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57.000000 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 330 "Plateau's problem is that of finding a surface of least area s
panning a given contour.  Plateau was a 19th century physicists who ex
perimented with soap films.  If one dips a contour made of wire into a
 soap solution, the surface tension will cause a soap film with least \+
surface area to span the contour when the wire is emerged." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Here we will con
sider a simple special case: when the contour consists of two parallel
 congruent rings of radius 1 a distance" }{TEXT 257 2 " l" }{TEXT -1 
167 " apart.  We will assume that the surface has the same symmetry as
 the contour, hence is a surface of revolution.  We will describe the \+
surface by its generating curve " }{XPPEDIT 18 0 "y=y(x)" "6#/%\"yG-F$
6#%\"xG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 41 "It is conven
ient to scale the parameters " }{TEXT 258 1 "x" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 39 " so that the rings are@th
e location " }{XPPEDIT 18 0 " x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 36 ".  The r
adius of the ring is then 1/" }{XPPEDIT 18 0 "l" "6#%\"lG" }{TEXT -1 
47 " .  For example, suppose the curve is given by:" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "y := 1-.2*cos(Pi*x/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG,&\"\"\"F&-%$cosG6#,$*&%#PiGF&%\"xGF&#F&\"\"#$!\"
#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(y,x=-1..1,0.
.1,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 
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jn7%F]o$!1(RF$>Jz!=&Ff]mF_o7%Fco$!1#GKp;O(p^Ff]mFeo7%F]oFd\\sFio7%FhnF
a\\sF\\p7%FYF^\\sF_p7%FTF[\\sFbp7%FOFh[sFep7%FhpFe[sFjp7%FEFb[sF]q7%F@
F_[sF`q7%F;F\\[sFcq7%Ffq$!1$QRVmjw7'Ff]mFhq7%F[r$!1xKu9PZ$H'Ff]mF]r7%F
)FcjrF`r-%(SCALINGG6#%,CONSTRAINEDG" 1 1 0 1 0 2 1 1 1 1 1.000000 
45.000000 45.000000 0 }}}}{PARA 0 "" 0 "" {TEXT -1 68 "Recall that the
 surface area of a surface of revolution is given by:" }}{PARA 258 "" 
0 "" {XPPEDIT 18 0 "A = 2*Pi*int(y*sqrt(1+(dy/dx)^2),x=-l..l)" "6#/%\"
AG*(\"\"#\"\"\"%#PiGF'-%$intG6$*&%\"yGF'-%%sqrtG6#,&\"\"\"F'*$*&%#dyGF
'%#dxG!\"\"\"\"#F'F'/%\"xG;,$%\"lGF7F=F'" }{TEXT -1 1 "," }}{PARA 0 "
" 0 "" {TEXT -1 24 "which after the scaling " }{XPPEDIT 18 0 "x->x/l" 
"6#R6#%\"xG7\"6$%)operatorG%&arrowG6\"*&F%\"\"\"%\"lG!\"\"F*F*F*" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "y->y/l" "6#R6#%\"yG7\"6$%)operatorG%&a
rrowG6\"*&F%\"\"\"%\"lG!\"\"F*F*F*" }{TEXT -1 10 ", becomes:" }}{PARA 
259 "" 0 "" {XPPEDIT 18 0 "A = 2*Pi*l^2*int(y*sqrt(1+(dy/dx)^2),x=-1..
