1 4 2 1 1.000000 71.000000 46.000000 0 0 "Cu
rve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 154 "As w
e remarked in class, the existence of TWO stationary solutions is surp
rising,\nand a bit unsettling.\n\n We can compute the areas, at least \+
numerically:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "I1:=evalf
(Int(2*Pi*y1*sqrt(1+diff(y1,x)^2),x=0..1));\nI2:=evalf(Int(2*Pi*y2*sqr
t(1+diff(y2,x)^2),x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G$
\"5:`.F-(oV$oi!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G$\"5$4$))*R
4Yxao&!#>" }}}{PARA 0 "" 0 "" {TEXT -1 261 "\nThe blue one looks sort \+
of plausible as the area minimizer.\nBut what kind of solution is the \+
red one (which has a larger area)? Is it a(nother) local minimum, or w
hat?\n\nTo try to figure this out, let us explore the neighborhoods of
 these two stationary points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 100 "First let's perturb away from them along
 the direction of sin(4*Pi*x) -- adding a bit of a wiggle --" }}{PARA 
0 "" 0 "" {TEXT -1 42 "and see what happens to the surface area:\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "I1:=evalf(Int(2*Pi*y1*sqrt(
1+diff(y1,x)^2),x=0..1));\nI2:=evalf(Int(2*Pi*y2*sqrt(1+diff(y2,x)^2),
x=0..1));\npert:=sin(4*Pi*x);\nunassign('y'):\nIy:=y->evalf(Int(2*Pi*y
*sqrt(1+diff(y,x)^2),x=0..1));\nepsvals:=[seq(0.05*i,i=-4..4)];\nplot(
[seq([eps,Iy(y1+eps*pert)],\n               eps=epsvals)],\n          \+
     labels=[epsilon,\"area\"],\n               style=point,color=red)
:\np1:=%:\nplot([seq([eps,Iy(y2+eps*pert)],\n               eps=epsval
s)],\n               labels=[epsilon,\"area\"],\n               style=
point,color=blue):\np2:=%:\nplots[display](p1,p2);\n\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#I1G$\"5:`.F-(oV$oi!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I2G$\"5$4$))*R4Yxao&!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%pertG-%$sinG6#,$*(\"\"%\"\"\"%#PiGF+%\"xGF+F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IyGj+6#%\"yG6\"6$%)operatorG%&arrow
GF(-%&evalfG6#-%$IntG6$,$**\"\"#\"\"\"%#PiGF59$F5-%%sqrtG6#,&F5F5*$)-%
%diffG6$F7%\"xGF4F5F5F5F5/FA;\"\"!F5F(F(F(6#FD" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(epsvalsG7+$!#?!\"#$!#:F($!#5F($!\"&F($\"\"!F0$\"\"&F
($\"#5F($\"#:F($\"#?F(" }}{PARA 13 "" 1 "" {GLPLOT2D 324 324 324 
{PLOTDATA 2 "6&-%'CURVESG6%7+7$$!37+++++++?!#=$\"33lYH:Qd'H(!#<7$$!3'*
*************\\\"F*$\"3;m$3l6o&\\oF-7$$!3/+++++++5F*$\"3r%>=k*f4BlF-7$
$!3G+++++++]!#>$\"33#=7l)e:KjF-7$$\"\"!F@$\"3R.F-(oV$oiF-7$$\"3G++++++
+]F;$\"3l/WRToENjF-7$$\"3/+++++++5F*$\"3z0M!)o$R'RlF-7$$\"3'**********
****\\\"F*$\"3&*=E87cWuoF-7$$\"37+++++++?F*$\"3U*Ru<IEcI(F--%'COLOURG6
&%$RGBG$\"*++++\"!\")F?F?-%&STYLEG6#%&POINTG-F$6%7+7$F($\"3bfb$yIp#Q5!
#;7$F/$\"3@W0[\\<dt')F-7$F4$\"39GUu])p'yrF-7$F9$\"3s*e>x'[2#4'F-7$F?$
\"31))*R4Yxao&F-7$FD$\"3:NOX>=)*)3'F-7$FI$\"3*4\"*fO\"[XhrF-7$FN$\"3!R
&fi<SNW')F-7$FS$\"3&[>*pa!*pM5Fbo-FX6&FZF?F?FenFhn-%+AXESLABELSG6%%(ep
silonGQ%area6\"-%%FONTG6#%(DEFAULTG-%%VIEWG6$FfqFfq" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 
0 "" 0 "" {TEXT -1 20 "What do you observe?" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Next, let's perturb away from t
hem along the direction of sin(Pi*x) -- just basically bowing" }}
{PARA 0 "" 0 "" {TEXT -1 14 "it up or down:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 481 "I1:=evalf(Int(2*Pi*y1*sqrt(1+diff(y1,x)^2),x=0..1)
);\nI2:=evalf(Int(2*Pi*y2*sqrt(1+diff(y2,x)^2),x=0..1));\npert:=sin(Pi
*x);\nunassign('y'):\nepsvals:=[seq(0.05*i,i=-4..4)];\nIy:=y->evalf(In
t(2*Pi*y*sqrt(1+diff(y,x)^2),x=0..1));\nplot([seq([eps,Iy(y1+eps*pert)
],eps=epsvals)],\n               labels=[epsilon,\"area\"],style=point
,color=red):\np1:=%:\nplot([seq([eps,Iy(y2+eps*pert)],eps=epsvals)],\n
               labels=[epsilon,\"area\"],style=point,color=blue):\np2:
=%:\nplots[display](p1,p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G$
\"5:`.F-(oV$oi!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G$\"5$4$))*R
4Yxao&!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pertG-%$sinG6#*&%#PiG
\"\"\"%\"xGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsvalsG7+$!#?!\"#
$!#:F($!#5F($!\"&F($\"\"!F0$\"\"&F($\"#5F($\"#:F($\"#?F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#IyGj+6#%\"yG6\"6$%)operatorG%&arrowGF(-%&eval
fG6#-%$IntG6$,$**\"\"#\"\"\"%#PiGF59$F5-%%sqrtG6#,&F5F5*$)-%%diffG6$F7
%\"xGF4F5F5F5F5/FA;\"\"!F5F(F(F(6#FD" }}{PARA 13 "" 1 "" {GLPLOT2D 
324 324 324 {PLOTDATA 2 "6&-%'CURVESG6%7+7$$!37+++++++?!#=$\"3[)))HQ#*
*f7h!#<7$$!3'**************\\\"F*$\"3ir*H^!p$H='F-7$$!3/+++++++5F*$\"3
)*fL*f7T9B'F-7$$!3G+++++++]!#>$\"3&HBh\\*QSfiF-7$$\"\"!F@$\"3R.F-(oV$o
iF-7$$\"3G+++++++]F;$\"3C8#Hv>^+E'F-7$$\"3/+++++++5F*$\"3x#f1>YUmB'F-7
$$\"3'**************\\\"F*$\"3mw#*o7-i+iF-7$$\"37+++++++?F*$\"37LFvED%
\\:'F--%'COLOURG6&%$RGBG$\"*++++\"!\")F?F?-%&STYLEG6#%&POINTG-F$6%7+7$
F($\"3uGB+4:p$*eF-7$F/$\"3**\\paquW;eF-7$F4$\"3X]cJ(eu-v&F-7$F9$\"3(f;
>C+AMq&F-7$F?$\"31))*R4Yxao&F-7$FD$\"3vJ![([m52dF-7$FI$\"3bdIUx&*RzdF-
7$FN$\"3(y(\\K90y7fF-7$FS$\"3<&yo%o:m:hF--FX6&FZF?F?FenFhn-%+AXESLABEL
SG6%%(epsilonGQ%area6\"-%%FONTG6#%(DEFAULTG-%%VIEWG6$FeqFeq" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}
}}{PARA 0 "" 0 "" {TEXT -1 226 "What do you observe this time?\n\nWhat
 \"kind\" of stationary points are each of these two solutions of the \+
Euler-Lagrange equation?\nHow can you be sure? What lessons does this \+
teach us for future applications of this technique?\n\n" }}}{MARK "12 \+
0" 128 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }