{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2
D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 256 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 
0 0 0 0 0 0 0 }{CSTYLE "" 19 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }
{CSTYLE "" 19 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 264 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 265 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 
0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple
 Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 
3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 
0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 33 "Appendix B
1--Superconducting Arcs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 
"" }{TEXT 261 26 "Aleksandar Donev, 12/17/00" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT -1 99 "This worksheet develops \+
the conjugate functions and related derivatives for the cost functions
 for " }{TEXT 262 20 "superconducting arcs" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "restart:" "6#%(restartG" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(plots):" "6#-%%withG6#%&plotsG" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The proposed form for the volta
ge-current characteristic is:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
263 1 "V:=i->R*i*(1+tanh((i-j)/xi)):" "6#>%\"VGR6#%\"iG7\"6$%)operator
G%&arrowG6\"*(%\"RG\"\"\"F'F/,&\"\"\"F/-%%tanhG6#*&,&F'F/%\"jG!\"\"F/%
#xiGF8F/F/F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Specific numb
ers are:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "numbers := \{R = \+
.5, j = 1.0, xi = .5e-1\}:" "6#>%(numbersG<%/%\"RG$\"\"&!\"\"/%\"jG$\"
#5!\"\"/%#xiG$\"\"&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "pl
ot(subs(numbers,V(i)),i=0..2,labels=[\"i\",\"v(i)\"]);" "6#-%%plotG6%-
%%subsG6$%(numbersG-%\"VG6#%\"iG/F-;\"\"!\"\"#/%'labelsG7$Q\"i6\"Q%v(i
)F6" }}{PARA 13 "" 1 "" {GLPLOT2D 453 168 168 {PLOTDATA 2 "6%-%'CURVES
G6$7in7$\"\"!F(7$$\"1LLLL3VfV!#<F(7$$\"1nmm\"H[D:)F,F(7$$\"1LLLe0$=C\"
!#;$\"1,Rs#RWN*o!#K7$$\"1LLL3RBr;F3$\"1eFQ[MkWe!#J7$$\"1mm;zjf)4#F3$\"
1LC.LzaPR!#I7$$\"1LL$e4;[\\#F3$\"1w0Pv<Z'G#!#H7$$\"1++]i'y]!HF3$\"1)p[
xwKTP\"!#G7$$\"1LL$ezs$HLF3$\"1vj(eeocf)FN7$$\"1++]7iI_PF3$\"17*3,lW%f
_!#F7$$\"1nmm;_M(=%F3$\"1)4tqdRWM$!#E7$$\"1LLL3y_qXF3$\"1)z'f)\\o/p\"!
#D7$$\"1+++]1!>+&F3$\"1lINkk$)Q5!#C7$$\"1+++]Z/NaF3$\"1:;Kb*pNQ'Feo7$$
\"1+++]$fC&eF3$\"1QhrhuC]O!#B7$$\"1LL$ez6:B'F3$\"1U$3\"3&[.x\"!#A7$$\"
1mmm;=C#o'F3$\"1U4wG9#=:\"!#@7$$\"1mmmm#pS1(F3$\"1Z@*3w@\"3cF\\q7$$\"1
++]i`A3vF3$\"1(oJ/G\"eAN!#?7$$\"1mmmm(y8!zF3$\"1t*=f0%='y\"!#>7$$\"1++
]i.tK$)F3$\"1HMUfabc5!#=7$$\"1+++DZ5Q&)F3$\"1hYycyudCFcr7$$\"1++](3zMu
)F3$\"1wES')y>-dFcr7$$\"1n;a)QA1&))F3$\"1e$Q$H-qH))Fcr7$$\"1MLe*olx&*)
F3$\"1Y/?99ak8F,7$$\"1+]i!**3\\1*F3$\"1m8%GVxD5#F,7$$\"1nmm\"H_?<*F3$
\"1R6<x+rDKF,7$$\"1nmT&GM)o$*F3$\"1v$y')*)Gn%pF,7$$\"1nm;zihl&*F3$\"1K
E7rzEJ9F37$$\"1L$e9EW<n*F3$\"1R%z$>-B]?F37$$\"1++vVA(yx*F3$\"17)*z.,W
\\GF37$$\"1n;/E-+%))*F3$\"14@s+`f:QF37$$\"1LLL3#G,***F3$\"1;s:_kX'*[F3
7$$\"1L3_Nl.55!#:$\"1o0$)\\+^]gF37$$\"1L$3-Dg5-\"Fcv$\"1f$GZ&3$p8(F37$
$\"1Le*['R3K5Fcv$\"1'e3xdL93)F37$$\"1LLezw5V5Fcv$\"1STXLan_))F37$$\"1n
mmJ+Ii5Fcv$\"1$H![\\==6)*F37$$\"1++v$Q#\\\"3\"Fcv$\"1$G>Ot)\\T5Fcv7$$
\"1LL$e\"*[H7\"Fcv$\"1#e$fN_z96Fcv7$$\"1+++qvxl6Fcv$\"16TQuACk6Fcv7$$
\"1++]_qn27Fcv$\"1h&yf6zt?\"Fcv7$$\"1++Dcp@[7Fcv$\"1=_!45c\"[7Fcv7$$\"
1++]2'HKH\"Fcv$\"1XJF!>>KH\"Fcv7$$\"1nmmwanL8Fcv$\"1\"f8gMtOL\"Fcv7$$
\"1+++v+'oP\"Fcv$\"13a,%ofoP\"Fcv7$$\"1LLeR<*fT\"Fcv$\"1:@`b;*fT\"Fcv7
$$\"1+++&)Hxe9Fcv$\"1u#e$pHxe9Fcv7$$\"1mm\"H!o-*\\\"Fcv$\"1%H/(*zE!*\\
\"Fcv7$$\"1++DTO5T:Fcv$\"1\"QO1k.6a\"Fcv7$$\"1nmmT9C#e\"Fcv$\"1J^aT9C#
e\"Fcv7$$\"1++D1*3`i\"Fcv$\"10xA1*3`i\"Fcv7$$\"1LLL$*zym;Fcv$\"1#)*GL*
zym;Fcv7$$\"1LL$3N1#4<Fcv$\"1:D$3N1#4<Fcv7$$\"1nm\"HYt7v\"Fcv$\"16l\"H
Yt7v\"Fcv7$$\"1+++q(G**y\"Fcv$\"1m****p(G**y\"Fcv7$$\"1nm;9@BM=Fcv$\"1
hm;9@BM=Fcv7$$\"1LLL`v&Q(=Fcv$\"1KLL`v&Q(=Fcv7$$\"1++DOl5;>FcvF_^l7$$
\"1++v.Uac>FcvFb^l7$$\"\"#F(Fe^l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXE
SLABELSG6$Q\"i6\"Q%v(i)Fb_l-%%VIEWG6$;F(Fe^l%(DEFAULTG" 1 2 0 1 0 2 9 
1 4 2 1.000000 44.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 32 "Here is the cost power function:" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "assume(xi>0,j>0,R>0,i>0):" "6#-%'assumeG6&2\"\"!%#xiG2F
'%\"jG2F'%\"RG2F'%\"iG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sim
plify(int(V(J),J=0..i));" "6#-%)simplifyG6#-%$intG6$-%\"VG6#%\"JG/F,;
\"\"!%\"iG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*(%#R|irG\"\"\")%$xi|i
rG\"\"#\"\"\"-%&dilogG6#*&,&-%$expG6#,$*&%#i|irGF*F(!\"\"F)F&-F16#,$*&
%#j|irGF*F(F6F)F&F&-F16#,$F:!\"#F&F&#F&F)**F%F*F(F&-%#lnG6#,&-F16#,$*&
,&F5F&F;!\"\"F*F(F6F)F&F&F&F&F5F&F&*(F%F*F'F*-F,6#*&,&F&F&F7F&F&F<F*F&
#FKF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "For some reason, I had \+
to simplify this further by hand (there may be further possible simpli
fications):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 256 1 "f:=i-> R*xi^2
/2*(dilog(exp(2*(i-j)/xi)+1)-dilog(exp(-2*j/xi)+1))+R*xi*ln(exp(2*(i-j
)/xi)+1)*i:" "6#>%\"fGR6#%\"iG7\"6$%)operatorG%&arrowG6\",&**%\"RG\"\"
\"*$%#xiG\"\"#F0\"\"#!\"\",&-%&dilogG6#,&-%$expG6#*(\"\"#F0,&F'F0%\"jG
F5F0F2F5F0\"\"\"F0F0-F86#,&-F<6#,$*(\"\"#F0FAF0F2F5F5F0\"\"\"F0F5F0F0*
*F/F0F2F0-%#lnG6#,&-F<6#*(\"\"#F0,&F'F0FAF5F0F2F5F0\"\"\"F0F0F'F0F0F,F
,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "And its plot:" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "plot(subs(numbers,f(i)),i = 0 .. 2,la
bels = [\"i\", \"f(i)\"]);" "6#-%%plotG6%-%%subsG6$%(numbersG-%\"fG6#%
\"iG/F-;\"\"!\"\"#/%'labelsG7$Q\"i6\"Q%f(i)F6" }}{PARA 13 "" 1 "" 
{GLPLOT2D 524 152 152 {PLOTDATA 2 "6%-%'CURVESG6$7U7$\"\"!F(7$$\"+M3Vf
V!#6F(7$$\"+#H[D:)F,F(7$$\"+e0$=C\"!#5F(7$$\"+3RBr;F3F(7$$\"+zjf)4#F3F
(7$$\"+'4;[\\#F3F(7$$\"+j'y]!HF3F(7$$\"+'zs$HLF3F(7$$\"+8iI_PF3F(7$$\"
+<_M(=%F3F(7$$\"+4y_qXF3F(7$$\"+]1!>+&F3$\"+X1!>v%!#?7$$\"+]Z/NaF3$\"+
Ko-6J!#>7$$\"+]$fC&eF3$\"+WBwO<!#=7$$\"+'z6:B'F3$\"+?bu$\\)Fgn7$$\"+<=
C#o'F3$\"+tveWb!#<7$$\"+n#pS1(F3$\"+iQ$[q#!#;7$$\"+j`A3vF3$\"+ekm-<!#:
7$$\"+n(y8!zF3$\"+I2J\\')F^p7$$\"+j.tK$)F3$\"+5M[F^!#97$$\"+)3zMu)F3$
\"+&*[^yF!#87$$\"+#H_?<*F3$\"+%GWnf\"!#77$$\"+!G;cc*F3$\"+$e!*4a(Feq7$
$\"+XA(yx*F3$\"+G-FP;F,7$$\"+4#G,***F3$\"+o@BlKF,7$$\"+^-1@5!\"*$\"+rW
:GfF,7$$\"+!o2J/\"Fhr$\"+`R#)y%*F,7$$\"+%Q#\\\"3\"Fhr$\"+/va'p\"F37$$
\"+;*[H7\"Fhr$\"+wRw$f#F37$$\"+qvxl6Fhr$\"+$Q'fqNF37$$\"+`qn27Fhr$\"+S
!HWc%F37$$\"+cp@[7Fhr$\"+c]#*fbF37$$\"+3'HKH\"Fhr$\"+D=(Qq'F37$$\"+xan
L8Fhr$\"+!>Ujw(F37$$\"+v+'oP\"Fhr$\"+'=vo$*)F37$$\"+S<*fT\"Fhr$\"+Xk(H
+\"Fhr7$$\"+&)Hxe9Fhr$\"+cC'f7\"Fhr7$$\"+.o-*\\\"Fhr$\"+)=D]C\"Fhr7$$
\"+TO5T:Fhr$\"+lU%HP\"Fhr7$$\"+U9C#e\"Fhr$\"+9=V,:Fhr7$$\"+1*3`i\"Fhr$
\"+tGdR;Fhr7$$\"+$*zym;Fhr$\"+Yg7w<Fhr7$$\"+^j?4<Fhr$\"+$=I$>>Fhr7$$\"
+jMF^<Fhr$\"+vD!\\1#Fhr7$$\"+q(G**y\"Fhr$\"+N))y,AFhr7$$\"+9@BM=Fhr$\"
+!G^BO#Fhr7$$\"+`v&Q(=Fhr$\"+ifG4DFhr7$$\"+Ol5;>Fhr$\"+!43%pEFhr7$$\"+
/Uac>Fhr$\"+b!4g#GFhr7$$\"\"#F($\"+KQ%z*HFhr-%'COLOURG6&%$RGBG$\"#5!\"
\"F(F(-%+AXESLABELSG6$Q\"i6\"Q%f(i)F^[l-%%VIEWG6$;F(F_z%(DEFAULTG" 1 
2 0 1 0 2 9 1 4 2 1.000000 44.000000 43.000000 0 }}}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 122 "The most important part of the whole calculation \+
is the calculation of the inverse of the voltage-current characteristi
c, " }{XPPEDIT 18 0 "I(V)=(V(I))^(-1)" "6#/-%\"IG6#%\"VG)-F'6#F%,$\"\"
\"!\"\"" }{TEXT -1 25 ". So long as there is no " }{XPPEDIT 18 0 "i^be
ta" "6#)%\"iG%%betaG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "V(I)" "6#-%\"
VG6#%\"IG" }{TEXT -1 83 ", the inverse is simple and expressible in te
rms of the LambertW special function: " }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "solve(V(i)=v,i);" "6#-%&solveG6$/-%\"VG6#%\"iG%\"vGF*" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*(%#R|irG\"\"\"%$xi|irG\"\"\"-
%)LambertWG6#*&*&%\"vGF*-%$expG6#*&,&*&F'F*%#j|irGF*\"\"#F0!\"\"F(*&F'
\"\"\"F)\"\"\"!\"\"F*F(*&F'\"\"\"F)\"\"\"F=F*F*F0F*F(F'F=#F*F8" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus:" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 258 1 "J:=v-> (v/R+xi*LambertW(v*exp((2*j-v/R)/xi)/(R*xi)))/2
:" "6#>%\"JGR6#%\"vG7\"6$%)operatorG%&arrowG6\"*&,&*&F'\"\"\"%\"RG!\"
\"F0*&%#xiGF0-%)LambertWG6#*(F'F0-%$expG6#*&,&*&\"\"#F0%\"jGF0F0*&F'F0
F1F2F2F0F4F2F0*&F1F0F4F0F2F0F0F0\"\"#F2F,F,F," }}}{EXCHG {PARA 0 "> " 
0 "" {XPPEDIT 19 1 "plot(subs(numbers,J(v)),v=0..2,labels = [\"v\", \"
i(v)\"]);" "6#-%%plotG6%-%%subsG6$%(numbersG-%\"JG6#%\"vG/F-;\"\"!\"\"
#/%'labelsG7$Q\"v6\"Q%i(v)F6" }}{PARA 13 "" 1 "" {GLPLOT2D 320 187 
187 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$\"\"!F(7$$\"+P@Ki8!#7$\"+,l^%R)!#5
7$$\"+tUkCFF,$\"+\"e>Kc)F/7$$\"+5k'p3%F,$\"+775i')F/7$$\"+X&)G\\aF,$\"
+!3%QK()F/7$$\"+=G$R<)F,$\"+wJoJ))F/7$$\"+4x&)*3\"!#6$\"+CdO-*)F/7$$\"
+klyM;FG$\"+h%RC+*F/7$$\"+<arz@FG$\"+i5)Q2*F/7$$\"+EJdpKFG$\"+=mXv\"*F
/7$$\"+M3VfVFG$\"+.wR[#*F/7$$\"+j&*)fD'FG$\"+aEXT$*F/7$$\"+#H[D:)FG$\"
+_h66%*F/7$$\"+e0$=C\"F/$\"+EHdD&*F/7$$\"+3RBr;F/$\"+@tV5'*F/7$$\"+zjf
)4#F/$\"+Zz#*y'*F/7$$\"+'4;[\\#F/$\"+JuoL(*F/7$$\"+j'y]!HF/$\"+L@[%y*F
/7$$\"+'zs$HLF/$\"+f0iK)*F/7$$\"+8iI_PF/$\"+^/\\x)*F/7$$\"+<_M(=%F/$\"
+U0T@**F/7$$\"+4y_qXF/$\"+>([)e**F/7$$\"+]1!>+&F/$\"+,\"=++\"!\"*7$$\"
+]Z/NaF/$\"+*)f8/5Fdr7$$\"+]$fC&eF/$\"+CP735Fdr7$$\"+'z6:B'F/$\"+aE!=,
\"Fdr7$$\"+<=C#o'F/$\"+WZI;5Fdr7$$\"+n#pS1(F/$\"+k+G?5Fdr7$$\"+j`A3vF/
$\"+-%p^-\"Fdr7$$\"+n(y8!zF/$\"+E@#)H5Fdr7$$\"+j.tK$)F/$\"+`.TN5Fdr7$$
\"+)3zMu)F/$\"+4?QT5Fdr7$$\"+#H_?<*F/$\"+@;e[5Fdr7$$\"+!G;cc*F/$\"+MeZ
c5Fdr7$$\"+4#G,***F/$\"+%=Or1\"Fdr7$$\"+!o2J/\"Fdr$\"+Q&R@3\"Fdr7$$\"+
%Q#\\\"3\"Fdr$\"+B.t+6Fdr7$$\"+;*[H7\"Fdr$\"+acJH6Fdr7$$\"+qvxl6Fdr$\"
+C#Gs;\"Fdr7$$\"+`qn27Fdr$\"+t:(z?\"Fdr7$$\"+cp@[7Fdr$\"+nwF[7Fdr7$$\"
+3'HKH\"Fdr$\"+@+C$H\"Fdr7$$\"+xanL8Fdr$\"+2wnL8Fdr7$$\"+v+'oP\"Fdr$\"
+m/'oP\"Fdr7$$\"+S<*fT\"Fdr$\"+C=*fT\"Fdr7$$\"+&)Hxe9Fdr$\"+,Ixe9Fdr7$
$\"+.o-*\\\"Fdr$\"+1o-*\\\"Fdr7$$\"+TO5T:Fdr$\"+UO5T:Fdr7$$\"+U9C#e\"F
drFcz7$$\"+1*3`i\"FdrFfz7$$\"+$*zym;FdrFiz7$$\"+^j?4<FdrF\\[l7$$\"+jMF
^<FdrF_[l7$$\"+q(G**y\"FdrFb[l7$$\"+9@BM=FdrFe[l7$$\"+`v&Q(=FdrFh[l7$$
\"+Ol5;>FdrF[\\l7$$\"+/Uac>FdrF^\\l7$$\"\"#F($\"+++++?Fdr-%'COLOURG6&%
$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"v6\"Q%i(v)F`]l-%%VIEWG6$;F(Fa\\l
%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Now, the conjugate function is sim
ply " }{XPPEDIT 18 0 "g(v)=J(v)*v-f(J(v))" "6#/-%\"gG6#%\"vG,&*&-%\"JG
6#F'\"\"\"F'F-F--%\"fG6#-F+6#F'!\"\"" }{TEXT -1 14 ", with a plot:" }}
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "g:=unapply(simplify(J(v)*v-f(
J(v))),v):" "6#>%\"gG-%(unapplyG6$-%)simplifyG6#,&*&-%\"JG6#%\"vG\"\"
\"F0F1F1-%\"fG6#-F.6#F0!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
19 1 "plot(subs(numbers,g(v)),v=0..2,labels=[\"v\",\"g(v)\"]);" "6#-%%
plotG6%-%%subsG6$%(numbersG-%\"gG6#%\"vG/F-;\"\"!\"\"#/%'labelsG7$Q\"v
6\"Q%g(v)F6" }}{PARA 13 "" 1 "" {GLPLOT2D 432 147 147 {PLOTDATA 2 "6%-
%'CURVESG6$7S7$\"\"!F(7$$\"+M3VfV!#6$\"+l7>BRF,7$$\"+#H[D:)F,$\"+_O1lu
F,7$$\"+e0$=C\"!#5$\"+-Ub]6F77$$\"+3RBr;F7$\"+5C[h:F77$$\"+zjf)4#F7$\"
+)y-P(>F77$$\"+'4;[\\#F7$\"+8!3$eBF77$$\"+j'y]!HF7$\"+;PqeFF77$$\"+'zs
$HLF7$\"+Kz)[<$F77$$\"+8iI_PF7$\"+S-q\"f$F77$$\"+<_M(=%F7$\"+C;PASF77$
$\"+4y_qXF7$\"+*)HE.WF77$$\"+]1!>+&F7$\"+!HaP$[F77$$\"+]Z/NaF7$\"+Z#)z
n_F77$$\"+]$fC&eF7$\"+D+x(o&F77$$\"+'z6:B'F7$\"+vmfqgF77$$\"+<=C#o'F7$
\"+AZlFlF77$$\"+n#pS1(F7$\"+%3hk\"pF77$$\"+j`A3vF7$\"+P#)pqtF77$$\"+n(
y8!zF7$\"+E+luxF77$$\"+j.tK$)F7$\"+_&\\+A)F77$$\"+)3zMu)F7$\"+6QaY')F7
7$$\"+#H_?<*F7$\"+TFN%4*F77$$\"+!G;cc*F7$\"+?4a3&*F77$$\"+4#G,***F7$\"
+s?>f**F77$$\"+!o2J/\"!\"*$\"+chGV5F`s7$$\"+%Q#\\\"3\"F`s$\"+a'f^3\"F`
s7$$\"+;*[H7\"F`s$\"+#z^88\"F`s7$$\"+qvxl6F`s$\"+P'30=\"F`s7$$\"+`qn27
F`s$\"+cCEI7F`s7$$\"+cp@[7F`s$\"+=$\\+G\"F`s7$$\"+3'HKH\"F`s$\"+F#\\sL
\"F`s7$$\"+xanL8F`s$\"+oJP!R\"F`s7$$\"+v+'oP\"F`s$\"+2***)[9F`s7$$\"+S
<*fT\"F`s$\"+'QWN]\"F`s7$$\"+&)Hxe9F`s$\"+&RP]c\"F`s7$$\"+.o-*\\\"F`s$
\"+i(oXi\"F`s7$$\"+TO5T:F`s$\"++$G&)o\"F`s7$$\"+U9C#e\"F`s$\"+u?x_<F`s
7$$\"+1*3`i\"F`s$\"+/E%=#=F`s7$$\"+$*zym;F`s$\"+\">>,*=F`s7$$\"+^j?4<F
`s$\"+f7sh>F`s7$$\"+jMF^<F`s$\"+bu]M?F`s7$$\"+q(G**y\"F`s$\"+&e]H5#F`s
7$$\"+9@BM=F`s$\"+3=B$=#F`s7$$\"+`v&Q(=F`s$\"+[\"*pcAF`s7$$\"+Ol5;>F`s
$\"+7-wOBF`s7$$\"+/Uac>F`s$\"+&pg]T#F`s7$$\"\"#F($\"+%3G5]#F`s-%'COLOU
RG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"v6\"Q%g(v)Fa[l-%%VIEWG6$;F(
Fbz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "d
iff(g(v),v)=J(v)" "6#/-%%diffG6$-%\"gG6#%\"vGF*-%\"JG6#F*" }{TEXT -1 
62 " according to the theory, and a comparison plot verifies this:" }}
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 265 1 "dg:=v->J(v):" "6#>%#dgGR6#%
\"vG7\"6$%)operatorG%&arrowG6\"-%\"JG6#F'F,F,F," }}}{EXCHG {PARA 0 "> \+
" 0 "" {XPPEDIT 19 1 "p1:=plot(subs(numbers,diff(g(v),v)),v=0..2):" "6
#>%#p1G-%%plotG6$-%%subsG6$%(numbersG-%%diffG6$-%\"gG6#%\"vGF2/F2;\"\"
!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p2:=plot(subs(numbe
rs,dg(v)),v=0..2,style=point,color=blue):" "6#>%#p2G-%%plotG6&-%%subsG
6$%(numbersG-%#dgG6#%\"vG/F/;\"\"!\"\"#/%&styleG%&pointG/%&colorG%%blu
eG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "display(p1,p2,labels = \+
[\"v\", \"D[g(v),v]\"]);" "6#-%(displayG6%%#p1G%#p2G/%'labelsG7$Q\"v6
\"Q*D[g(v),v]F," }}{PARA 13 "" 1 "" {GLPLOT2D 410 192 192 {PLOTDATA 2 
"6&-%'CURVESG6$7S7$$\"+M3VfV!#6$\"+/wR[#*!#57$$\"+j&*)fD'F*$\"+_EXT$*F
-7$$\"+#H[D:)F*$\"+_h66%*F-7$$\"+e0$=C\"F-$\"+BHdD&*F-7$$\"+3RBr;F-$\"
+BtV5'*F-7$$\"+zjf)4#F-$\"+_z#*y'*F-7$$\"+'4;[\\#F-$\"+EuoL(*F-7$$\"+j
'y]!HF-$\"+L@[%y*F-7$$\"+'zs$HLF-$\"+h0iK)*F-7$$\"+8iI_PF-$\"+\\/\\x)*
F-7$$\"+<_M(=%F-$\"+O0T@**F-7$$\"+4y_qXF-$\"+G([)e**F-7$$\"+]1!>+&F-$
\"+,\"=++\"!\"*7$$\"+]Z/NaF-$\"+*)f8/5Fdo7$$\"+]$fC&eF-$\"+CP735Fdo7$$
\"+'z6:B'F-$\"+aE!=,\"Fdo7$$\"+<=C#o'F-$\"+WZI;5Fdo7$$\"+n#pS1(F-$\"+k
+G?5Fdo7$$\"+j`A3vF-$\"+-%p^-\"Fdo7$$\"+n(y8!zF-$\"+D@#)H5Fdo7$$\"+j.t
K$)F-$\"+b.TN5Fdo7$$\"+)3zMu)F-$\"+5?QT5Fdo7$$\"+#H_?<*F-$\"+@;e[5Fdo7
$$\"+!G;cc*F-$\"+MeZc5Fdo7$$\"+4#G,***F-$\"+&=Or1\"Fdo7$$\"+!o2J/\"Fdo
$\"+Q&R@3\"Fdo7$$\"+%Q#\\\"3\"Fdo$\"+B.t+6Fdo7$$\"+;*[H7\"Fdo$\"+acJH6
Fdo7$$\"+qvxl6Fdo$\"+C#Gs;\"Fdo7$$\"+`qn27Fdo$\"+t:(z?\"Fdo7$$\"+cp@[7
Fdo$\"+mwF[7Fdo7$$\"+3'HKH\"Fdo$\"+@+C$H\"Fdo7$$\"+xanL8Fdo$\"+2wnL8Fd
o7$$\"+v+'oP\"Fdo$\"+l/'oP\"Fdo7$$\"+S<*fT\"Fdo$\"+D=*fT\"Fdo7$$\"+&)H
xe9Fdo$\"+,Ixe9Fdo7$$\"+.o-*\\\"Fdo$\"+2o-*\\\"Fdo7$$\"+TO5T:Fdo$\"+VO
5T:Fdo7$$\"+U9C#e\"Fdo$\"+T9C#e\"Fdo7$$\"+1*3`i\"Fdo$\"+3*3`i\"Fdo7$$
\"+$*zym;Fdo$\"+%*zym;Fdo7$$\"+^j?4<FdoFbx7$$\"+jMF^<FdoFex7$$\"+q(G**
y\"FdoFhx7$$\"+9@BM=FdoF[y7$$\"+`v&Q(=FdoF^y7$$\"+Ol5;>FdoFay7$$\"+/Ua
c>FdoFdy7$$\"\"#\"\"!$\"+++++?Fdo-%'COLOURG6&%$RGBG$\"#5!\"\"FiyFiy-F$
6%7gn7$FiyFiy7$$\"+P@Ki8!#7$\"+,l^%R)F-7$$\"+tUkCFFjz$\"+\"e>Kc)F-7$$
\"+5k'p3%Fjz$\"+775i')F-7$$\"+X&)G\\aFjz$\"+!3%QK()F-7$$\"+=G$R<)Fjz$
\"+wJoJ))F-7$$\"+4x&)*3\"F*$\"+CdO-*)F-7$$\"+klyM;F*$\"+h%RC+*F-7$$\"+
<arz@F*$\"+i5)Q2*F-7$$\"+EJdpKF*$\"+=mXv\"*F-7$F($\"+.wR[#*F-7$F/$\"+a
EXT$*F-F37$F9$\"+EHdD&*F-7$F>$\"+@tV5'*F-7$FC$\"+Zz#*y'*F-7$FH$\"+JuoL
(*F-FL7$FR$\"+f0iK)*F-7$FW$\"+^/\\x)*F-7$Ffn$\"+U0T@**F-7$F[o$\"+>([)e
**F-F_oFeoFjoF_pFdpFipF^q7$Fdq$\"+E@#)H5Fdo7$Fiq$\"+`.TN5Fdo7$F^r$\"+4
?QT5FdoFbrFgr7$F]s$\"+%=Or1\"FdoFasFfsF[tF`tFet7$F[u$\"+nwF[7FdoF_uFdu
7$Fju$\"+m/'oP\"Fdo7$F_v$\"+C=*fT\"FdoFcv7$Fiv$\"+1o-*\\\"Fdo7$F^w$\"+
UO5T:Fdo7$FcwFcw7$FhwFhw7$F]xF]xFaxFdxFgxFjxF]yF`yFcyFfy-F]z6&F_zFiyFi
y$\"*++++\"!\")-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"v6\"%!G-%%VIEWG6$
;FiyFgy%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The second derivative accordin
g to theory is " }{XPPEDIT 18 0 "diff(g(v),v$2)=1/diff(V(J(v)),i)" "6#
/-%%diffG6$-%\"gG6#%\"vG-%\"$G6$F*\"\"#*&\"\"\"\"\"\"-F%6$-%\"VG6#-%\"
JG6#F*%\"iG!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 264 1 "d2g:=unapply(1/eval(diff(V(i),i),i=J(v)),v):" "6#>%$d2
gG-%(unapplyG6$*&\"\"\"\"\"\"-%%evalG6$-%%diffG6$-%\"VG6#%\"iGF4/F4-%
\"JG6#%\"vG!\"\"F9" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p1:=plo
t(subs(numbers,diff(g(v),v$2)),v=0..2):" "6#>%#p1G-%%plotG6$-%%subsG6$
%(numbersG-%%diffG6$-%\"gG6#%\"vG-%\"$G6$F2\"\"#/F2;\"\"!\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p2:=plot(subs(numbers,d2g(v)),
v=0..2,style=point,color=blue):" "6#>%#p2G-%%plotG6&-%%subsG6$%(number
sG-%$d2gG6#%\"vG/F/;\"\"!\"\"#/%&styleG%&pointG/%&colorG%%blueG" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "display(p1,p2,labels = [\"v\",
 \"D2[g(v),v]\"]);" "6#-%(displayG6%%#p1G%#p2G/%'labelsG7$Q\"v6\"Q+D2[
g(v),v]F," }}{PARA 13 "" 1 "" {GLPLOT2D 431 197 197 {PLOTDATA 2 "6&-%'
CURVESG6$7_o7$$\"+M3VfV!#6$\"+&zdB&e!#57$$\"+<0dL[F*$\"+r=l0`F-7$$\"+*
>5xI&F*$\"+3^xc[F-7$$\"+#))\\=y&F*$\"+h@n\"[%F-7$$\"+j&*)fD'F*$\"+c&yN
;%F-7$$\"+G*oU?(F*$\"+^GU`OF-7$$\"+#H[D:)F*$\"+P$yCE$F-7$$\"+%pU&G5F-$
\"+SpD[EF-7$$\"+e0$=C\"F-$\"+La8ZAF-7$$\"+LA`c9F-$\"+&y**R'>F-7$$\"+3R
Br;F-$\"+%)Q^b<F-7$$\"+zjf)4#F-$\"+9)4DZ\"F-7$$\"+'4;[\\#F-$\"+WVX-8F-
7$$\"+j'y]!HF-$\"+@e/\"=\"F-7$$\"+'zs$HLF-$\"+))pG$4\"F-7$$\"+8iI_PF-$
\"+!\\vA.\"F-7$$\"+<_M(=%F-$\"*%Qq)*)*F-7$$\"+4y_qXF-$\"*dU7m*F-7$$\"+
]1!>+&F-$\"*g\"[B&*F-7$$\"+]Z/NaF-$\"*?>4^*F-7$$\"+]$fC&eF-$\"*BZ^h*F-
7$$\"+'z6:B'F-$\"*SOJ\")*F-7$$\"+<=C#o'F-$\"+nC:>5F-7$$\"+n#pS1(F-$\"*
&R`l5!\"*7$$\"+j`A3vF-$\"*)>JS6F[s7$$\"+n(y8!zF-$\"*TP5B\"F[s7$$\"+j.t
K$)F-$\"*JbwO\"F[s7$$\"+)3zMu)F-$\"*2c0b\"F[s7$$\"+#H_?<*F-$\"*C)pF=F[
s7$$\"+!G;cc*F-$\"*w>/@#F[s7$$\"+XA(yx*F-$\"*k`t\\#F[s7$$\"+4#G,***F-$
\"*Sww'GF[s7$$\"+^-1@5F[s$\"*m$HuLF[s7$$\"+!o2J/\"F[s$\"*6,/0%F[s7$$\"
+cQq_5F[s$\"*'Q45WF[s7$$\"+K+Ii5F[s$\"*(3(Q\"[F[s7$$\"+3i*=2\"F[s$\"*K
*>i_F[s7$$\"+%Q#\\\"3\"F[s$\"*<%y^dF[s7$$\"+<l&=4\"F[s$\"*a-oJ'F[s7$$
\"+]1A-6F[s$\"*iz1!pF[s7$$\"+$y%e76F[s$\"*,'ywuF[s7$$\"+;*[H7\"F[s$\"*
p9a,)F[s7$$\"+VKOW6F[s$\"+?Q.5*)F-7$$\"+qvxl6F[s$\"+9KKj%*F-7$$\"+7ts'
=\"F[s$\"+WJX[(*F-7$$\"+`qn27F[s$\"+ZT(f))*F-7$$\"+0q%zA\"F[s$\"+KIyZ*
*F-7$$\"+cp@[7F[s$\"+!4gi(**F-7$$\"+#GB2F\"F[s$\"+)GX,***F-7$$\"+3'HKH
\"F[s$\"+$><f***F-7$$\"+xanL8F[s$\"+fQ;****F-7$$\"+v+'oP\"F[s$\"+no%)*
***F-7$$\"+S<*fT\"F[s$\"+9q'*****F-7$$\"+&)Hxe9F[s$\"+HT******F-7$$\"+
.o-*\\\"F[s$\"+2%)******F-7$$\"+TO5T:F[s$\"+I'*******F-7$$\"+U9C#e\"F[
s$\"+%*********F-7$$\"+1*3`i\"F[s$\"+-+++5F[s7$$\"+$*zym;F[s$\"+l*****
***F-7$$\"+^j?4<F[s$\"+\"*********F-7$$\"+jMF^<F[s$\"+)*********F-7$$
\"+q(G**y\"F[s$\"+++++5F[s7$$\"+9@BM=F[sFh^l7$$\"+`v&Q(=F[sFh^l7$$\"+O
l5;>F[sFh^l7$$\"+/Uac>F[sFh^l7$$\"\"#\"\"!Fh^l-%'COLOURG6&%$RGBG$\"#5!
\"\"Fi_lFi_l-F$6%7_o7$F($\"+=xN_eF-7$F/$\"+9>l0`F-7$F4$\"+\\]xc[F-7$F9
$\"+8@n\"[%F-7$F>$\"+9&yN;%F-7$FC$\"+*yAMl$F-7$FH$\"+5$yCE$F-7$FM$\"+O
pD[EF-7$FR$\"+$RNrC#F-7$FW$\"+q(**R'>F-7$Ffn$\"+IQ^b<F-7$F[o$\"+#z4DZ
\"F-7$F`o$\"+5VX-8F-7$Feo$\"+ee/\"=\"F-7$Fjo$\"+#)pG$4\"F-7$F_p$\"+%[v
A.\"F-7$Fdp$\"+)*Rq)*)*F*7$Fip$\"+>GCh'*F*7$F^q$\"+gB[B&*F*7$Fcq$\"+'f
?4^*F*7$Fhq$\"+]m9:'*F*7$F]r$\"+_h88)*F*7$Fbr$\"+WC:>5F-7$Fgr$\"+3S`l5
F-7$F]s$\"+P=JS6F-7$Fbs$\"+(RP5B\"F-7$Fgs$\"+Ialn8F-7$F\\t$\"+xfb]:F-7
$Fat$\"+D%)pF=F-7$Fft$\"+O*>/@#F-7$F[u$\"+[MN(\\#F-7$F`u$\"+zjnnGF-7$F
eu$\"+qMHuLF-7$Fju$\"+!3,/0%F-7$F_v$\"+FR45WF-7$Fdv$\"+(*3(Q\"[F-7$Fiv
$\"+3#*>i_F-7$F^w$\"+,Vy^dF-7$Fcw$\"+sC!oJ'F-7$Fhw$\"+N'z1!pF-7$F]x$\"
+igywuF-7$Fbx$\"+2ZT:!)F-7$Fgx$\"+zN.5*)F-7$F\\y$\"+(GBLY*F-7$Fay$\"+i
IX[(*F-7$Ffy$\"+<T(f))*F-7$F[z$\"+aGyZ**F-7$F`z$\"+W(fi(**F-7$Fez$\"+s
[9!***F-7$Fjz$\"+Qq\"f***F-7$F_[l$\"+]Q;****F-7$Fd[l$\"+ik%)****F-7$Fi
[l$\"+!)p'*****F-7$F^\\l$\"+gQ******F-7$Fc\\l$\"+I()******F-7$Fh\\l$\"
+g(*******F-7$F]]l$\"+S********F-7$Fb]lFh^l7$Fg]lFh^l7$F\\^lFh^l7$Fa^l
Fh^lFe^lFj^lF]_lF`_lFc_lFf_l-F[`l6&F]`lFi_lFi_l$\"*++++\"!\")-%&STYLEG
6#%&POINTG-%+AXESLABELSG6$Q\"v6\"Q+D2[g(v),v]F`\\m-%%VIEWG6$;Fi_lFg_l%
(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}}
{MARK "25 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }