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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "elas37.ms" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "interface(labelling=false):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "grtw();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorI
I~Version~1.70~(R5)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,31~May~1998G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Den
is~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyri
ght~1994-1998~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\o
Latest~version~available~from:~http://astro.queensu.ca/|irgrtensor/G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PDefaults~read~from~c:/Grtii(5)/Lib
/grtensor.iniG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(5)/Metric
sG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grOptionTermSize:=0:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grlib(elasticity):" }}
{PARA 6 "" 1 "" {TEXT -1 22 "The Elasticity Package" }}{PARA 6 "" 1 "
" {TEXT -1 23 "Last Modified May, 1998" }}{PARA 6 "" 1 "" {TEXT -1 53 
"Created by P. Musgrave and K. Lake. Copyright 1996-98" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Conventio
ns: Deformed state $B$ with metric $G_\{ij\}$ generated from coordinat
es $(y1,y2,y3)$, " }}{PARA 0 "" 0 "" {TEXT -1 86 "undeformed state $B_
\{0\}$ with metric $g_\{ij\}$ generated from coordinates $(x1,x2,x3)$.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "\{\\b
f 3.7 Extension, inflation, and shear of a cylindrical tube\}" }}
{PARA 0 "" 0 "" {TEXT -1 41 "Take the body coordinates $(r,\\theta,z)$
." }}{PARA 0 "" 0 "" {TEXT -1 21 "To generate $G_\{ij\}$:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "qload(ey);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%2Default~spacetimeG%#eyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6For~
the~ey~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%/)%#x~G%\"1G%#y1G/)F%%\"2G%#y2G/)F%%\"3G%#y3G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*$%$~dsG\"\"#,(*&%#~dG\"\"\")%#y1G%#2~GF*F**&F)F*)%#y2
GF-F*F**&F)F*)%#y3GF-F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "xform1:=[y1(r,theta)=r*cos(theta),y
2(r,theta)=r*sin(theta),y3(z)=z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%'xform1G7%/-%#y1G6$%\"rG%&thetaG*&F*\"\"\"-%$cosG6#F+F-/-%#y2GF)*&F*F
--%$sinGF0F-/-%#y3G6#%\"zGF;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "grtransform(ey,deformed,xform1):" }}{PARA 6 "" 1 "" {TEXT -1 35 
"The new default metric is: deformed" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "gralter(g(dn,dn),trig);" }}{PARA 6 "" 1 "" {TEXT -1 
103 "Component simplification of a GRTensorII object:\n \nApplying rou
tine `simplify[trig]` to object g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%*CPU~Time~G$\"$J\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~def
ormed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metri
c~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F)F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thetaG\"\"\"&%!GF'F)*$%\"rG\"\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F)F)
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "To generate $g_\{ij\}$:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "qload(ex);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%2Default~spacetimeG%#exG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%6For~the~ex~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/)%#x~G%\"1G%#x1G/)F%%\"2G%#x2G/)F
%%\"3G%#x3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$%$~dsG\"\"#,(*&%#~dG\"\"\")%#x1G%#2
~GF*F**&F)F*)%#x2GF-F*F**&F)F*)%#x3GF-F*F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 110 "xform2:=[x1(r,theta,z)=r*Q(r)*cos(theta-phi(r)),x2
(r,theta,z)=r*Q(r)*sin(theta-phi(r)),x3(z)=(z-w(r))/lambda];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'xform2G7%/-%#x1G6%%\"rG%&thetaG%\"zG*(F*
\"\"\"-%\"QG6#F*F.-%$cosG6#,&F+!\"\"-%$phiGF1F.F./-%#x2GF),$*(F*F.F/F.
-%$sinGF4F.F6/-%#x3G6#F,*&,&F,F.-%\"wGF1F6F.%'lambdaGF6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grtransform(ex,undeformed,xform2):
" }}{PARA 6 "" 1 "" {TEXT -1 37 "The new default metric is: undeformed
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gralter(g(dn,dn),trig,e
xpand);" }}{PARA 6 "" 1 "" {TEXT -1 146 "Component simplification of a
 GRTensorII object:\n \nApplying routine `simplify[trig]` to object g(
dn,dn)\nApplying routine expand to object g(dn,dn)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%*CPU~Time~G$\"#!*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 
38 "Applying routine autoAlias to g(dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>
For~the~undeformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Co
variant~metric~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dn
GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F),
,*(F(\"\"#-%\"QGF'F.&%$phiGF'F.F)*$F/F.F)*(F/F)F(F)&F0F'F)F.*&F(F.F5F.
F)*&&%\"wGF'F.%'lambdaG!\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%
#g~G6#%\"rG\"\"\"&%!G6#%&thetaGF),$*(F(\"\"#-%\"QGF'F0&%$phiGF'F)!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!G6#%\"zGF
),$*&&%\"wGF'F)%'lambdaG!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*&&%#g~G6#%&thetaG\"\"\"&%!GF'F)*&%\"rG\"\"#-%\"QG6#F-F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F)*$%'lambdaG!\"#" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "body(g=undeformed,G=deform
ed):" }}{PARA 6 "" 1 "" {TEXT -1 36 "The default metric is now: deform
ed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "grcalc(strain(dn,dn)
,I1,I2,I3,B(up,up),stress(up,up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/%*CPU~Time~G$\"$,%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
gralter(_,simplify,factor);" }}{PARA 6 "" 1 "" {TEXT -1 562 "Component
 simplification of a GRTensorII object:\n \nApplying routine simplify \+
to object strain(dn,dn)\nApplying routine simplify to object I1\nApply
ing routine simplify to object I2\nApplying routine simplify to object
 I3\nApplying routine simplify to object B(up,up)\nApplying routine si
mplify to object stress(up,up)\nApplying routine factor to object stra
in(dn,dn)\nApplying routine factor to object I1\nApplying routine fact
or to object I2\nApplying routine factor to object I3\nApplying routin
e factor to object B(up,up)\nApplying routine factor to object stress(
up,up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$6)!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`):" }}
{PARA 6 "" 1 "" {TEXT -1 225 "Applying routine autoAlias to strain(dn,
dn)\nApplying routine autoAlias to I1\nApplying routine autoAlias to I
2\nApplying routine autoAlias to I3\nApplying routine autoAlias to B(u
p,up)\nApplying routine autoAlias to stress(up,up)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%<For~the~deformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%.Strain~TensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'strainG6$%#dn
GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G
F'F),$*&,.*$%'lambdaG\"\"#!\"\"**F(F1-%\"QGF'F1&%$phiGF'F1F0F1F)*&F4F1
F0F1F)**F4F)F(F)&F5F'F)F0F1F1*(F(F1F:F1F0F1F)*$&%\"wGF'F1F)F)F0!\"##F2
F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6
#%&thetaGF),$*(F(\"\"#-%\"QGF'F0&%$phiGF'F)#F)F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6#%\"zGF),$*&&%\"wGF'F)%
'lambdaG!\"##F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G
6#%&thetaG\"\"\"&%!GF'F),$*(%\"rG\"\"#,&-%\"QG6#F.F)!\"\"F)F),&F1F)F)F
)F)#F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"
\"&%!GF'F),$*(,&%'lambdaGF)!\"\"F)F),&F/F)F)F)F)F/!\"##F)\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%$I1~G*(,2*$-%\"QG6#%\"rG\"\"#F,*(F+F,F(F,&
%$phiGF*F,\"\"\"*(F(F0F+F0&F)F*F0F,*&F+F,F2F,F0*&F(\"\"%%'lambdaGF,F0*
*F(\"\"$F+F0F2F0F6F,F,**F(F,F+F,F2F,F6F,F0*&F(F,&%\"wGF*F,F0F0F(!\"#,&
F(F0*&F+F0F2F0F0F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Inv
ariant~I2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2~G*(,.**%\"rG\"\"#-
%\"QG6#F(F)&%$phiGF,F)%'lambdaGF)\"\"\"*&F*F)F/F)F)**F*F0F(F0&F+F,F0F/
F)F)*(F(F)F3F)F/F)F0*$&%\"wGF,F)F0F0F0F0F*!\"#,&F*F0*&F(F0F3F0F0F8" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%$I3~G*(-%\"QG6#%\"rG!\"#%'lambdaG\"\"#,&F&
\"\"\"*&F)F.&F'F(F.F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Elasticit
y~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"BG6$%#upGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!GF'F(*(,&*&-%\"
QG6#F'\"\"#%'lambdaGF1F(F(F(F(F.!\"#,&F.F(*&F'F(&F/F0F(F(F3" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!G%&thetaGF(*(%'lambda
G\"\"#&%$phiG6#F'F(,&-%\"QGF1F(*&F'F(&F4F1F(F(!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!G%\"zGF(*(&%\"wG6#F'F(-%\"QGF/
!\"#,&F0F(*&F'F(&F1F/F(F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B
~G%&thetaG\"\"\")%!GF'F(**,.**%\"rG\"\"#-%\"QG6#F.F/&%$phiGF2F/%'lambd
aGF/F(*&F0F/F5F/F(**F0F(F.F(&F1F2F(F5F/F/*$&%\"wGF2F/F(F(F(*(F.F/F8F/F
5F/F(F(F0!\"#F.F=,&F0F(*&F.F(F8F(F(F=" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/*&)%#B~G%\"zG\"\"\")%!GF'F(*(,,**%\"rG\"\"#-%\"QG6#F.F/&%$phiGF2F/
%'lambdaGF/F(*&F0F/F5F/F/**F0F(F.F(&F1F2F(F5F/F/*(F.F/F8F/F5F/F(*$&%\"
wGF2F/F(F(F0!\"#,&F0F(*&F.F(F8F(F(F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%.stress~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#up
GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!GF'F(*(
,.*&-%$PhiG6#F'F(-%\"QGF0\"\"#F(*(-%$psiGF0F(F1F3%'lambdaGF3F(F5F(*&-%
\"pGF0F(F1\"\"%F(**F9F(F1\"\"$F'F(&F2F0F(F3**F9F(F1F3F'F3F>F3F(F(F1!\"
#,&F1F(*&F'F(F>F(F(F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"
rG\"\"\")%!G%&thetaGF(*(&%$phiG6#F'F(,&-%$PhiGF/F(*&-%$psiGF/F(%'lambd
aG\"\"#F(F(,&-%\"QGF/F(*&F'F(&F:F/F(F(!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%\"zGF(**&%\"wG6#F'F(,&*&-%$Ph
iGF/F(-%\"QGF/\"\"#F(-%$psiGF/F(F(F4!\"#,&F4F(*&F'F(&F5F/F(F(F9" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&thetaG\"\"\")%!GF'F(**,<**
-%$PhiG6#%\"rGF(F1\"\"#-%\"QGF0F2&%$phiGF0F2F(*&F.F(F3F2F(**F.F(F3F(F1
F(&F4F0F(F2*(F.F(F1F2F9F2F(*,-%$psiGF0F(F1F2F3F2F5F2%'lambdaGF2F(*(F<F
(F3F2F>F2F(*,F<F(F3F(F1F(F9F(F>F2F2*&F<F(&%\"wGF0F2F(F<F(**F<F(F1F2F9F
2F>F2F(*&-%\"pGF0F(F3\"\"%F(**FFF(F3\"\"$F1F(F9F(F2**FFF(F3F2F1F2F9F2F
(F(F1!\"#F3FL,&F3F(*&F1F(F9F(F(FL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*&)%$tauG%&thetaG\"\"\")%!G%\"zGF(**-%$PhiG6#%\"rGF(&%$phiGF/F(&%\"wGF
/F(,&-%\"QGF/F(*&F0F(&F7F/F(F(!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
/*&)%$tauG%\"zG\"\"\")%!GF'F(*(,:*(-%$PhiG6#%\"rGF(-%\"QGF0\"\"%%'lamb
daG\"\"#F(*,F.F(F2\"\"$F1F(&F3F0F(F5F6F6*,F.F(F2F6F1F6F9F6F5F6F(*(F.F(
F2F6&%\"wGF0F6F(*,-%$psiGF0F(F1F6F2F6&%$phiGF0F6F5F6F(*(F?F(F2F6F5F6F6
*,F?F(F2F(F1F(F9F(F5F6F6**F?F(F1F6F9F6F5F6F(*&F?F(F<F6F(*&-%\"pGF0F(F2
F4F(**FHF(F2F8F1F(F9F(F6**FHF(F2F6F1F6F9F6F(F(F2!\"#,&F2F(*&F1F(F9F(F(
FL" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Now the incompressibility c
ondition $I_\{3\} = 1$ can be solved explicitly for $Q(r)$" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 34 "dsolve(grcomponent(I3,[])=1,Q(r));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$/*$-%\"QG6#%\"rG\"\"#*&F(!\"#,&*&F(F)%
'lambdaG\"\"\"!\"\"%$_C1GF/F//F$*&F(F+,&F-F/F1F/F/" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 100 "Following Green and Zerna we first removed deriv
atives of $Q(r)$ and leave $Q(r)$ in the components." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 121 "grmap(g(dn,dn)[undeformed],strain(dn,dn),I1,I2,
I3,B(up,up),stress(up,up),subs,diff(Q(r),r)=(lambda-Q(r)^2)/(Q(r)*r),`
x`);" }}{PARA 6 "" 1 "" {TEXT -1 229 "Applying routine subs to g(dn,dn
)\nApplying routine subs to strain(dn,dn)\nApplying routine subs to I1
\nApplying routine subs to I2\nApplying routine subs to I3\nApplying r
outine subs to B(up,up)\nApplying routine subs to stress(up,up)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,expand);" }}{PARA 
6 "" 1 "" {TEXT -1 343 "Component simplification of a GRTensorII objec
t:\n \nApplying routine expand to object g(dn,dn)\nApplying routine ex
pand to object strain(dn,dn)\nApplying routine expand to object I1\nAp
plying routine expand to object I2\nApplying routine expand to object \+
I3\nApplying routine expand to object B(up,up)\nApplying routine expan
d to object stress(up,up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Ti
me~G$\"#I!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(
_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>For~the~undeformed~spacetime:
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metric~tensorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F),(*(F(\"\"#-%\"QGF'F.&%$p
hiGF'F.F)*&F/!\"#%'lambdaGF.F)*&&%\"wGF'F.F5F4F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!G6#%&thetaGF),$*(F(\"\"#-%\"QG
F'F0&%$phiGF'F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"
rG\"\"\"&%!G6#%\"zGF),$*&&%\"wGF'F)%'lambdaG!\"#!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thetaG\"\"\"&%!GF'F)*&%\"rG\"\"#-%\"Q
G6#F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF
'F)*$%'lambdaG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~deform
ed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!GF'F),*#F)\"\"#F)*(F(F.
-%\"QGF'F.&%$phiGF'F.#!\"\"F.*&F0!\"#%'lambdaGF.F4*&&%\"wGF'F.F8F7F4" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6#%&t
hetaGF),$*(F(\"\"#-%\"QGF'F0&%$phiGF'F)#F)F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6#%\"zGF),$*&&%\"wGF'F)%
'lambdaG!\"##F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G
6#%&thetaG\"\"\"&%!GF'F),&*&%\"rG\"\"#-%\"QG6#F.F/#!\"\"F/*$F.F/#F)F/
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"\"&%!GF'F
),&#F)\"\"#F)*$%'lambdaG!\"##!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%8Elasticity~Invariant~I1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~
G,,**%'lambdaG!\"#%\"rG\"\"#-%\"QG6#F)F*&%$phiGF-F*\"\"\"*$F+F(F0*&F'F
(F+F*F0*$F'F*F0*(F'F(F+F*&%\"wGF-F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%8Elasticity~Invariant~I2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2
~G,,*(%\"rG\"\"#-%\"QG6#F'F(&%$phiGF+F(\"\"\"*$F)F(F.*&F)!\"#%'lambdaG
F(F.*&&%\"wGF+F(F2F1F.*$F2F1F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8El
asticity~Invariant~I3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I3~G\"\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Elasticity~Tensor~BG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%\"BG6$%#upGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!GF'F(,&*$-%\"QG6#F'\"\"#F(*$%'la
mbdaG!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!
G%&thetaGF(*&&%$phiG6#F'F(-%\"QGF/\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&)%#B~G%\"rG\"\"\")%!G%\"zGF(*&&%\"wG6#F'F(%'lambdaG!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"\")%!GF'F(,**&-%
\"QG6#%\"rG\"\"#&%$phiGF/F1F(*(F0!\"#%'lambdaGF5&%\"wGF/F1F(*&F0F5F6F5
F(*(F0F5F6F1F-F5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"zG\"
\"\")%!GF'F(,**(%\"rG\"\"#-%\"QG6#F-F.&%$phiGF1F.F(*$F/F.F(*&F/!\"#%'l
ambdaGF.F(*&&%\"wGF1F.F7F6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stre
ss~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#upGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!GF'F(,**(%'lam
bdaG!\"#-%$PhiG6#F'F(-%\"QGF1\"\"#F(*&-%$psiGF1F(F2F4F(*&F-F.F6F(F(-%
\"pGF1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%
&thetaGF(,&**&%$phiG6#F'F(-%\"QGF0\"\"#%'lambdaG!\"#-%$PhiGF0F(F(*(F.F
(F1F3-%$psiGF0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG
\"\"\")%!G%\"zGF(,&**&%\"wG6#F'F(%'lambdaG!\"#-%$PhiGF0F(-%\"QGF0\"\"#
F(*(F.F(F1F2-%$psiGF0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tau
G%&thetaG\"\"\")%!GF'F(,0**%'lambdaG!\"#-%$PhiG6#%\"rGF(-%\"QGF1\"\"#&
%$phiGF1F5F(*(F2F.F/F(F3F.F(*(-%$psiGF1F(F3F5F6F5F(**F2F.F-F.F:F(&%\"w
GF1F5F(*(F2F.F-F.F:F(F(**F2F.F-F5F:F(F3F.F(*&F2F.-%\"pGF1F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&thetaG\"\"\")%!G%\"zGF(*,-
%$PhiG6#%\"rGF(&%$phiGF/F(&%\"wGF/F(-%\"QGF/\"\"#%'lambdaG!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"zG\"\"\")%!GF'F(,0*&-%$Ph
iG6#%\"rGF(%'lambdaG\"\"#F(**F1!\"#F-F(-%\"QGF/F2&%\"wGF/F2F(**-%$psiG
F/F(F0F2F5F2&%$phiGF/F2F(*&F:F(F5F2F(*(F:F(F5F4F1F2F(*(F1F4F:F(F7F2F(-
%\"pGF/F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We now define the ve
ctor $T^\{a\} = \\tau^\{ab\}||-\{b\}$." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "grdefine(`T\{^b\}`,\{\},`stress\{^a^b;a\}`):" }}{PARA 6 "" 1 "
" {TEXT -1 74 "Created a definition for    stress(up,up,cdn)\nCreated \+
definition for T(up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grc
alc(T(up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$@\"!\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`)
;" }}{PARA 6 "" 1 "" {TEXT -1 35 "Applying routine autoAlias to T(up)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,expand):" }}
{PARA 6 "" 1 "" {TEXT -1 90 "Component simplification of a GRTensorII \+
object:\n \nApplying routine expand to object T(up)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%*CPU~Time~G$\"#5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<Fo
r~the~deformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&T(up)G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"TG6#%#upG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/)%\"TG%\"rG,<*(%'lambdaG!\"#-%\"QG6#F&\"\"#&%$PhiGF-\"
\"\"F1**F)F*F+F1-F0F-F1&F,F-F1F.*&F+F.&%$psiGF-F1F1*(F+F1-F7F-F1F4F1F.
*&F)F*F6F1F1&%\"pGF-F1**F&!\"\"F)F*F+F.F3F1F1*(F&F>F+F.F9F1F1*,F&F1F)F
*F+F.F3F1&%$phiGF-F.F>*(F&F>F+F*F3F1F>**F&F1F+F.F9F1FAF.F>**F&F>F)F*F9
F1&%\"wGF-F.F>**F&F>F)F.F+F*F9F1F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/)%\"TG%&thetaG,2**-%\"QG6#%\"rG\"\"#%'lambdaG!\"#&%$phiG6$F,F,\"\"\"-
%$PhiGF+F3F3*,F)F3F.F/&F1F+F3F4F3&F*F+F3F-**F)F-F.F/F7F3&F5F+F3F3*(F)F
-F0F3-%$psiGF+F3F3**F)F3F7F3F<F3F8F3F-*(F)F-F7F3&F=F+F3F3*,F)F-F.F/F,!
\"\"F7F3F4F3\"\"$**F)F-F,FBF7F3F<F3FC" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/)%\"TG%\"zG,0**%'lambdaG!\"#&%\"wG6$%\"rGF.\"\"\"-%$PhiG6#F.F/-%\"
QGF2\"\"#F/**F)F*&F,F2F/&F1F2F/F3F5F/*,F)F*F7F/F0F/F3F/&F4F2F/F5*(F)F*
F+F/-%$psiGF2F/F/*(F)F*F7F/&F=F2F/F/*,F)F*F.!\"\"F7F/F0F/F3F5F/**F)F*F
.FAF7F/F<F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The equi
librium condition $T^\{a\} = \\tau^\{ab\}||_\{b\} = 0$  gives $p(r)$.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "expand(dsolve(grcomponent(T(up)
,[r])=0,p(r)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"rG,&*&%
'lambdaG!\"#-%$intG6$,:*&&%$PhiGF&\"\"\"-%\"QGF&\"\"#!\"\"*(-F2F&F3F4F
3&F5F&F3F+*(&%$psiGF&F3F4F6F*F6F7**-F=F&F3F4F3F:F3F*F6F+F<F7*(F'F7F4F6
F9F3F7**F'F7F4F6F?F3F*F6F7**F'F3F4F6F9F3&%$phiGF&F6F3**F'F7F4F+F9F3F*F
6F3*,F'F3F4F6F?F3FCF6F*F6F3*(F'F7F?F3&%\"wGF&F6F3**F'F7F4F+F?F3F*\"\"%
F3F'F3F3%$_C1GF3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 96 "Because we have solved for $Q(r)$ we can go back a
nd write the entire analysis in explicit form." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "grmap(g(dn,dn)[undeformed],strain(dn,dn),I1,I2,I3,B(
up,up),stress(up,up),T(up),subs,Q(r)=(lambda*(r^2+K))^(1/2)/r,`x`);" }
}{PARA 6 "" 1 "" {TEXT -1 260 "Applying routine subs to g(dn,dn)\nAppl
ying routine subs to strain(dn,dn)\nApplying routine subs to I1\nApply
ing routine subs to I2\nApplying routine subs to I3\nApplying routine \+
subs to B(up,up)\nApplying routine subs to stress(up,up)\nApplying rou
tine subs to T(up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gralt
er(_,simplify,factor);" }}{PARA 6 "" 1 "" {TEXT -1 732 "Component simp
lification of a GRTensorII object:\n \nApplying routine simplify to ob
ject g(dn,dn)\nApplying routine simplify to object strain(dn,dn)\nAppl
ying routine simplify to object I1\nApplying routine simplify to objec
t I2\nApplying routine simplify to object I3\nApplying routine simplif
y to object B(up,up)\nApplying routine simplify to object stress(up,up
)\nApplying routine simplify to object T(up)\nApplying routine factor \+
to object g(dn,dn)\nApplying routine factor to object strain(dn,dn)\nA
pplying routine factor to object I1\nApplying routine factor to object
 I2\nApplying routine factor to object I3\nApplying routine factor to \+
object B(up,up)\nApplying routine factor to object stress(up,up)\nAppl
ying routine factor to object T(up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%*CPU~Time~G$\"%V?!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>For~the~undeformed
~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metric~ten
sorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F)*(,.*(%'lambdaG\"
\"$&%$phiGF'\"\"#F(\"\"%F)**F/F0F1F3F(F3%\"KGF)F3*(F/F0F1F3F6F3F)*&F(F
3F/F0F)*&&%\"wGF'F3F(F3F)*&F:F3F6F)F)F),&*$F(F3F)F6F)!\"\"F/!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!G6#%&thetaGF)
,$*(%'lambdaGF),&*$F(\"\"#F)%\"KGF)F)&%$phiGF'F)!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!G6#%\"zGF),$*&&%\"wGF'F)%
'lambdaG!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thet
aG\"\"\"&%!GF'F)*&%'lambdaGF),&*$%\"rG\"\"#F)%\"KGF)F)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F)*$%'lambdaG!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*&&%(strain~G6#%\"rG\"\"\"&%!GF'F),$*(,2*&%'lambdaG\"\"#F(F1!\"\"*&F0
F1%\"KGF)F2*(F0\"\"$&%$phiGF'F1F(\"\"%F)**F0F6F7F1F(F1F4F)F1*(F0F6F7F1
F4F1F)*&F(F1F0F6F)*&&%\"wGF'F1F(F1F)*&F>F1F4F)F)F),&*$F(F1F)F4F)F2F0!
\"##F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"
\"&%!G6#%&thetaGF),$*(%'lambdaGF),&*$F(\"\"#F)%\"KGF)F)&%$phiGF'F)#F)F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6#
%\"zGF),$*&&%\"wGF'F)%'lambdaG!\"##F)\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%&thetaG\"\"\"&%!GF'F),(*&%\"rG\"\"#%'l
ambdaGF)#!\"\"F/*&F0F)%\"KGF)F1*$F.F/#F)F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"\"&%!GF'F),$*(,&%'lambdaGF)!\"
\"F)F),&F/F)F)F)F)F/!\"##F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8
Elasticity~Invariant~I1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~G**,
8*&&%$phiG6#%\"rG\"\"#F+\"\"'\"\"\"*(F(F,F+\"\"%%\"KGF.F,*(F(F,F+F,F1F
,F.*$F+F0F,*&F+F,F1F.F,*$F1F,F.*&F+F0%'lambdaG\"\"$F.*(F+F,F7F8F1F.F.*
&&%\"wGF*F,F+F0F.*(F;F,F+F,F1F.F,*&F;F,F1F,F.F.F7!\"\",&*$F+F,F.F1F.F?
F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I2G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2~G**,6*(%'lambdaG\"\"$&%$phiG6#
%\"rG\"\"#F-\"\"'\"\"\"**F(F)F*F.F-\"\"%%\"KGF0F.**F(F)F*F.F-F.F3F.F0*
&F-F2F(F)F.*(F-F.F(F)F3F0F.*&F(F)F3F.F0*&&%\"wGF,F.F-F2F0*(F9F.F-F.F3F
0F0*&F-F.F3F0F0*$F-F2F0F0F-!\"#,&*$F-F.F0F3F0!\"\"F(F>" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I3~G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Elas
ticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"BG6$%#upGF&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!GF'F(*(,(*&F'
\"\"#%'lambdaG\"\"$F(*&F/F0%\"KGF(F(*$F'F.F(F(F/!\"#F'F4" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!G%&thetaGF(**&%$phiG6#F'F
(%'lambdaGF(,&*$F'\"\"#F(%\"KGF(F(F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&)%#B~G%\"rG\"\"\")%!G%\"zGF(*&&%\"wG6#F'F(%'lambdaG!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"\")%!GF'F(**,2*(%
'lambdaG\"\"$&%$phiG6#%\"rG\"\"#F3\"\"%F(**F.F/F0F4F3F4%\"KGF(F4*(F.F/
F0F4F7F4F(*&&%\"wGF2F4F3F4F(*&F:F4F7F(F(*$F3F4F(F7F(*&F3F4F.F/F(F(F3!
\"#,&F=F(F7F(!\"\"F.F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"
zG\"\"\")%!GF'F(**,2*(%'lambdaG\"\"$&%$phiG6#%\"rG\"\"#F3\"\"'F(**F.F/
F0F4F3\"\"%%\"KGF(F4**F.F/F0F4F3F4F8F4F(*&F3F7F.F/F4*(F3F4F.F/F8F(F4*&
F.F/F8F4F(*&&%\"wGF2F4F3F7F(*(F>F4F3F4F8F(F(F(F3!\"#,&*$F3F4F(F8F(!\"
\"F.FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stress~tensorG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#upGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!GF'F(*(,.*(-%$PhiG6#F'F(%'lambd
aGF(F'\"\"#F(*(F.F(F1F(%\"KGF(F(*(-%$psiGF0F(F1\"\"$F'F2F(*(F6F(F1F8F4
F(F(*&F6F(F'F2F(*(-%\"pGF0F(F1F2F'F2F(F(F1!\"#F'F>" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%&thetaGF(*,&%$phiG6#F'F(,&*
$F'\"\"#F(%\"KGF(F(,&-%$PhiGF/F(*&-%$psiGF/F(%'lambdaGF2F(F(F'!\"#F:!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%\"zG
F(**&%\"wG6#F'F(,(*(-%$PhiGF/F(%'lambdaGF(F'\"\"#F(*(F2F(F4F(%\"KGF(F(
*&-%$psiGF/F(F'F5F(F(F4!\"#F'F;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&
)%$tauG%&thetaG\"\"\")%!GF'F(**,>**-%$PhiG6#%\"rGF(&%$phiGF0\"\"#%'lam
bdaGF(F1\"\"%F(*,F.F(F2F4F5F(F1F4%\"KGF(F4**F.F(F2F4F5F(F8F4F(*(F.F(F5
F(F1F4F(**-%$psiGF0F(F5\"\"$F2F4F1F6F(*,F<F(F5F>F2F4F1F4F8F(F4**F<F(F5
F>F2F4F8F4F(*(F<F(&%\"wGF0F4F1F4F(*(F<F(FBF4F8F(F(*&F<F(F1F4F(*&F<F(F8
F(F(*(F<F(F5F>F1F4F(*(-%\"pGF0F(F5F4F1F4F(*(FIF(F5F4F8F(F(F(F1!\"#F5FL
,&*$F1F4F(F8F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&the
taG\"\"\")%!G%\"zGF(*.-%$PhiG6#%\"rGF(&%$phiGF/F(&%\"wGF/F(%'lambdaG!
\"\",&*$F0\"\"#F(%\"KGF(F(F0!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*
&)%$tauG%\"zG\"\"\")%!GF'F(**,@*(-%$PhiG6#%\"rGF(%'lambdaG\"\"%F1F3F(*
*F.F(F2F3F1\"\"#%\"KGF(F(**F.F(&%\"wGF0F5F2F(F1F3F(*,F.F(F8F5F2F(F1F5F
6F(F5**F.F(F8F5F2F(F6F5F(**-%$psiGF0F(F2\"\"$&%$phiGF0F5F1\"\"'F(*,F=F
(F2F?F@F5F1F3F6F(F5*,F=F(F2F?F@F5F1F5F6F5F(*(F=F(F2F?F1F3F5**F=F(F2F?F
1F5F6F(F5*(F=F(F2F?F6F5F(*(F=F(F8F5F1F3F(**F=F(F8F5F1F5F6F(F(*(-%\"pGF
0F(F2F5F1F3F(**FKF(F2F5F1F5F6F(F(F(F1!\"#F2FN,&*$F1F5F(F6F(!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&T(up)G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%\"TG6#%#upG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"rG,$*
*,J**-%$psiG6#F&\"\"\"&%\"wGF-\"\"#F&F1%\"KGF.F.**F+F.%'lambdaG\"\"$&%
$phiGF-F1F&\"\"'F.*,F+F.F4F5F6F1F&\"\"%F2F.F1*,F+F.F4F5F6F1F&F1F2F1F.*
(F+F.F4F5F2F1F.*(F+F.F/F1F&F:F.*(&%$PhiGF-F.F&\"\"&F4F.!\"\"*(F4F5&F,F
-F.F&FAFB*(FDF.F&F5F2F.FB*(&%\"pGF-F.F&FAF4F1FB*(-F@F-F.F4F.F2F1F.**F4
F5FDF.F&F5F2F.!\"#**F4F5FDF.F&F.F2F1FB**FGF.F&F5F4F1F2F.FB**FJF.F6F1F4
F.F&F8F.*,FJF.F6F1F4F.F&F:F2F.F1*,FJF.F6F1F4F.F&F1F2F1F.**F?F.F&F5F4F.
F2F.FL**F?F.F&F.F4F.F2F1FB*&FDF.F&FAFBF.F&!\"$F4FL,&*$F&F1F.F2F.FBFB" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#/)%\"TG%&thetaG*(%'lambdaG!\"\",:*(&
%$phiG6$%\"rGF/\"\"\"-%$PhiG6#F/F0F/\"\"$F0**F,F0F1F0F/F0%\"KGF0F0*(&F
-F3F0F1F0F/\"\"#F4*(F8F0F1F0F6F0F0*(F8F0&F2F3F0F/F4F0**F8F0F<F0F/F0F6F
0F0**F(F9F,F0-%$psiGF3F0F/F4F0*,F(F9F,F0F?F0F/F0F6F0F0**F8F0F?F0F(F9F/
F9F4**F(F9F8F0F?F0F6F0F0**F(F9F8F0&F@F3F0F/F4F0*,F(F9F8F0FEF0F/F0F6F0F
0F0F/!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"zG*(,4**&%\"wG6
$%\"rGF-\"\"\"-%$PhiG6#F-F.F-\"\"$%'lambdaGF.F.*,F*F.F/F.F-F.F3F.%\"KG
F.F.**&F+F1F.&F0F1F.F-F2F3F.F.*,F7F.F8F.F-F.F3F.F5F.F.**F7F.F/F.F-\"\"
#F3F.F.**F7F.F/F.F3F.F5F.!\"\"*(F*F.-%$psiGF1F.F-F2F.*(F7F.&F@F1F.F-F2
F.*(F7F.F?F.F-F;F.F.F-!\"$F3!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "expand(dsolve(grcomponent(T(up),[r])=0,p(r)));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"rG,&*&%'lambdaG!\"#-%$intG
6$,F*,F'!\"\",&*$F'\"\"#\"\"\"%\"KGF5F1-%$psiGF&F5&%\"wGF&F4F6F5F5*,F'
\"\"$F2F1F7F5F*F<&%$phiGF&F4F5*.F'F5F2F1F7F5F*F<F=F4F6F5F4*.F'F1F2F1F7
F5F*F<F=F4F6F4F5*,F'!\"$F2F1F7F5F*F<F6F4F5**F'F5F2F1F7F5F9F4F5**F'F4F2
F1&%$PhiGF&F5F*F5F1**F'F4F2F1F*F<&F8F&F5F1*(F2F1FHF5F6F5F1*,F'FBF2F1-F
FF&F5F*F5F6F4F5**F2F1F*F<FHF5F6F5F+*,F'F+F2F1F*F<FHF5F6F4F1*,F'F<F2F1F
KF5F=F4F*F5F5*.F'F5F2F1FKF5F=F4F*F5F6F5F4*.F'F1F2F1FKF5F=F4F*F5F6F4F5*
*F2F1FEF5F*F5F6F5F+*,F'F+F2F1FEF5F*F5F6F4F1*(F'F4F2F1FHF5F1F'F5F5%$_C1
GF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "At this point Green and Z
erna introduce the assumption $W = C_\{1\}(I_\{1\}-3)+C_\{2\}(I_\{2\}-
3)$ which gives" }}{PARA 0 "" 0 "" {TEXT -1 38 "$\\Phi = 2*C_\{1\}$ an
d $\\Psi = 2*C_\{2\}$:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "grmap(str
ain(dn,dn),I1,I2,I3,B(up,up),stress(up,up),T(up),subs,Phi(r)=2*c[1],ps
i(r)=2*c[2],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 226 "Applying routine su
bs to strain(dn,dn)\nApplying routine subs to I1\nApplying routine sub
s to I2\nApplying routine subs to I3\nApplying routine subs to B(up,up
)\nApplying routine subs to stress(up,up)\nApplying routine subs to T(
up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*&&%(strain~G6#%\"rG\"\"\"&%!GF'F),$*(,2*&%'lambdaG\"\"#F(F1!\"\"*&F0
F1%\"KGF)F2*(F0\"\"$&%$phiGF'F1F(\"\"%F)**F0F6F7F1F(F1F4F)F1*(F0F6F7F1
F4F1F)*&F(F1F0F6F)*&&%\"wGF'F1F(F1F)*&F>F1F4F)F)F),&*$F(F1F)F4F)F2F0!
\"##F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"
\"&%!G6#%&thetaGF),$*(%'lambdaGF),&*$F(\"\"#F)%\"KGF)F)&%$phiGF'F)#F)F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!G6#
%\"zGF),$*&&%\"wGF'F)%'lambdaG!\"##F)\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%&thetaG\"\"\"&%!GF'F),(*&%\"rG\"\"#%'l
ambdaGF)#!\"\"F/*&F0F)%\"KGF)F1*$F.F/#F)F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"\"&%!GF'F),$*(,&%'lambdaGF)!\"
\"F)F),&F/F)F)F)F)F/!\"##F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8
Elasticity~Invariant~I1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~G**,
8*&&%$phiG6#%\"rG\"\"#F+\"\"'\"\"\"*(F(F,F+\"\"%%\"KGF.F,*(F(F,F+F,F1F
,F.*$F+F0F,*&F+F,F1F.F,*$F1F,F.*&F+F0%'lambdaG\"\"$F.*(F+F,F7F8F1F.F.*
&&%\"wGF*F,F+F0F.*(F;F,F+F,F1F.F,*&F;F,F1F,F.F.F7!\"\",&*$F+F,F.F1F.F?
F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I2G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2~G**,6*(%'lambdaG\"\"$&%$phiG6#
%\"rG\"\"#F-\"\"'\"\"\"**F(F)F*F.F-\"\"%%\"KGF0F.**F(F)F*F.F-F.F3F.F0*
&F-F2F(F)F.*(F-F.F(F)F3F0F.*&F(F)F3F.F0*&&%\"wGF,F.F-F2F0*(F9F.F-F.F3F
0F0*&F-F.F3F0F0*$F-F2F0F0F-!\"#,&*$F-F.F0F3F0!\"\"F(F>" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I3~G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Elas
ticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"BG6$%#upGF&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!GF'F(*(,(*&F'
\"\"#%'lambdaG\"\"$F(*&F/F0%\"KGF(F(*$F'F.F(F(F/!\"#F'F4" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"\"\")%!G%&thetaGF(**&%$phiG6#F'F
(%'lambdaGF(,&*$F'\"\"#F(%\"KGF(F(F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&)%#B~G%\"rG\"\"\")%!G%\"zGF(*&&%\"wG6#F'F(%'lambdaG!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"\")%!GF'F(**,2*(%
'lambdaG\"\"$&%$phiG6#%\"rG\"\"#F3\"\"%F(**F.F/F0F4F3F4%\"KGF(F4*(F.F/
F0F4F7F4F(*&&%\"wGF2F4F3F4F(*&F:F4F7F(F(*$F3F4F(F7F(*&F3F4F.F/F(F(F3!
\"#,&F=F(F7F(!\"\"F.F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"
zG\"\"\")%!GF'F(**,2*(%'lambdaG\"\"$&%$phiG6#%\"rG\"\"#F3\"\"'F(**F.F/
F0F4F3\"\"%%\"KGF(F4**F.F/F0F4F3F4F8F4F(*&F3F7F.F/F4*(F3F4F.F/F8F(F4*&
F.F/F8F4F(*&&%\"wGF2F4F3F7F(*(F>F4F3F4F8F(F(F(F3!\"#,&*$F3F4F(F8F(!\"
\"F.FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stress~tensorG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#upGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!GF'F(*(,.*(&%\"cG6#F(F(%'lambda
GF(F'\"\"#F2*(F.F(F1F(%\"KGF(F2*(&F/6#F2F(F1\"\"$F'F2F2*(F6F(F1F8F4F(F
2*&F6F(F'F2F2*(-%\"pG6#F'F(F1F2F'F2F(F(F1!\"#F'F?" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%&thetaGF(*,&%$phiG6#F'F(,&*$F
'\"\"#F(%\"KGF(F(,&&%\"cG6#F(F2*&&F66#F2F(%'lambdaGF2F2F(F'!\"#F;!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!G%\"zGF(**
&%\"wG6#F'F(,(*(&%\"cG6#F(F(%'lambdaGF(F'\"\"#F6*(F2F(F5F(%\"KGF(F6*&&
F36#F6F(F'F6F6F(F5!\"#F'F<" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&)%$ta
uG%&thetaG\"\"\")%!GF'F(**,>**&%\"cG6#F(F(&%$phiG6#%\"rG\"\"#%'lambdaG
F(F4\"\"%F5*,F.F(F1F5F6F(F4F5%\"KGF(F7**F.F(F1F5F6F(F9F5F5*(F.F(F6F(F4
F5F5**&F/6#F5F(F6\"\"$F1F5F4F7F5*,F=F(F6F?F1F5F4F5F9F(F7**F=F(F6F?F1F5
F9F5F5*(F=F(&%\"wGF3F5F4F5F5*(F=F(FCF5F9F(F5*&F=F(F4F5F5*&F=F(F9F(F5*(
F=F(F6F?F4F5F5*(-%\"pGF3F(F6F5F4F5F(*(FJF(F6F5F9F(F(F(F4!\"#F6FM,&*$F4
F5F(F9F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&thetaG\"
\"\")%!G%\"zGF(,$*.&%\"cG6#F(F(&%$phiG6#%\"rGF(&%\"wGF3F(%'lambdaG!\"
\",&*$F4\"\"#F(%\"KGF(F(F4!\"#F;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*
&)%$tauG%\"zG\"\"\")%!GF'F(**,@*(&%\"cG6#F(F(%'lambdaG\"\"%%\"rGF2\"\"
#**F.F(F1F2F3F4%\"KGF(F4**F.F(&%\"wG6#F3F4F1F(F3F2F4*,F.F(F8F4F1F(F3F4
F6F(F2**F.F(F8F4F1F(F6F4F4**&F/6#F4F(F1\"\"$&%$phiGF:F4F3\"\"'F4*,F>F(
F1F@FAF4F3F2F6F(F2*,F>F(F1F@FAF4F3F4F6F4F4*(F>F(F1F@F3F2F2**F>F(F1F@F3
F4F6F(F2*(F>F(F1F@F6F4F4*(F>F(F8F4F3F2F4**F>F(F8F4F3F4F6F(F4*(-%\"pGF:
F(F1F4F3F2F(**FLF(F1F4F3F4F6F(F(F(F3!\"#F1FO,&*$F3F4F(F6F(!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&T(up)G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%\"TG6#%#upG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"rG,$*
*,:**&%\"cG6#\"\"#\"\"\"&%\"wG6#F&F.F&F.%\"KGF/F.**F+F/%'lambdaG\"\"$&
%$phiGF2F.F&\"\"'F.*,F+F/F5F6F7F.F&\"\"%F3F/F;*,F+F/F5F6F7F.F&F.F3F.F.
*(F+F/F5F6F3F.F.*(F+F/F0F.F&F;F.*(&%\"pGF2F/F&\"\"&F5F.!\"\"*(&F,6#F/F
/F5F/F3F.F.**F@F/F&F6F5F.F3F/FC**FEF/F7F.F5F/F&F9F.*,FEF/F7F.F5F/F&F;F
3F/F;*,FEF/F7F.F5F/F&F.F3F.F.F/F&!\"$F5!\"#,&*$F&F.F/F3F/FCFC" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"TG%&thetaG*(%'lambdaG!\"\",2*(&%$
phiG6$%\"rGF/\"\"\"&%\"cG6#F0F0F/\"\"$\"\"#**F,F0F1F0F/F0%\"KGF0F5*(&F
-6#F/F0F1F0F/F5\"\"'*(F9F0F1F0F7F0F5**F(F5F,F0&F26#F5F0F/F4F5*,F(F5F,F
0F>F0F/F0F7F0F5**F9F0F>F0F(F5F/F5F;**F(F5F9F0F>F0F7F0F5F0F/!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"zG*(,.**&%\"wG6$%\"rGF-\"\"
\"&%\"cG6#F.F.F-\"\"$%'lambdaGF.\"\"#*,F*F.F/F.F-F.F3F.%\"KGF.F4**&F+6
#F-F.F/F.F-F4F3F.F4**F8F.F/F.F3F.F6F.!\"#*(F*F.&F06#F4F.F-F2F4*(F8F.F=
F.F-F4F4F.F-!\"$F3F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "We can no
w solve \{\\em explicitly\} for $w(r)$, $\\phi(r)$, and $p(r)$" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(grcomponent(T(up),[z])=0,w(r
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"wG6#%\"rG,&%$_C1G\"\"\"*&%
$_C2GF*-%#lnG6#,(*(&%\"cG6#F*F*%'lambdaGF*F'\"\"#F**(F2F*F5F*%\"KGF*F*
*&&F36#F6F*F'F6F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "W1
(r):=c[3]+c[4]*ln(c[1]*lambda*r^2+c[1]*lambda*K+c[2]*r^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%#W1G6#%\"rG,&&%\"cG6#\"\"$\"\"\"*&&F*6#\"
\"%F--%#lnG6#,(*(&F*6#F-F-%'lambdaGF-F'\"\"#F-*(F7F-F9F-%\"KGF-F-*&&F*
6#F:F-F'F:F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve(
grcomponent(T(up),[theta])=0,phi(r));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%$phiG6#%\"rG,&%$_C1G\"\"\"*&%$_C2GF*,&-%#lnGF&!\"#-F/6#,&*$F'\"
\"#F*%\"KGF*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "W2(r)
:=c[5]+c[6]*(2*ln(r)-ln(r^2+K));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-
%#W2G6#%\"rG,&&%\"cG6#\"\"&\"\"\"*&&F*6#\"\"'F-,&-%#lnGF&\"\"#-F46#,&*
$F'F5F-%\"KGF-!\"\"F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "
grmap(T(up),subs,w(r)=W1(r),phi(r)=W2(r),`x`);" }}{PARA 6 "" 1 "" 
{TEXT -1 30 "Applying routine subs to T(up)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "expand(grcomponent(T(up),[r]));" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,Z*,%\"rG\"\"&%'lambdaG!\"\",&*$F%\"\"#\"\"\"%\"KGF,!
\"$&%\"cG6#F,F,&F06#\"\"'F+!\")*.F%\"\"$F'F(F)F.F/F,F2F+F-F,!#;*.F%F7F
'!\"#F)F(&F06#F+F7&F06#\"\"%F+,(*(F/F,F'F,F%F+F,*(F/F,F'F,F-F,F,*&F;F,
F%F+F,F:F5*.F%F(F'F,F)F(F;F,F2F+F-F,F8*(F%F+F)F(&%\"pG6#F%F,F,*(F)F(FF
F,F-F,F,*,F%F.F'F(F)F(F/F,F-F+F:*,F%F.F'F,F)F(F;F,F-F+F:*0F%F,F)F(F;F,
F=F+F@F:F-F,F/F+F5*2F%F,F'F(F)F(F;F+F=F+F@F:F-F,F/F,F8*0F%F,F'F:F)F(F;
F7F=F+F@F:F-F,F5*,F%F,F'F,F)F(F;F,F2F+F5*,F%F7F'F,F)F:F;F,F2F+\"#;*,F%
F&F'F,F)F.F;F,F2F+F5*.F%F,F'F,F)F:F;F,F2F+F-F,\"#K*.F%F7F'F,F)F.F;F,F2
F+F-F,F8*.F%F.F'F,F)F(F;F,F2F+F-F+F5*.F%F(F'F,F)F:F;F,F2F+F-F+FQ*.F%F,
F'F,F)F.F;F,F2F+F-F+F5*.F%F7F)F(F;F,F=F+F@F:F/F+F5*0F%F7F'F(F)F(F;F+F=
F+F@F:F/F,F8*,F%F,F'F(F)F(F/F,F2F+F5*,F%F7F'F(F)F:F/F,F2F+FQ*.F%F(F'F(
F)F(F/F,F2F+F-F,F8*.F%F,F'F(F)F:F/F,F2F+F-F,FT*.F%F.F'F(F)F(F/F,F2F+F-
F+F5*.F%F(F'F(F)F:F/F,F2F+F-F+FQ*.F%F,F'F(F)F.F/F,F2F+F-F+F5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(grcomponent(T(up),[r]
)=0,p(r));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"rG,$**,^p*.%
'lambdaG\"\"%-%#lnG6#,&*$F'\"\"#\"\"\"%\"KGF4F4&%\"cG6#F3F4&F76#\"\"'F
3F'F-&F76#F4F4!\"%*.F,F--F/F&F4F6F4F'F3F<F4F5F4F3*.F,F-F@F4F6F4F9F3F'F
-F<F4\"\")*,F@F4F<F3F,F3F'F3F5F4F3*,F@F4F<F4F,F4F'F-F6F4F3**F,\"\"$F@F
4F6F3F'F-F3*0F,F-F.F4F6F4F9F3F'F3F<F4F5F4F>*,F,FFF.F4F6F3F9F3F'F-F>*.F
,F-F.F4F6F4F'F3F<F4F5F4!\"\"*,F.F4F<F3F,F3F'F3F5F4FJ*,F.F4F<F4F,F4F'F-
F6F4FJ*,F,F-F.F4F6F4F'F-F<F4FJ**F,FFF.F4F6F3F'F-FJ**F.F4F<F3F,F3F'F-FJ
*.F.F4F<F3F9F3F,F3F'F3F5F4F>*,F.F4F<F3F9F3F,F3F'F-F>*0F,F-F@F4F6F4F9F3
F'F3F<F4F5F4FB*,F,FFF@F4F6F3F9F3F'F-FB*.F@F4F<F3F9F3F,F3F'F3F5F4FB*.F@
F4F<F4F9F3F,F4F'F-F6F4FB*,F@F4F<F3F9F3F,F3F'F-FB**F@F4F<F3F,F3F'F-F3**
F<F3F,F3F'F3F5F4F4*(F<F3F,F3F5F3F4*,F6F4&F76#F-F3F<F4F,F4F'F3F-*(F6F3F
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