1)" "6#/%\"AG**\"\"#\"\"\"%#PiGF'%\"lG\"\"#-%$intG6$*&%\"yGF'-%%sqrtG6
#,&\"\"\"F'*$*&%#dyGF'%#dxG!\"\"\"\"#F'F'/%\"xG;,$\"\"\"F9\"\"\"F'" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "arclength :=
 sqrt(1+diff(y(x),x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*arcleng
thG*$,&\"\"\"F'*$-%%diffG6$-%\"yG6#%\"xGF/\"\"#F'#F'F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "L := y(x)*arclength;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"LG*&-%\"yG6#%\"xG\"\"\",&F*F**$-%%diffG6$F&F
)\"\"#F*#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A := 2*Pi*
l^2*Int(L,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*(%#PiG
\"\"\"%\"lG\"\"#-%$IntG6$*&-%\"yG6#%\"xGF(,&F(F(*$-%%diffG6$F/F2F*F(#F
(F*/F2;!\"\"F(F(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 55 "The surface is to have least area. Thus, if we replace \+
" }{TEXT 259 1 "y" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "y+t*z" "6#,&%\"y
G\"\"\"*&%\"tGF%%\"zGF%F%" }{TEXT -1 67 ", where z vanishes@the end
point, the area will be smallest when " }{XPPEDIT 18 0 " t=0" "6#/%\"t
G\"\"!" }{TEXT -1 78 ".  We use ordinary calculus to derive a necessar
y condition for this to occur." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "subs(y(x)=y(x)+t*z(x),A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*
(%#PiG\"\"\"%\"lG\"\"#-%$IntG6$*&,&-%\"yG6#%\"xGF&*&%\"tGF&-%\"zGF0F&F
&F&,&F&F&*$-%%diffG6$F-F1F(F&#F&F(/F1;!\"\"F&F&F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "diff(%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,$*(%#PiG\"\"\"%\"lG\"\"#-%$IntG6$,&*&-%\"zG6#%\"xGF&,&F&F&*$,&-%%di
ffG6$-%\"yGF0F1F&*&%\"tGF&-F66$F.F1F&F&F(F&#F&F(F&**,&F8F&*&F;F&F.F&F&
F&F2#!\"\"F(F4F&F<F&F&/F1;FCF&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "subs(t=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%
#PiG\"\"\"%\"lG\"\"#-%$IntG6$,&*&-%\"zG6#%\"xGF&,&F&F&*$-%%diffG6$-%\"
yGF0F1F(F&#F&F(F&**F7F&F2#!\"\"F(F4F&-F56$F.F1F&F&/F1;F<F&F&F(" }}}
{PARA 0 "" 0 "" {TEXT -1 86 "If we integrate the second term by parts,
 we obtain that the variation of the area is:" }}{PARA 260 "" 0 "" 
{XPPEDIT 18 0 "2*Pi*l^2*int(z(x)*(sqrt(1+diff(y(x),x)^2)-diff(y(x)*dif
f(y(x),x)/sqrt(1+diff(y(x),x)^2),x)),x=-1..1)" "6#**\"\"#\"\"\"%#PiGF%
%\"lG\"\"#-%$intG6$*&-%\"zG6#%\"xGF%,&-%%sqrtG6#,&\"\"\"F%*$-%%diffG6$
-%\"yG6#F0F0\"\"#F%F%-F96$*(-F<6#F0F%-F96$-F<6#F0F0F%-F36#,&\"\"\"F%*$
-F96$-F<6#F0F0\"\"#F%!\"\"F0FRF%/F0;,$\"\"\"FR\"\"\"F%" }}{PARA 0 "" 
0 "" {TEXT -1 62 "By our discussion above, this must be zero for all v
ariations " }{XPPEDIT 18 0 "z(x)" "6#-%\"zG6#%\"xG" }{TEXT -1 155 " wh
ich vanish@the end points.  It is not hard to see from this that th
e expression in parenthesis must be zero.  This gives a differential e
quation for " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "eq := diff(y(x)*diff(y(x),x)/arclength,x)-arclength=0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,**&-%%diffG6$-%\"yG6#%\"xGF.\"\"#,&
\"\"\"F1*$F(F/F1#!\"\"F/F1*(F+F1-F)6$F(F.F1F0F3F1**F+F1F(F/F0#!\"$F/F6
F1F4*$F0#F1F/F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sim
plify(eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&,(*$-%%diffG6$-%\"y
G6#%\"xGF.\"\"#\"\"\"*&F+F0-F)6$F(F.F0!\"\"F0F0F0,&F0F0F'F0#!\"$F/F4\"
\"!" }}}{PARA 0 "" 0 "" {TEXT -1 48 "We may multiply the equation by t
he denominator:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eq := %*a
rclength^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*$-%%diffG6$-%
\"yG6#%\"xGF.\"\"#!\"\"*&F+\"\"\"-F)6$F(F.F2F2F0F2\"\"!" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We now ask Maple V
 to solve this equation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 
"### WARNING: `dsolve` has been extensively rewritten, many new result
 forms can occur and options are slightly different, see help page for
 details\nsols := dsolve(eq,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%solsG6$/%\"xG,&*&%$_C1G#!\"\"\"\"#-%#lnG6#,&*&F*#\"\"\"F--%\"yG6#F'
F4F4*$,&F,F4*&F5F-F*F4F4F3F4F4F4%$_C2GF,/F',&F)F,F;F," }}}{PARA 0 "" 
0 "" {TEXT -1 55 "Note that the solution involves two arbitray constan
ts " }{TEXT 262 3 "_C1" }{TEXT -1 5 " and " }{TEXT 263 4 "_C2," }
{TEXT -1 25 " and is given implicitly." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 17 "We now solve for " }{TEXT 260 1 "y" }
{TEXT -1 13 " in terms of " }{TEXT 261 1 "x" }{TEXT -1 56 ".  We first
 separate the two solutions Maple V returned." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "sol1 := rhs(sols[1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%sol1G,&*&%$_C1G#!\"\"\"\"#-%#lnG6#,&*&F'#\"\"\"F*-%
\"yG6#%\"xGF1F1*$,&F)F1*&F2F*F'F1F1F0F1F1F1%$_C2GF)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "sol2 := rhs(sols[2]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%sol2G,&*&%$_C1G#!\"\"\"\"#-%#lnG6#,&*&F'#\"\"\"F*-
%\"yG6#%\"xGF1F1*$,&F)F1*&F2F*F'F1F1F0F1F1F)%$_C2GF)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x=sol1,y(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(,&\"\"\"F&*$-%$expG6#*&%$_C1G#F&\"\"#,&%\"xGF&%$_C2
GF&F&F.F&F&F(!\"\"F,#F2F.F-" }}}{PARA 0 "" 0 "" {TEXT -1 11 "We may se
t " }{XPPEDIT 18 0 "_C2=0" "6#/%$_C2G\"\"!" }{TEXT -1 124 " for the so
lution has to be even (otherwise, we would replace half of the solutio
n with the half which has the lesser area)." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "subs(_C2=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$*(,&\"\"\"F&*$-%$expG6#*&%\"xGF&%$_C1G#F&\"\"#F/F&F&F(!\"\"F-#F0F/F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a>0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(subs(_C1=a^2,%));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&-%$expG6#,$*&%\"xG\"\"\"%#a|ir
GF,!\"#F,F,F,F,-F'6#F*F,F-!\"\"#F,\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 
28 "This is simply the function:" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "y
(x) = cosh(a*x)/a" "6#/-%\"yG6#%\"xG*&-%%coshG6#*&%\"aG\"\"\"F'F.F.F-!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "The surface of re
volution generated by this solution is called the " }{TEXT 268 8 "cate
noid" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot
3d([cosh(t),theta,t], t=-.7.. .7, theta=0..2*Pi, coords=cylindrical);
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h[KF/$\"1)\\'*)e,S77F+Fghr7%$!1&pa\"G]%eF'F/FQFghr7%$!1Q\"oT:&Qv))F/$
\"1y\"oT:&Qv))F/Fghr7%FY$\"1WZ:G]%eF'F/Fghr7%Fhn$\"1[ozMkh[KF/Fghr7%F]
o$\"1)>9v(ybbSFJFghr7%FboFdoFghr7%Fgo$!1uY:G]%eF'F/Fghr7%$!1&>oT:&Qv))
F/$!1A\"oT:&Qv))F/Fghr7%$!1gZ:G]%eF'F/FcpFghr7%$!1lozMkh[KF/FhnFghr7%$
!1H=I5Lj/eFJF]oFghr7%$\"1`nzMkh[KF/$!1*\\'*)e,S77F+Fghr7%$\"1gY:G]%eF'
F/FcqFghr7%$\"13\"oT:&Qv))F/$!12#oT:&Qv))F/Fghr7%F[r$!1\"ya\"G]%eF'F/F
ghr7%F`rFbrFghr7%F)$!1'RG]v:66)FJFghr" 1 1 0 1 0 2 1 1 1 2 1.000000 
45.000000 45.000000 0 }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 75 "We now go through the same steps with the second sol
ution Maple V returned." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s
olve(x=sol2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&\"\"\"F&*$
-%$expG6#,$*&%$_C1G#F&\"\"#,&%\"xGF&%$_C2GF&F&!\"\"F/F&F&F(F3F-#F3F/F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(_C2=0,%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&\"\"\"F&*$-%$expG6#,$*&%\"xGF&%$
_C1G#F&\"\"#!\"\"F0F&F&F(F1F.#F1F0F/" }}}{PARA 0 "" 0 "" {TEXT -1 46 "
Note that this is the same solution as before." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We now verify that the so
lution indeed satisfies the equation." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "a := 'a';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y := cosh(a*x)/a;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*&-%%coshG6#*&%\"aG\"\"\"%\"xGF+
F+F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalb(simplify
(subs(y(x)=Y,eq)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{PARA 0 "" 0 "" {TEXT -1 49 "The solution now involves one arbitrary c
onstant " }{TEXT 265 1 "a" }{TEXT -1 134 ".  This constant is determin
ed by the boundary conditions.  In order to satisfy the boundary condi
tions, we must satisfy the equation:" }}{PARA 262 "" 0 "" {XPPEDIT 18 
0 "a/cosh(a) = l" "6#/*&%\"aG\"\"\"-%%coshG6#F%!\"\"%\"lG" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 146 "This is a nonlinear equation who
se solutions cannot be written explicitly.  In order to investigate th
e number of solutions, we plot the function " }{XPPEDIT 18 0 "a/cosh(a
)" "6#*&%\"aG\"\"\"-%%coshG6#F$!\"\"" }{TEXT -1 1 ":" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "k := a/cosh(a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG*&%\"aG\"\"\"-%%coshG6#F&!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "plot(k,a=0..3,y=0..1);" }}{PARA 13 "" 1 "
" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6%-%'CURVESG6$7U7$\"\"!F(7$$\"1++
+DJdpK!#<$\"18A_)HEyE$F,7$$\"1+++]i9RlF,$\"1.`sz->DlF,7$$\"1++vVV)RQ*F
,$\"19JU6\"=GM*F,7$$\"1++vVA)GA\"!#;$\"10%>H8&z87F<7$$\"1++]Peui=F<$\"
1<O#*y&*)3$=F<7$$\"1++]i3&o]#F<$\"1'eXj'H4ICF<7$$\"1++voX*y9$F<$\"1`]J
&[=\")*HF<7$$\"1++vVTAUPF<$\"1P'[/6bY\\$F<7$$\"1++v$*zhdVF<$\"1@s3#)[F
uRF<7$$\"1++v$>fS*\\F<$\"1)3d(>9/IWF<7$$\"1++v=$f%GcF<$\"1K%y]yp6%[F<7
$$\"1+++Dy,\"G'F<$\"1AK;9Ga<_F<7$$\"1++]7<zboF<$\"1(Hc)oL\\4bF<7$$\"1,
++v4&G](F<$\"1Vyb%>vSz&F<7$$\"1+++Drc_\")F<$\"1WK7.-%Q.'F<7$$\"1+++D!*
oy()F<$\"13!pTu`GA'F<7$$\"1++v$pnsM*F<$\"1BlaFIRgjF<7$$\"1++]siL-5!#:$
\"1\"G?,>GT['F<7$$\"1+++!R5'f5F]q$\"1/beVZKdlF<7$$\"1+]P/QBE6F]q$\"1mN
!)ej#)3mF<7$$\"1+++:o?&=\"F]q$\"1N\"H;gMni'F<7$$\"1+]Pa&4*\\7F]q$\"1Z*
*)=\\0$>mF<7$$\"1+]7j=_68F]q$\"1s)eeAH&)e'F<7$$\"1++vVy!eP\"F]q$\"1`Z)
p4%\\MlF<7$$\"1+](=WU[V\"F]q$\"1<Pj\\cUnkF<7$$\"1++DJ#>&)\\\"F]q$\"1,X
+)H)oyjF<7$$\"1+]P>:mk:F]q$\"1\"39z0=4F'F<7$$\"1+]iv&QAi\"F]q$\"1a*QAO
9g;'F<7$$\"1++vtLU%o\"F]q$\"1Zpc`&*)G/'F<7$$\"1+++bjm[<F]q$\"1c:Cm'>n!
fF<7$$\"1++vyb^6=F]q$\"1W&[Bj\"HmdF<7$$\"1+]PMaKs=F]q$\"12_%>,\"*\\i&F
<7$$\"1++D6W%)R>F]q$\"11x%fHxKY&F<7$$\"1+++:K^+?F]q$\"1=:t3&yZJ&F<7$$
\"1++]7,Hl?F]q$\"1Wwlm\"pR:&F<7$$\"1+]P4w)R7#F]q$\"1%3/ZFbq+&F<7$$\"1+
+]x%f\")=#F]q$\"1cHGLe$f%[F<7$$\"1+]P/-a[AF]q$\"1J7EQ%3Xp%F<7$$\"1+](=
Yb;J#F]q$\"1>]x:4,PXF<7$$\"1++]i@OtBF]q$\"1UQ#zJAVQ%F<7$$\"1+]PfL'zV#F
]q$\"1=]#>3DjA%F<7$$\"1+++!*>=+DF]q$\"1o9=!eXj2%F<7$$\"1++DE&4Qc#F]q$
\"1Fy([rRb#RF<7$$\"1+]P%>5pi#F]q$\"1RQIKy$)yPF<7$$\"1+++bJ*[o#F]q$\"1g
x_&pZnk$F<7$$\"1++Dr\"[8v#F]q$\"1$z)p@mv)\\$F<7$$\"1+++Ijy5GF]q$\"12uB
'o7'pLF<7$$\"1+]P/)fT(GF]q$\"1EZHL5RNKF<7$$\"1+]i0j\"[$HF]q$\"1H(H\\an
.6$F<7$$\"\"$F($\"1'*HeAy$)zHF<-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXES
LABELSG6$%\"aG%\"yG-%%VIEWG6$;F(F\\[l;F($\"\"\"F(" 1 2 0 1 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "diff(k,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%
coshG6#%\"aG!\"\"\"\"\"*(F(F*F%!\"#-%%sinhGF'F*F)" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "solve(%=0,a);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RootOfG6#,**&%#_ZG\"\"\"-%$expG6#F(\"\"#F)F(!\"\"*$F*F-F.F.F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(%%=0,a=1..2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S'y'*>\"!\"*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "evalf(subs(a=%,k));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+%>Mui'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Note tha
t there is a value " }{XPPEDIT 18 0 "l[0]" "6#&%\"lG6#\"\"!" }{TEXT 
-1 25 " = .6627... such that if " }{TEXT 264 1 "l" }{TEXT -1 3 " > " }
{XPPEDIT 18 0 "l[0]" "6#&%\"lG6#\"\"!" }{TEXT -1 35 ", the equation ha
s no solution, if " }{XPPEDIT 18 0 "l < l[0]" "6#2%\"lG&F$6#\"\"!" }
{TEXT -1 33 " there are two solutions, and if " }{XPPEDIT 18 0 "l=l[0]
" "6#/%\"lG&F$6#\"\"!" }{TEXT -1 25 " there is one solution.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "There is \+
always another solution consisting of two flat disks:  " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "plot3d(\{[r,theta,.4],[r,theta,-.4
]\},r=0..1,theta=0..2*Pi, coords=cylindrical, scaling=constrained, ori
entation=[43,64]);" }}{PARA 13 "" 1 "" {GLPLOT3D 303 303 303 
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a:;FdhlF*7%$!1Y`WJ\"H'H[F,$!1(e7bA&4%H\"F,F*7%$!1.A#*=q7IVF,$!1$)*****
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$\"1\"ommmm;a$F,F*7%$!1%*3)Ggu>%oF,$\"13iZp:IL=F,F*7%$!1JLLLLL$3(F,$\"
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5!orpVWC(F,$\"1.*o#QG9T>F,F*7%$\"1!H$QG0>&\\'F,$\"1'***********\\PF,F*
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,F*7%$\"19*o#QG9T>F,$\"13!orpVWC(F,F*7%$\"1`Z79pl67FdhlFierF*7%$!1\"*)
o#QG9T>F,$\"19!orpVWC(F,F*7%$!1')**********\\PF,$\"1'H$QG0>&\\'F,F*7%$
!1$4**)e3I.`F,$\"1<\"**)e3I.`F,F*7%$!1zKQG0>&\\'F,$\"1;+++++]PF,F*7%$!
10!orpVWC(F,$\"1E*o#QG9T>F,F*7%$!1)************\\(F,$\"11&\\#GQJBCFdhl
F*7%$!1<!orpVWC(F,$!1z)o#QG9T>F,F*7%$!1-LQG0>&\\'F,$!1u**********\\PF,
F*7%$!1F\"**)e3I.`F,$!1$3**)e3I.`F,F*7%$!1C+++++]PF,$!1sKQG0>&\\'F,F*7
%$!1P*o#QG9T>F,$!1-!orpVWC(F,F*7%$!1'[!o)GP%oMFdhlFairF*7%$\"1p)o#QG9T
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F^gtF`gtFbiu7%FcgtFegtFbiu7%FhgtFjgtFbiu7%F]htF_htFbiu7%FbhtFdftFbiu7%
FehtFghtFbiu7%FjhtF\\itFbiu7%F_itFaitFbiu7%FditFfitFbiu7%FiitF[jtFbiu7
%F\\ctF^jtFbiu7;7%FbjtF)Fbiu7%FejtFgjtFbiu7%FjjtF\\[uFbiu7%F_[uFa[uFbi
u7%Fd[uFf[uFbiu7%Fi[uF[\\uFbiu7%F^\\uFbjtFbiu7%Fa\\uFc\\uFbiu7%Ff\\uFh
\\uFbiu7%F[]uF]]uFbiu7%F`]uFb]uFbiu7%Fe]uFg]uFbiu7%Fj]uF\\^uFbiu7%F_^u
Fa^uFbiu7%Fd^uFf^uFbiu7%Fi^uF[_uFbiu7%F^_uF`_uFbiu7%Fc_uFe_uFbiu7%Fh_u
Fj]uFbiu7%F[`uF]`uFbiu7%F``uFb`uFbiu7%Fe`uFg`uFbiu7%Fj`uF\\auFbiu7%F_a
uFaauFbiu7%FbjtFdauFbiu7;7%FhauF)Fbiu7%F\\buF^buFbiu7%FabuFcbuFbiu7%Ff
buFhbuFbiu7%F[cuF]cuFbiu7%F`cuFbcuFbiu7%FecuFhauFbiu7%FhcuFjcuFbiu7%F]
duF_duFbiu7%FbduFdduFbiu7%FgduFiduFbiu7%F\\euF^euFbiu7%FaeuFceuFbiu7%F
feuFheuFbiu7%F[fuF]fuFbiu7%F`fuFbfuFbiu7%FefuFgfuFbiu7%FjfuF\\guFbiu7%
F_guFaeuFbiu7%FbguFdguFbiu7%FgguFiguFbiu7%F\\huF^huFbiu7%FahuFchuFbiu7
%FfhuFhhuFbiu7%FhauF[iuFbiu-%(SCALINGG6#%,CONSTRAINEDG-%+PROJECTIONG6%
$\"#VF)$\"#kF)\"\"\"" 1 1 0 1 0 2 1 1 1 1 1.000000 64.000000 
43.000000 1 }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 29 "The area of this solution is " }{XPPEDIT 18 0 "A=2*Pi" "6#/%\"A
G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 3 ".  " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "Area[1] := 2*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%%AreaG6#\"\"\",$%#PiG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "We compute the area of the solutions we f
ound." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(value(subs
(y(x)=Y,A)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,%#PiG\"\"\"%\"lG
\"\"#-%$expG6#%\"aGF&,**$*&,(-F*6#,$F,!\"#F(F&F&-F*6#,$F,!\"%F&F&-F*6#
,$F,F(F&#F&F(F&*&F/F<F1F&!\"\"*&-%#lnG6#*&,(F&F&F1F>*&F/F<-F*6#,$F,F>F
&F&F&F)F&F&FFF&F(*&-FA6#*&,(F1F&F>F&FEF&F&F)F&F&FFF&F4F&F,F4F<" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Maple ne
eds a little help in finding a simpler expression for this integral.  \+
We start by simplifying the integrand " }{TEXT 272 1 "L" }{TEXT -1 
106 ".  It is typically preferable in working with Maple to simplify a
n expression before using it any further." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "simplify(subs(y(x)=Y,L));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(-%%coshG6#*&%\"aG\"\"\"%\"xGF)\"\"#F(!\"\"-%%csgnG6#F
$F)" }}}{PARA 0 "" 0 "" {TEXT -1 27 "We mutliply by the sign of " }
{XPPEDIT 18 0 "cosh(a*x)" "6#-%%coshG6#*&%\"aG\"\"\"%\"xGF(" }{TEXT 
-1 58 " (which is positive!), and then integrate to get the area:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(%*csgn(cosh(a*x)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%coshG6#*&%\"aG\"\"\"%\"xGF)\"
\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "2*Pi*l^2*int(
%, x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,%#PiG\"\"\"%\"lG\"
\"#,(-%$expG6#,$%\"aG\"\"%#F&\"\")#!\"\"F1F&*&F.F&-F+6#,$F.F(F&#F&F(F&
F.!\"#-F+6#,$F.F9F&F/" }}}{PARA 0 "" 0 "" {TEXT -1 122 "This expressio
n is already quite better than before.  We will identify it as an even
 simpler expression.  Call the result " }{TEXT 0 2 "Ar" }{TEXT -1 1 ".
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Ar := simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ArG,$*,%#PiG\"\"\"%\"lG\"\"#,(-%$ex
pG6#,$%\"aG\"\"%F(!\"\"F(*&F0F(-F-6#,$F0F*F(F1F(F0!\"#-F-6#,$F0F7F(#F(
F*" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Recognize this function as " }
{XPPEDIT 18 0 "Pi*l^2*(sinh(2*a)+2*a)/a^2" "6#**%#PiG\"\"\"*$%\"lG\"\"
#F%,&-%%sinhG6#*&\"\"#F%%\"aGF%F%*&\"\"#F%F/F%F%F%*$F/\"\"#!\"\"" }
{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Br := Pi*l^
2/(a^2)*(sinh(2*a)+2*a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BrG**%#
PiG\"\"\"%\"lG\"\"#%\"aG!\"#,&-%%sinhG6#,$F*F)F'F*F)F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalb(simplify(Ar=Br));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 59 "Ag
ain Maple needs a little help, as if it didn't know that " }{XPPEDIT 
18 0 "sinh(2*a)=(exp(2*a)-exp(-2*a))/2" "6#/-%%sinhG6#*&\"\"#\"\"\"%\"
aGF)*&,&-%$expG6#*&\"\"#F)F*F)F)-F.6#,$*&\"\"#F)F*F)!\"\"F7F)\"\"#F7" 
}{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Br1 := sub
s(sinh(2*a)=(exp(2*a)-exp(-2*a))/2,Br): evalb(simplify(Ar=Br1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 
20 "Finally, substitute " }{XPPEDIT 18 0 "l = a/cosh(a)" "6#/%\"lG*&%
\"aG\"\"\"-%%coshG6#F&!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "Area[2] := subs(l=a/cosh(a),Br);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%%AreaG6#\"\"#*(%#PiG\"\"\"-%%coshG6#%\"aG!\"#,&-%%
sinhG6#,$F.F'F*F.F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pl
ot(\{Area[1],Area[2]\},a=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 
303 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!$\"1'ezrI&=$G'!#:7$$\"1+++]i9R
l!#<F)7$$\"1++vVA)GA\"!#;F)7$$\"1++]Peui=F3F)7$$\"1++]i3&o]#F3F)7$$\"1
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[V\"F+F)7$$\"1++DJ#>&)\\\"F+F)7$$\"1+]P>:mk:F+F)7$$\"1+]iv&QAi\"F+F)7$
$\"1++vtLU%o\"F+F)7$$\"1+++bjm[<F+F)7$$\"1++vyb^6=F+F)7$$\"1+]PMaKs=F+
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7#F+F)7$$\"1++]x%f\")=#F+F)7$$\"1+]P/-a[AF+F)7$$\"1+](=Yb;J#F+F)7$$\"1
++]i@OtBF+F)7$$\"1+]PfL'zV#F+F)7$$\"1+++!*>=+DF+F)7$$\"1++DE&4Qc#F+F)7
$$\"1+]P%>5pi#F+F)7$$\"1+++bJ*[o#F+F)7$$\"1++Dr\"[8v#F+F)7$$\"1+++Ijy5
GF+F)7$$\"1+]P/)fT(GF+F)7$$\"1+]i0j\"[$HF+F)7$$\"\"$F(F)-%'COLOURG6&%$
RGBG$\"#5!\"\"F(F(-F$6$7W7$F(F(7$$\"1+++DJdpKF/$\"1q,X,0u0TF37$F-$\"12
%QKPoR>)F37$$\"1++vVV)RQ*F/$\"1v$Rk()RB<\"F+7$F1$\"1>q<G]`@:F+7$$\"1+]
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\"1&e<VuIV&[F+7$FD$\"1#zd8$)[(p`F+7$FG$\"1hYIK^G@eF+7$FJ$\"1N%pX$*Q8A'
F+7$FM$\"1W!p6Cm7_'F+7$FP$\"1M$p:i@K!oF+7$FS$\"1Z^62=AJqF+7$FV$\"1NhoE
zV.sF+7$FY$\"1()[5$=$fBtF+7$Ffn$\"1.m;96$pU(F+7$Fin$\"1V$f$HCA&[(F+7$F
\\o$\"1Owgm!\\V_(F+7$F_o$\"1&)Q@BOJPvF+7$Fbo$\"1)[$pJdJKvF+7$Feo$\"1]K
(e'fU7vF+7$Fho$\"1G/O\"H6$zuF+7$F[p$\"1X#y]=[.W(F+7$F^p$\"1Qr/-d`\"R(F
+7$Fap$\"1(>oB6DcL(F+7$Fdp$\"1\\f``c=%G(F+7$Fgp$\"1q$*zI=1FsF+7$Fjp$\"
11+sc7]nrF+7$F]q$\"18d\"RUR'4rF+7$F`q$\"1^ex#**GZ0(F+7$Fcq$\"1zT@(=9c*
pF+7$Ffq$\"1x*HEflX%pF+7$Fiq$\"1L:KTm_#*oF+7$F\\r$\"1k#4(yDuZoF+7$F_r$
\"1g-'ens9!oF+7$Fbr$\"12\\l;\"[0w'F+7$Fer$\"12Wg;gZ?nF+7$Fhr$\"1'R8Q=L
Ro'F+7$F[s$\"1\"*fA$=%R[mF+7$F^s$\"1`5[/5q;mF+7$Fas$\"1zcj.2v'e'F+7$Fd
s$\"1JqXOOQflF+7$Fgs$\"1-\"\\v-wh`'F+7$Fjs$\"1$H5Z3F<^'F+7$F]t$\"1YH`*
>%o\"\\'F+7$F`t$\"1=n^#R%3skF+7$Fct$\"1?Oc&3>\\X'F+7$Fft$\"17wZFQ3QkF+
-Fit6&F[uF(F\\uF(-%+AXESLABELSG6$%\"aG%!G-%%VIEWG6$;F(Fft%(DEFAULTG" 
1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "fsolve(Area[1]=Area[2],a);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+9FK#R'!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "evalf(subs(a=%,k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+rR(pF&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Note that there is a \+
value " }{XPPEDIT 18 0 "l[1]" "6#&%\"lG6#\"\"\"" }{TEXT -1 80 " = .527
7... for which the catenoid has the same surface area as the two disks
.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "No
te that Area[2] has a maximum@the same value of " }{TEXT 273 1 "a" 
}{TEXT -1 7 " where " }{XPPEDIT 18 0 "a/cosh(a)" "6#*&%\"aG\"\"\"-%%co
shG6#F$!\"\"" }{TEXT -1 15 " has a maximum." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "simplify(diff(Area[2],a));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(%#PiG\"\"\",**&-%%sinhG6#%\"aGF&-F*6#,$F,\"\"#F&!\"
\"*&F,F&F)F&!\"#*&-%%coshGF+F&-F6F.F&F&F5F&F&F5!\"$F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(%=0,a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RootOfG6#,**&%#_ZG\"\"\"-%$expG6#F(\"\"#F)F(!\"\"*$F
*F-F.F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fsolve(%%=0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S'y'*>\"!\"*" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 
18 0 "l[0]<>l[1]" "6#0&%\"lG6#\"\"!&F%6#\"\"\"" }{TEXT -1 12 ".  For e
ach " }{XPPEDIT 18 0 "l < l[1]" "6#2%\"lG&F$6#\"\"\"" }{TEXT -1 28 ", \+
the catenoid with smaller " }{TEXT 270 1 "a" }{TEXT -1 26 " has the le
ast area.  For " }{TEXT 266 1 "l" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "l[
1]" "6#&%\"lG6#\"\"\"" }{TEXT -1 74 ", the least area solution is the \+
two disks, but the catenoid with smaller " }{TEXT 267 1 "a" }{TEXT -1 
48 " has area smaller than the catenoid with larger " }{TEXT 271 1 "a
" }{TEXT -1 8 ".  When " }{XPPEDIT 18 0 "l=l[0]" "6#/%\"lG&F$6#\"\"!" 
}{TEXT -1 38 ", the two catenoid solutions coincide." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Note that when " }
{XPPEDIT 18 0 "l->0" "6#R6#%\"lG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*
" }{TEXT -1 26 ", the catenoid with large " }{TEXT 274 1 "a" }{TEXT 
-1 66 " approaches the two disks solution.  As an indication, we compu
te " }{XPPEDIT 18 0 "limit(Area[2],a=infinity)" "6#-%&limitG6$&%%AreaG
6#\"\"#/%\"aG%)infinityG" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "limit(Area[2],a=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%#PiG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 46 "In fact the approximation is quite good since \+
" }{XPPEDIT 18 0 "Area[2]-Area[1]" "6#,&&%%AreaG6#\"\"#\"\"\"&F%6#\"\"
\"!\"\"" }{TEXT -1 20 " tends to zero like " }{XPPEDIT 18 0 "a*exp(-2*
a)" "6#*&%\"aG\"\"\"-%$expG6#,$*&\"\"#F%F$F%!\"\"F%" }{TEXT -1 1 ":" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Limit(('Area[2]-Area[1]')/(
a*exp(-2*a)),a=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG
6$*(,&&%%AreaG6#\"\"#\"\"\"&F)6#F,!\"\"F,%\"aGF/-%$expG6#,$F0!\"#F//F0
%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 118 "plot3d([subs(a=3.5,Y),theta,x], theta=0..2*Pi, \+
x=-1..1, coords=cylindrical, scaling=constrained, orientation=[49,79])
;" }}{PARA 13 "" 1 "" {GLPLOT3D 303 303 303 {PLOTDATA 3 "6%-%%MESHG6#7
;7;7%$\"1zC#Hw#4NZ!#:\"\"!$!\"\"F,7%$\"1$o&**Q%\\(RNF+F,$!1mmmmmmm\"*!
#;7%$\"1)*zv.9nZEF+F,$!1LLLLLLL$)F47%$\"1q5?j0V#)>F+F,$!1*************
\\(F47%$\"1sc4CH.([\"F+F,$!1lmmmmmmmF47%$\"128\"ynN!>6F+F,$!1JLLLLLLeF
47%$\"10D%fX4\"p%)F4F,$!1)*************\\F47%$\"1_.+?lWtkF4F,$!1lmmmmm
mTF47%$\"1M+c&4\"RK]F4F,$!1KLLLLLLLF47%$\"1baVw=[ASF4F,$!1************
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