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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 261 "" 0 "" {TEXT 273 44 "MAPS from amd to spherical coordinates
 in 3D" }}{PARA 0 "" 0 "" {TEXT -1 18 "The 2 sphere in E3" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "SPHER3D.mws --  up
dated 02/02/2000" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "R. M. Kiehn." }}{PARA 0 "" 0 "" {TEXT -1 51 "Notes at htt
p://www.uh.edu/~rkiehn/pdf/defects2.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "with(linalg):with(plo
ts):with(liesymm):with(difforms): setup(u,v,r):\ndefform(Z=0,u=0,v=0,r
=0,theta=0,phi=0,p=const,q=const,a=const,Y=0,X=0,E=0,ss=const);\n" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for close" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for `&^`" }}
{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for d" }}{PARA 7 
"" 1 "" {TEXT -1 34 "Warning, new definition for mixpar" }}{PARA 7 "" 
1 "" {TEXT -1 35 "Warning, new definition for wdegree" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Specify the components of the P
osition vector" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "X:=r*sin(
theta)*cos(phi);Y:=r*sin(theta)*sin(phi);Z:=r*cos(theta);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"XG*(%\"rG\"\"\"-%$sinG6#%&thetaGF'-%$cosG6#
%$phiGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG*(%\"rG\"\"\"-%$sinG
6#%&thetaGF'-F)6#%$phiGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG*&%
\"rG\"\"\"-%$cosG6#%&thetaGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "
Define a position vector in R3 as three functions:" }}{PARA 0 "" 0 "" 
{TEXT -1 73 "It is a map from an initial state [r,theta,phi] to a fina
l state [X,Y,Z]." }}{PARA 0 "" 0 "" {TEXT -1 69 "The Jacobian matrix o
f the map defines a Frame Field at all points, R" }}{PARA 0 "" 0 "" 
{TEXT -1 80 "(This is the classic fiber bundle 0f n+n^2 coordinates ma
pped to n coordinates.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dim:=3;coord:=[r,theta,phi];R:=[X,
Y,Z];`dR`=d(R);FFJ:=jacobian(R,coord);FFJINV:=simplify(inverse(FFJ));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dimG\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&coordG7%%\"rG%&thetaG%$phiG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG7%*(%\"rG\"\"\"-%$sinG6#%&thetaGF(-%$cosG6#%$phiG
F(*(F'\"\"\"F)F2-F*F/F(*&F'F2-F.F+F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#/%#dRG7%,(*(-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF,-%\"dG6#%\"rGF,F
,**F4F,F-\"\"\"-F.F*F,-F2F*F,F,**F4F6F(F6-F)F/F,-F2F/F,!\"\",(*(F(F6F:
F6F1F6F,**F4F6F:F6F7F6F8F6F,**F4F6F(F6F-F6F;F6F,,&*&F7F6F1F6F,*(F4F6F(
F6F8F6F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FFJG-%'matrixG6#7%7%*&-
%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF/*(%\"rGF/F0\"\"\"-F1F-F/,$*(F5F
6F+F6-F,F2F/!\"\"7%*&F+F6F:F6*(F5F6F:F6F7F6*(F5F6F+F6F0F67%F7,$*&F5F6F
+F6F;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'FFJINVG-%'matrixG6#7%
7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF/*&F+\"\"\"-F,F2F/-F1F-7%*&
*&F0F5F7F/F5%\"rG!\"\"*&*&F6F5F7F5F5F;F<,$*&F+F5F;F<!\"\"7%,$*&F6F5*&F
;\"\"\"F+\"\"\"F<FA*&F0F5*&F;\"\"\"F+\"\"\"F<\"\"!" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 82 "For later use compute the pull back metric on \{r
,theta,phi\}, and the scale factors" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 186 "GMN:=simplify(innerprod(transpose(FFJ),FFJ));Scalefa
ctor:=array([[1,0,0],[0,1/r,0],[0,0,1/(r*sin(u))]]):OTH:=simplify(inne
rprod(FFJ,Scalefactor)):simplify(innerprod(transpose(OTH),OTH)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$GMNG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)%\"rG\"\"#\"\"\"F+7%F+F+,&F
-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}}{EXCHG {PARA 12 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "The Repere Mobil
e or Jacobian FRAME MATRIX,  FFJ. note that the frame matrix is not ne
cessarily  orthonormal.!!  In the example it is orthogonal, but not no
rmalized.   Compute the determinant of the JAcobian matrix for later u
se." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "DETFFJ:=simplify(fa
ctor(det(FFJ)));E:=1/(DETFFJ)^(1/3);Gmn:=simplify(innerprod(transpose(
FFJ),FFJ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DETFFJG*&-%$sinG6#%&
thetaG\"\"\")%\"rG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"E
G*&\"\"\"F&*$)*&-%$sinG6#%&thetaG\"\"\")%\"rG\"\"#F&#\"\"\"\"\"$F&!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GmnG-%'matrixG6#7%7%\"\"\"\"
\"!F+7%F+*$)%\"rG\"\"#\"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1
!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 464 "When the metric is diag
onal it is easy to renormalize each vector of the Frame to make the Fr
ame Orthonormal.  The procedure is to post multiply the Jacobian by th
e reciprocal scale factors (the reciprocals of the square roots of the
 diagonal metric elements).  The Frame matrix could be normalized (for
cing the determinant to 1) by multiplying the Frame by an arbitrary fu
nction E(r,theta,phi).  This procedure is not the same as the orthonor
malization procedure." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 30 "Frame M
atrix = Jacobian matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "
FF:=evalm(FFJ);DETFRAME:=simplify(det(FF));FFINVD:=simplify(evalm(FF^(
-1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#FFG-%'matrixG6#7%7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG6
#%$phiGF/*(%\"rGF/F0\"\"\"-F1F-F/,$*(F5F6F+F6-F,F2F/!\"\"7%*&F+F6F:F6*
(F5F6F:F6F7F6*(F5F6F+F6F0F67%F7,$*&F5F6F+F6F;\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)DETFRAMEG*&-%$sinG6#%&thetaG\"\"\")%\"rG\"\"#\"\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'FFINVDG-%'matrixG6#7%7%*&-%$s
inG6#%&thetaG\"\"\"-%$cosG6#%$phiGF/*&F+\"\"\"-F,F2F/-F1F-7%*&*&F0F5F7
F/F5%\"rG!\"\"*&*&F6F5F7F5F5F;F<,$*&F+F5F;F<!\"\"7%,$*&F6F5*&F;\"\"\"F
+\"\"\"F<FA*&F0F5*&F;\"\"\"F+\"\"\"F<\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 68 "GG:=evalm(FFINVD);inducedmet:=simplify(innerprod(tr
anspose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG-%'matrixG6
#7%7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF/*&F+\"\"\"-F,F2F/-F1F-7
%*&*&F0F5F7F/F5%\"rG!\"\"*&*&F6F5F7F5F5F;F<,$*&F+F5F;F<!\"\"7%,$*&F6F5
*&F;\"\"\"F+\"\"\"F<FA*&F0F5*&F;\"\"\"F+\"\"\"F<\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%+inducedmetG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)
%\"rG\"\"#\"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Compute the vector of induced 1-fo
rms [F]|dr,dtheta,dphi>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "
sigma:=(innerprod(FF,[d(r),d(theta),d(phi)])):sigma1:=sigma[1];sigma2:
=sigma[2];sigma3:=sigma[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigm
a1G,(*(-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF+-%\"dG6#%\"rGF+F+**F3F+
F,\"\"\"-F-F)F+-F1F)F+F+**F3F5F'F5-F(F.F+-F1F.F+!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'sigma2G,(*(-%$sinG6#%&thetaG\"\"\"-F(6#%$phiGF+
-%\"dG6#%\"rGF+F+**F2F+F,\"\"\"-%$cosGF)F+-F0F)F+F+**F2F4F'F4-F6F-F+-F
0F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G,&*&-%$cosG6#%&th
etaG\"\"\"-%\"dG6#%\"rGF+F+*(F/F+-%$sinGF)F+-F-F)F+!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 22 "Compute the vector of " }{TEXT 286 30 "Cl
osure Affine torsion 2 forms" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "dsi
gma1:=wcollect(d(sigma1));dsigma2:=wcollect(d(sigma2));dsigma3:=wcolle
ct(d(sigma3));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dsigma1G\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dsigma2G\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(dsigma3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "Compute the vector of topological torsion 3-forms" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "TTsigm
a1:=wcollect(sigma1&^d(sigma1));TTsigma2:=wcollect(sigma2&^d(sigma2));
TTsigma3:=wcollect(sigma3&^d(sigma3));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%)TTsigma1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)TTsigma2
G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)TTsigma3G\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 82 "Now compute the Cartan matrix of connection 1-forms from \+
C=[F(inverse)] times d[F]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
129 "dFF:=simplify(array([[d(FF[1,1]),d(FF[1,2]),d(FF[1,3])],[d(FF[2,1
]),d(FF[2,2]),d(FF[2,3])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3])]])):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "rightcartanconnection:=simplify(evalm(FFINVD&*dFF));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6rightcartanconnectionG-%'matrix
G6#7%7%\"\"!,$*&%\"rG\"\"\"-%\"dG6#%&thetaGF.!\"\",&*&F-\"\"\"-F06#%$p
hiGF.F3*(F-F6F7F6)-%$cosGF1\"\"#F6F.7%*&F/F6F-!\"\"*&-F06#F-F6F-FA,$*(
F<F.-%$sinGF1F.F7F6F37%*&F7F6F-FA*&*&F<F6F7F6F6FGFA*&,&*&FGF6FCF.F.*(F
-F6F<F6F/F6F.F6*&F-\"\"\"FG\"\"\"FA" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "leftcartanconnection:=(evalm(-dFF&^FFINVD));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%5leftcartanconnectionG-%'matrixG6#7%
7%*&,.*,)-%$sinG6#%&thetaG\"\"$\"\"\"-%$cosG6#%$phiG\"\"\"%\"rGF8-F/F6
F8-%\"dGF6F8F8**)F4\"\"#F3)-F5F0F?F3F.F8-F<6#F9F8!\"\"*.F4F3F@F3F.F3F9
F3F:F3F;F3F8*(F.F3)F:F?F3FBF3FD**F9F3FGF3FAF8-F<F0F8FD*,F:F3F9F3F.F3F4
F3F;F3FDF3*&F9\"\"\"F.\"\"\"!\"\"*&,.**F-F3FGF3F9F3F;F3F8*,F:F3F@F3F.F
3F4F3FBF3FD*,FGF3F@F3F.F3F9F3F;F3F8**F4F3F.F3F:F3FBF3F8*,F4F3F9F3F:F3F
AF3FIF3F8**F9F3F.F3F>F3F;F3F8F3*&F9\"\"\"F.\"\"\"FN,$*&*&F4F3,(*(F9F3F
@F3FIF3F8*(FAF3F.F3FBF3FD*(F9F3)F.F?F3FIF3F8F8F3F9FNFD7%,$*&,.**F-F3F>
F3F9F3F;F3F8FRF8*,F>F3F@F3F.F3F9F3F;F3F8FTFDFUFD**F9F3F.F3FGF3F;F3F8F3
*&F9\"\"\"F.\"\"\"FNFD,$*&,.F,F8**FGF3F@F3F.F3FBF3F8FEF8*(F.F3F>F3FBF3
F8**F9F3F>F3FAF3FIF3F8FJFDF3*&F9\"\"\"F.\"\"\"FNFD,$*&*&F:F3FgnF3F3F9F
NFD7%*&*&F4F3,(FjnF8FinF8FhnF8F8F3F9FN*&*&F:F3FepF3F3F9FN,$*&*&F[oF3FB
F3F3F9FNFD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "ZZ1:=simplify(innerprod(FFINVD,leftcartanconnection,
FF));TESTSIMILARITY:=simplify(evalm(ZZ1+rightcartanconnection));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ZZ1G-%'matrixG6#7%7%\"\"!*&%\"rG\"
\"\"-%\"dG6#%&thetaGF-,&*&F,\"\"\"-F/6#%$phiGF-F-*(F,F4F5F4)-%$cosGF0
\"\"#F4!\"\"7%,$*&F.F4F,!\"\"F=,$*&-F/6#F,F4F,FAF=*(F:F--%$sinGF0F-F5F
47%,$*&F5F4F,FAF=,$*&*&F:F4F5F4F4FGFAF=,$*&,&*&FGF4FDF-F-*(F,F4F:F4F.F
4F-F4*&F,\"\"\"FG\"\"\"FAF=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/TEST
SIMILARITYG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "The above comput
ation demonstrates the similarity transform between the left and the r
ight cartan connection matrices.  The matrix TESTSIMILARITY should be \+
the null matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(wcolle
ct(factor(leftcartanconnection[1,1])));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,(*&*&,(*,-%$cosG6#%$phiG\"\"\")-F)6#%&thetaG\"\"#\"\"\"-%$sinGF
/F,%\"rGF,-F4F*F,F,**)F3\"\"$F2F(F2F5F2F6F2F,**F6F2F5F2F3F2F(F2!\"\"F,
-%\"dGF*F,F2*&F5\"\"\"F3\"\"\"!\"\"F,*&*&,&*&F3F2)F6F1F2F;*(F3F2F-F2)F
(F1F2F;F,-F=6#F5F,F2*&F5\"\"\"F3\"\"\"FAF,*&*(FFF2F.F,-F=F/F,F2F3FAF;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The above 11 component of th
e Left Cartan connection demonstrates the complexity of the  matrix fo
r the Jacobian mapping." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "NOW us
e tensor methods" }}{PARA 0 "" 0 "" {TEXT -1 53 "First compute the dif
ferentials of the inverse matrix" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "for i from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do d1GG[i,j,k] := (diff(GG[i,j],coord[k])) od od od: " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Compute the elements of the matr
ix product of - d[G][F] " }}{PARA 0 "" 0 "" {TEXT 259 32 "which is the
 right Cartan matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "fo
r b from 1 to dim do for a from 1 to dim do for k from 1 to dim do  s:
=0;for m from 1 to dim do s := s+(d1GG[a,m,k]*FF[m,b]); CC[a,b,k]:=sim
plify(-s) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "for b from 1 to dim do for \+
a from 1 to dim do for k from 1 to dim do if CC[a,b,k]=0 then else pri
nt(`Cabk`(a,b,k)=CC[a,b,k]) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 
257 44 "THE non zero CARTAN CONNECTION coefficients." }}{PARA 0 "" 0 "
" {TEXT 258 52 "                            C(a,b,c) index (1,-1,-1)" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#\"\"\"F'*&\"\"\"F*%
\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$\"\"\"F'*
&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"
\"\"\"#F(,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"
\"#F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
%CabkG6%\"\"$\"\"#F'*&-%$cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"\"\"$F(,&%\"rG!\"\"*&F*F')-
%$cosG6#%&thetaG\"\"#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%C
abkG6%\"\"#\"\"$F(,$*&-%$sinG6#%&thetaG\"\"\"-%$cosGF-F/!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"\"*&\"\"\"F*%\"r
G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"#*&-%$
cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 45 "These results agree with matrix method above." }}{PARA 0 "" 0 "
" {TEXT -1 72 "Now compute the Anti symmetric [bk] components of the C
artan connection:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for j
 from 1 to dim do for i from 1 to dim do for k from 1 to dim do  s := \+
(CC[i,j,k]-CC[i,k,j])/2; TTCCS[i,j,k]:=s od od od  ;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 to dim do for j from 1 to d
im do for k from 1 to dim do if TTCCS[i,j,k]=0 then else print(`Cartan
affineTorsion`(i,k,j)=TTCCS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 60 "If no entries appear in t
he area, the affine torsion is ZERO" }}{PARA 0 "" 0 "" {TEXT -1 68 "Ne
xt construct the induced metric on the initial state (r,theta,phi)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 60 "Christoffel Connection coefficients from \+
the induced metric " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "metr
ic:=simplify(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'metricG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)%\"rG\"\"
#\"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "metricinverse:=inverse(metric):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "for i from 1 to dim do for
 j from 1 to dim do for k from 1 to dim do d1gun[i,j,k] := (diff(metri
c[i,j],coord[k])) od od od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 241 "for i from 1 to dim do for j from i to dim do for k from 1 to d
im do C1S[i,j,k] := 0  od od od; for i from 1 to dim do for j from 1 t
o dim do for k from 1 to dim do C1S[i,j,k] := 1/2*d1gun[i,k,j]+1/2*d1g
un[j,k,i]-1/2*d1gun[i,j,k]  od od od; " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 184 " for k from 1 to dim do for i from 1 to dim do for j
 from 1 to dim do s := 0; for m to dim do s := s+metricinverse[k,m]*C1
S[i,j,m] od; C2S[k,i,j] := simplify(factor(s),trig) od od od; " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "for i from 1 to dim do for \+
j from 1 to dim do for k from 1 to dim do if C2S[i,j,k]=0 then  else p
rint(`Gamma2`(i,j,k)=C2S[i,j,k]) fi od od od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"#F(,$%\"rG!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"$F(,&%\"rG!\"\"*&F*F')-%$co
sG6#%&thetaG\"\"#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma
2G6%\"\"#\"\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%'Gamma2G6%\"\"#F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"#\"\"$F(,$*&-%$sinG6#%&thetaG\"\"\"-%$
cosGF-F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"
\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamm
a2G6%\"\"$\"\"#F',$*&*&-%$sinG6#%&thetaG\"\"\"-%$cosGF.F0\"\"\",&!\"\"
F0*$)F1F(F3F0!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%
\"\"$F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%'Gamma2G6%\"\"$F'\"\"#,$*&*&-%$sinG6#%&thetaG\"\"\"-%$cosGF.F0\"\"\"
,&!\"\"F0*$)F1F(F3F0!\"\"F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "F
or the Jacobian mapping, the Christoffel coefficientsGamma(a,b,c) are \+
the same as the right Cartan connection coefficients.  There is no aff
ine torsion." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The Left Cartan m
atrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 127 "for a from 1 to dim do for j from 1 to d
im do for k from 1 to dim do d1GG[a,j,k] := simplify(diff(GG[a,j],coor
d[k])) od od od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Compute the \+
elements of the matrix product of [F]d[G]" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 35 "LEFT CARTA
N CONNECTION coefficients" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 26 "Delta(ijk) index (1,-1,-1)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 170 "for i from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do  s:=0;for m to dim do s := s+FF[i,m]*(d1GG[m,j,k]); D
D[i,j,k]:=simplify(factor(s),trig) od od od od ;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 139 "for i from 1 to dim do for j from 1 to dim d
o for k from 1 to dim do if DD[i,j,k]=0 then else print(`Delta`(i,j,k)
=DD[i,j,k]) fi od od od ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Delta
G6%\"\"\"F'F',$*&,(*&)-%$cosG6#%$phiG\"\"#\"\"\")-F.6#%&thetaGF1F2F'F'
F'*$F,F2!\"\"F2%\"rG!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Del
taG6%\"\"\"F'\"\"#,$*&*(-%$sinG6#%&thetaGF'-%$cosGF.F',&!\"\"F'*$)-F16
#%$phiGF(\"\"\"F'F'F9,&F3F'*$)F0F(F9F'!\"\"F3" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F',$*&*(-%$cosG6#%$phiGF'-%$sinG
F.F',&!\"\"F'*$)-F-6#%&thetaGF(\"\"\"F'F'F9%\"rG!\"\"F3" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F(,$*&**-%$sinG6#%&thetaG
F'-F-6#%$phiGF'-%$cosGF1F'-F4F.F'\"\"\",&!\"\"F'*$)F5F(F6F'!\"\"F8" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#\"\"$F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"$F'*&*(-%$cosG6#
%$phiGF'-F,6#%&thetaGF'-%$sinGF0F'\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"$\"\"#,$-%$cosG6#%$phiG!\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"\"F(,$*&*(-%$cos
G6#%$phiGF(-%$sinGF.F(,&!\"\"F(*$)-F-6#%&thetaGF'\"\"\"F(F(F9%\"rG!\"
\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"\"F',$*&*
*-%$sinG6#%&thetaGF(-F-6#%$phiGF(-%$cosGF1F(-F4F.F(\"\"\",&!\"\"F(*$)F
5F'F6F(!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"
\"\"\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#F'
\"\"\"*&,(*$)-%$cosG6#%&thetaGF'\"\"\"!\"\"*&)-F.6#%$phiGF'F1F,F1F(*$F
4F1F2F1%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#
F'F'*&*(-%$sinG6#%&thetaG\"\"\")-%$cosG6#%$phiGF'\"\"\"-F1F,F.F4,&!\"
\"F.*$)F5F'F4F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"
\"#\"\"$\"\"\"*&*(-%$sinG6#%$phiGF)-%$cosG6#%&thetaGF)-F-F2F)\"\"\"%\"
rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"$F',$-
%$sinG6#%$phiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"
\"$\"\"\"F(*&*(-%$cosG6#%$phiGF(-F,6#%&thetaGF(-%$sinGF0F(\"\"\"%\"rG!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"\"\"\"#-%
$cosG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"
#\"\"\"*&*(-%$sinG6#%$phiGF)-%$cosG6#%&thetaGF)-F-F2F)\"\"\"%\"rG!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"#F(-%$sinG6#%
$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"\"*&,&
!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F3%\"rG!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 41 "The anti-symmetric part of the Connection" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "for j from 1 to dim do for \+
i from 1 to dim do for k from 1 to dim do  s := (DD[i,j,k]-DD[i,k,j])/
2; TTS[i,j,k]:=simplify(s) od od od  ;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "for i from 1 t
o dim do for j from 1 to dim do for k from 1 to dim do if TTS[i,j,k]=0
 then else print(`LeftTorsion`(i,k,j)=TTS[i,k,j]) fi od od od ;" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"
\"\"\"#F',$*&,,*(-%$sinG6#%&thetaGF'-%$cosGF/F'%\"rGF'!\"\"**F-\"\"\"F
1F6F3F6)-F26#%$phiGF(F6F'*&F8F'-F.F9F'F4*(F<F6)F1F(F6F8F6F(*(F8F6F<F6)
F1\"\"%F6F4F6*&,&F4F'*$F>F6F'\"\"\"F3\"\"\"!\"\"#F'F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"\"\"$F',$*&*(-%$cosG6#%$ph
iGF'-F-6#%&thetaGF'-%$sinGF1F'\"\"\"%\"rG!\"\"#F'\"\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"F'\"\"#,$*&,,*(-%$sinG6#%&
thetaGF'-%$cosGF/F'%\"rGF'!\"\"**F-\"\"\"F1F6F3F6)-F26#%$phiGF(F6F'*&F
8F'-F.F9F'F4*(F<F6)F1F(F6F8F6F(*(F8F6F<F6)F1\"\"%F6F4F6*&,&F4F'*$F>F6F
'\"\"\"F3\"\"\"!\"\"#F4F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftT
orsionG6%\"\"\"\"\"$\"\"#,&-%$cosG6#%$phiG#!\"\"F)F/F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"F'\"\"$,$*&*(-%$cosG6#%$ph
iGF'-F-6#%&thetaGF'-%$sinGF1F'\"\"\"%\"rG!\"\"#!\"\"\"\"#" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"\"\"#\"\"$,&#F'F(F'-%$c
osG6#%$phiGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"
\"#F'\"\"\",$*&,.*,-%$sinG6#%&thetaGF(-%$cosG6#%$phiGF(-F.F3F(-F2F/F(%
\"rGF(F(*$)F6F'\"\"\"F(*$)F6\"\"%F:!\"\"*&)F1F'F:F9F:!\"#*&F@F:F<F:F(*
$F@F:F(F:*&,&F>F(F8F(\"\"\"F7\"\"\"!\"\"#F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#\"\"$\"\"\",$*&,&%\"rGF)*(-%$sin
G6#%$phiGF)-%$cosG6#%&thetaGF)-F0F5F)F)\"\"\"F-!\"\"#F)F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#\"\"\"F',$*&,.*,-%$sinG6
#%&thetaGF(-%$cosG6#%$phiGF(-F.F3F(-F2F/F(%\"rGF(F(*$)F6F'\"\"\"F(*$)F
6\"\"%F:!\"\"*&)F1F'F:F9F:!\"#*&F@F:F<F:F(*$F@F:F(F:*&,&F>F(F8F(\"\"\"
F7\"\"\"!\"\"#F>F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG
6%\"\"#\"\"$F',$-%$sinG6#%$phiG#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%,LeftTorsionG6%\"\"#\"\"\"\"\"$,$*&,&%\"rGF(*(-%$sinG6#%$phiG
F(-%$cosG6#%&thetaGF(-F0F5F(F(\"\"\"F-!\"\"#!\"\"F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#F'\"\"$,$-%$sinG6#%$phiG#\"\"
\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$\"\"#\"
\"\",$*&,&*&-%$cosG6#%$phiGF)%\"rGF)!\"\"*(-%$sinGF0F)-F/6#%&thetaGF)-
F6F8F)F)\"\"\"F2!\"\"#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Left
TorsionG6%\"\"$F'\"\"\",$*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F
4%\"rG!\"\"#F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%
\"\"$\"\"\"\"\"#,$*&,&*&-%$cosG6#%$phiGF(%\"rGF(!\"\"*(-%$sinGF0F(-F/6
#%&thetaGF(-F6F8F(F(\"\"\"F2!\"\"#F3F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%,LeftTorsionG6%\"\"$\"\"\"F',$*&,&!\"\"F(*$)-%$cosG6#%&thetaG
\"\"#\"\"\"F(F4%\"rG!\"\"#F,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 
-1 93 "***************************************************************
******************************" }}{PARA 271 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 54 "Frame matrix = Jacobian matrix times a factor (det =1)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 153 "with(linalg):with(plots):with(liesymm):with(dif
forms): setup(u,v,r):\ndefform(Z=0,u=0,v=0,r=0,theta=0,phi=0,p=const,q
=const,a=const,Y=0,X=0,E=0,ss=const):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 "X:=r*sin(theta)*cos(phi):Y:=r*sin(theta)*sin(phi):Z:=r*cos(theta
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dim:=3:coord:=[r,theta,phi]:
R:=[X,Y,Z]:`dR`=d(R):FFJ:=jacobian(R,coord):FFJINV:=simplify(inverse(F
FJ)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "GMN:=simplify(innerprod(t
ranspose(FFJ),FFJ)):Scalefactor:=array([[1,0,0],[0,1/r,0],[0,0,1/(r*si
n(u))]]):OTH:=simplify(innerprod(FFJ,Scalefactor)):simplify(innerprod(
transpose(OTH),OTH)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "DETFFJ:=s
implify(factor(det(FFJ))):E:=1/(DETFFJ)^(1/3):Gmn:=simplify(innerprod(
transpose(FFJ),FFJ)):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new def
inition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definiti
on for trace" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition f
or close" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for `
&^`" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for d" }}
{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for mixpar" }}
{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition for wdegree" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 281 0 "" }}{PARA 0 "" 0 "" {TEXT 282 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "For the Jacobian mapping, th
e Christoffel coefficientsGamma(a,b,c) are the same as the right Carta
n connection coefficients.  There is no affine torsion.  " }{TEXT 283 
149 "But for the normalized Jacobian matrix the Christoffel symbols ar
e NOT = to the coefficients of the right Cartan matrix.  The Frame exh
ibits torsion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 78 "FF:=evalm(E*FFJ);DETFRAME:=simplify(det(FF));FFINVD
:=simplify(evalm(FF^(-1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%'matrixG6#7%7%*&*&-%$sinG6#%&t
hetaG\"\"\"-%$cosG6#%$phiG\"\"\"F0*$)*&F,F5)%\"rG\"\"#F0#\"\"\"\"\"$F0
!\"\"*&*(F:F5F1F0-F2F.F5F0*$)F8#\"\"\"F>F0F?,$*&*(F:F0F,F0-F-F3F5F0*$)
F8#\"\"\"F>F0F?!\"\"7%*&*&F,F0FJF0F0*$)F8#\"\"\"F>F0F?*&*(F:F0FJF0FBF0
F0*$)F8#\"\"\"F>F0F?*&*(F:F0F,F0F1F0F0*$)F8#\"\"\"F>F0F?7%*&FBF0*$)F8#
\"\"\"F>F0F?,$*&*&F:F0F,F0F0*$)F8#\"\"\"F>F0F?FO\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%)DETFRAMEG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'FFINVDG-%'matrixG6#7%7%,$*&*()%\"rG\"\"#\"\"\",&!\"\"\"\"\"*$
)-%$cosG6#%&thetaGF/F0F3F3-F76#%$phiGF3F0*$)*&-%$sinGF8F3F-F0#\"\"#\"
\"$F0!\"\"F2,$*&*(F-F0F1F0-FAF;F3F0*$)F?#\"\"#FDF0FEF2*&*(F-F0F6F3F@F0
F0*$)F?#\"\"#FDF0FE7%*&**F.F3F@F0F:F0F6F0F0*$)F?#\"\"#FDF0FE*&**F.F0F@
F0FIF0F6F0F0*$)F?#\"\"#FDF0FE*&*&F.F0F1F0F0*$)F?#\"\"#FDF0FE7%,$*&*&FI
F0F.F0F0*$)F?#\"\"#FDF0FEF2*&*&F:F0F.F0F0*$)F?#\"\"#FDF0FE\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "GG:=evalm(FFINVD);inducedmet
:=simplify(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#GGG-%'matrixG6#7%7%,$*&*()%\"rG\"\"#\"\"\",&!\"\"\"\"\"*$)-%$
cosG6#%&thetaGF/F0F3F3-F76#%$phiGF3F0*$)*&-%$sinGF8F3F-F0#\"\"#\"\"$F0
!\"\"F2,$*&*(F-F0F1F0-FAF;F3F0*$)F?#\"\"#FDF0FEF2*&*(F-F0F6F3F@F0F0*$)
F?#\"\"#FDF0FE7%*&**F.F3F@F0F:F0F6F0F0*$)F?#\"\"#FDF0FE*&**F.F0F@F0FIF
0F6F0F0*$)F?#\"\"#FDF0FE*&*&F.F0F1F0F0*$)F?#\"\"#FDF0FE7%,$*&*&FIF0F.F
0F0*$)F?#\"\"#FDF0FEF2*&*&F:F0F.F0F0*$)F?#\"\"#FDF0FE\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%+inducedmetG-%'matrixG6#7%7%*&\"\"\"F+*$)*
&-%$sinG6#%&thetaG\"\"\")%\"rG\"\"#F+#\"\"#\"\"$F+!\"\"\"\"!F;7%F;*&*$
F4F+F+*$)F.#\"\"#F9F+F:F;7%F;F;,$*&*&F4F+,&!\"\"F3*$)-%$cosGF1F6F+F3F3
F+*$)F.#\"\"#F9F+F:FH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Compute \+
the vector of induced 1-forms [F]|dr,dtheta,dphi>" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 97 "sigma:=(innerprod(FF,[d(r),d(theta),d(phi)])
):sigma1:=sigma[1];sigma2:=sigma[2];sigma3:=sigma[3];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'sigma1G,$*&,(*(-%$sinG6#%&thetaG\"\"\"-%$cosG6#
%$phiGF--%\"dG6#%\"rGF-!\"\"**F5F-F.\"\"\"-F/F+F--F3F+F-F6**F5F8F)F8-F
*F0F--F3F0F-F-F8*$)*&F)F8)F5\"\"#F8#\"\"\"\"\"$F8!\"\"F6" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'sigma2G*&,(*(-%$sinG6#%&thetaG\"\"\"-F)6#%$ph
iGF,-%\"dG6#%\"rGF,F,**F3F,F-\"\"\"-%$cosGF*F,-F1F*F,F,**F3F5F(F5-F7F.
F,-F1F.F,F,F5*$)*&F(F5)F3\"\"#F5#\"\"\"\"\"$F5!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'sigma3G,$*&,&*&-%$cosG6#%&thetaG\"\"\"-%\"dG6#%\"r
GF-!\"\"*(F1F--%$sinGF+F--F/F+F-F-\"\"\"*$)*&F4F7)F1\"\"#F7#\"\"\"\"\"
$F7!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Compute the vector \+
of " }{TEXT 285 30 "Closure Affine torsion 2 forms" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 87 "dsigma1:=wcollect(d(sigma1));dsigma2:=wcollect(d(si
gma2));dsigma3:=wcollect(d(sigma3));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(dsigma1G,(*&**)%\"rG\"\"#\"\"\")-%$sinG6#%&thetaGF*F
+-F.6#%$phiG\"\"\"-%#&^G6$-%\"dGF2-F96#F)F4F+*$)*&F-F4F(F+#\"\"%\"\"$F
+!\"\"#!\"#FA*&*,F-F+-%$cosGF2F4F(F+-FHF/F4-F66$-F9F/F:F4F+*$)F>#\"\"%
FAF+FB#F4FA*&*,)F)FAF+F-F+F1F+FIF+-F66$F8FLF4F+*$)F>#\"\"%FAF+FB#!\"\"
FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dsigma2G,(*&**)%\"rG\"\"#\"\"
\")-%$sinG6#%&thetaGF*F+-%$cosG6#%$phiG\"\"\"-%#&^G6$-%\"dGF3-F:6#F)F5
F+*$)*&F-F5F(F+#\"\"%\"\"$F+!\"\"#F*FB*&*,F-F+-F.F3F5F(F+-F2F/F5-F76$-
F:F/F;F5F+*$)F?#\"\"%FBF+FC#F5FB*&*,)F)FBF+F-F+F1F+FHF+-F76$F9FKF5F+*$
)F?#\"\"%FBF+FCFP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dsigma3G,$*&*(
)%\"rG\"\"#\"\"\",&*$)-%$cosG6#%&thetaGF*F+\"\"\"*$)-%$sinGF1F*F+F*F3-
%#&^G6$-%\"dGF1-F<6#F)F3F+*$)*&F6F3F(F+#\"\"%\"\"$F+!\"\"#!\"\"FD" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Compute the vector of topological \+
torsion 3-forms" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "TTsigma1:=wcollect(sigma1&^d(sigma1));TTsigma2:=wcol
lect(sigma2&^d(sigma2));TTsigma3:=wcollect(sigma3&^d(sigma3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)TTsigma1G\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)TTsigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
)TTsigma3G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now compute the Cartan matrix of c
onnection 1-forms from C=[F(inverse)] times d[F]" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 129 "dFF:=simplify(array([[d(FF[1,1]),d(FF[1,2]),
d(FF[1,3])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3])],[d(FF[3,1]),d(FF[3,2]),
d(FF[3,3])]])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rightcartanconnection:=simplify(eva
lm(FFINVD&*dFF),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6rightcart
anconnectionG-%'matrixG6#7%7%,$*&,(-%\"dG6#%\"rG\"\"#*&F-\"\"\")-%$cos
G6#%&thetaGF1\"\"\"!\"#**F0F3-%$sinGF7F3F5F3-F.F7F3F3F9*&F0\"\"\",&!\"
\"F3*$F4F9F3\"\"\"!\"\"#F3\"\"$,$*&F0F9F>F9FB*(FAF3F0F9-F.6#%$phiGF37%
*&F>F9F0FE,$*&,(F2F3F;F3F-FBF9*&F0\"\"\"FA\"\"\"FEFF,$*(F5F9FKF9F<F9FB
7%*&FKF9F0FE*&*&F5F9FKF9F9F<FE,$*&,(F-F3F2FBF;F1F9*&F0\"\"\"FA\"\"\"FE
#FBFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "RCL:=rightcartanco
nnection;wcollect(simplify(wcollect(RCL[1,1]),trig));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$RCLG%6rightcartanconnectionG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*&*&,&\"\"#\"\"\"*$)-%$cosG6#%&thetaGF'\"\"\"!\"#F(
-%\"dG6#%\"rGF(F/*&F4\"\"\",&!\"\"F(F)F(\"\"\"!\"\"#F(\"\"$*&*(-%$sinG
F-F(F+F(-F2F-F(F/F7F:F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "leftcartanconnection:=(
evalm(-dFF&^FFINVD));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%5leftcartan
connectionG-%'matrixG6#7%7%,$*&,8*&)-%$cosG6#%$phiG\"\"#\"\"\"-%\"dG6#
%\"rG\"\"\"F3*(F.F4F5F4)-F06#%&thetaGF3F4!\"%*,-%$sinGF1F9F8F9F/F9-F6F
1F9F;F4!\"$*,-FBF=F9F.F4F8F4F<F9-F6F=F9F9*()F<\"\"%F4F.F4F5F4F3*,F/F4F
IF4F8F4FAF4FCF4\"\"$**F.F4F;F4)FFF3F4F5F4!\"\"*.F/F4F;F4FNF4FAF4FCF4F8
F4FL*&)FAF3F4F5F4FO*(FRF4F5F4F;F4F9*,FFF4FRF4F8F4F<F4FGF4!\"#F4*&F8\"
\"\")FF\"\"#F4!\"\"#F9FL,$*&,8*(F/F4FAF4F5F4FL**F/F4FAF4F5F4F;F4!\"&*(
F8F4FRF4FCF4FL**F8F4FRF4FCF4F;F4!\"'*.F/F4FFF4FAF4F8F4F<F4FGF4FL**FAF4
FIF4F/F4F5F4F3**F8F4FCF4FIF4FRF4FL*,FAF4F;F4FNF4F/F4F5F4FO*,FRF4F;F4FN
F4FCF4F8F4FL*(F8F4F.F4FCF4FL**F8F4F.F4FCF4F;F4FDF4*&F8\"\"\")FF\"\"#F4
FZFen,$*&*&F/F4,.*(F<F4FFF4F5F4FD*()F<FLF4FFF4F5F4FL**F;F4FNF4F8F4FGF4
F3*(F8F4FGF4F;F4F[o*&F8F4FGF4FL*(FIF4F8F4FGF4F3F9F4*&F8\"\"\")FF\"\"#F
4FZ#FOFL7%,$*&,8FinFLFjnF[oFdoFDFeo\"\"'F_oFLF`oF3**F8F4FCF4FIF4F.F4FD
FboFO*,F.F4F;F4FNF4FCF4F8F4FDF\\oFDF]oFLF4*&F8\"\"\")FF\"\"#F4FZFen,$*
&,8FQFUFSFJF@FDFTFO*(FIF4FRF4F5F4FUFKFL**FRF4F;F4FNF4F5F4F9FPFLF-F9F:F
OFEF3F4*&F8\"\"\")FF\"\"#F4FZFip,$*&*&FAF4F]pF4F4*&F8\"\"\")FF\"\"#F4F
ZFip7%,$*&*&F/F4,.F^pFLFbpF[oFcpFLF_pFDFdpF3FapF3F9F4*&F8\"\"\")FF\"\"
#F4FZFen,$*&*&FAF4FirF4F4*&F8\"\"\")FF\"\"#F4FZFen,$*&,,*(F;F4FNF4F5F4
F3**F8F4FFF4F<F4FGF4F9F5FO*&F5F4F;F4F3*&FIF4F5F4FOF4*&F8\"\"\")FF\"\"#
F4FZFen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "ZZ1:=simplify(innerprod(FFINVD,leftcartanconnection,
FF));TESTSIMILARITY:=simplify(evalm(ZZ1+rightcartanconnection));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ZZ1G-%'matrixG6#7%7%,$*&,&*(-%$cosG
6#%&thetaG\"\"\"%\"rGF2-%\"dGF0F2F2*&-%$sinGF0F2-F56#F3F2\"\"#\"\"\"*&
F3\"\"\"F7\"\"\"!\"\"#F2\"\"$*&F3F<F4F<,$*(,&!\"\"F2*$)F.F;F<F2F2F3F<-
F56#%$phiGF2FG7%,$*&F4F<F3F@FG,$*&,&F-F2F6FGF<*&F3\"\"\"F7\"\"\"F@FA*(
F.F<FJF<F7F<7%,$*&FJF<F3F@FG,$*&*&F.F<FJF<F<F7F@FG,$*&,(F9F2*&F9F<FIF<
FG**F3F<F7F<F.F<F4F<F;F<*&F3\"\"\"FF\"\"\"F@FA" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%/TESTSIMILARITYG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 172 "The above computation demonstrates the similarity transform be
tween the left and the right cartan connection matrices.  The matrix T
ESTSIMILARITY should be the null matrix." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "(wcollect(factor(leftcartanconnection[1,1])));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&*&,.*$)-%$cosG6#%$phiG\"\"#\"\"\"F
-*&F(F.)-F*6#%&thetaGF-F.!\"%*&F(F.)F1\"\"%F.F-*&)-%$sinGF+F-F.F0F.\"
\"\"*(F(F.F0F.)-F;F2F-F.!\"\"*$F9F.F@F<-%\"dG6#%\"rGF<F.*&FE\"\"\")F?
\"\"#F.!\"\"#F<\"\"$*&*&,(*,F)F<F0F.F>F.F:F<FEF<FL**F)F.F6F.FEF.F:F.FL
**F:F.FEF.F)F.F0F.!\"$F<-FCF+F<F.*&FE\"\"\")F?\"\"#F.FJFK*&*&,&**F?F<F
(F.FEF.F1F<F<**F?F.F9F.FEF.F1F.!\"#F<-FCF2F<F.*&FE\"\"\")F?\"\"#F.FJFK
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "The above 1,1 component of t
he Left Cartan connection demonstrates the complexity of the  matrix f
or the Jacobian mapping." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "NOW u
se tensor methods" }}{PARA 0 "" 0 "" {TEXT -1 53 "First compute the di
fferentials of the inverse matrix" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "for i from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do d1GG[i,j,k] := (diff(GG[i,j],coord[k])) od od od: " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Compute the elements of the matr
ix product of - d[G][F] " }}{PARA 0 "" 0 "" {TEXT 259 32 "which is the
 right Cartan matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "fo
r b from 1 to dim do for a from 1 to dim do for k from 1 to dim do  s:
=0;for m from 1 to dim do s := s+(d1GG[a,m,k]*FF[m,b]); CC[a,b,k]:=sim
plify(-s) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "for b from 1 to dim do for \+
a from 1 to dim do for k from 1 to dim do if CC[a,b,k]=0 then else pri
nt(`Cabk`(a,b,k)=CC[a,b,k]) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 
257 44 "THE non zero CARTAN CONNECTION coefficients." }}{PARA 0 "" 0 "
" {TEXT 258 52 "                            C(a,b,c) index (1,-1,-1)" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"F'F',$*&\"\"\"F*%\"
rG!\"\"#!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"
F'\"\"#,$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#!\"\"\"\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#\"\"\"F'*&\"\"\"F*%\"rG!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$\"\"\"F'*&\"\"\"F*
%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"\"\"#F(
,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#F'\"\"
\",$*&\"\"\"F+%\"rG!\"\"#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%%CabkG6%\"\"#F'F',$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"#!\"\"\"\"$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$\"\"#F'*&-%$cosG6#
%&thetaG\"\"\"-%$sinGF,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Ca
bkG6%\"\"\"\"\"$F(,&%\"rG!\"\"*&F*F')-%$cosG6#%&thetaG\"\"#\"\"\"F'" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#\"\"$F(,$*&-%$cosG6#%
&thetaG\"\"\"-%$sinGF-F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%C
abkG6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"#,$*&-%$cosG6#%&thetaG\"\"\"-%$si
nGF-!\"\"#F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "These results a
gree with matrix method above." }}{PARA 0 "" 0 "" {TEXT -1 72 "Now com
pute the Anti symmetric [bk] components of the Cartan connection:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for j from 1 to dim do for \+
i from 1 to dim do for k from 1 to dim do  s := (CC[i,j,k]-CC[i,k,j])/
2; TTCCS[i,j,k]:=s od od od  ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 159 "for i from 1 to dim do for j from 1 to dim do for k from 1 to
 dim do if TTCCS[i,j,k]=0 then else print(`CartanaffineTorsion`(i,k,j)
=TTCCS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 260 60 "If no entries appear in the area, the affine torsion \+
is ZERO" }}{PARA 0 "" 0 "" {TEXT -1 68 "Next construct the induced met
ric on the initial state (r,theta,phi)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%4CartanaffineTorsionG6%\"\"\"\"\"#F',$*&-%$cosG6#%&thetaG\"\"
\"-%$sinGF-!\"\"#F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4Cartan
affineTorsionG6%\"\"\"F'\"\"#,$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"
#!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTorsion
G6%\"\"#F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#!\"\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"#\"\"\"F',$*&\"\"\"F+%\"r
G!\"\"#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTors
ionG6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#!\"\"F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"$F'\"\"#,$*&-%$cosG6#%&th
etaG\"\"\"-%$sinGF-!\"\"#!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%4CartanaffineTorsionG6%\"\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"#F(F'" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"$\"\"#F
',$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#\"\"\"\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 60 "Christoffel Connection coefficients from the in
duced metric " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "metric:=si
mplify(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%'metricG-%'matrixG6#7%7%*&\"\"\"F+*$)*&-%$sinG6#%&thetaG\"\"\")%\"
rG\"\"#F+#\"\"#\"\"$F+!\"\"\"\"!F;7%F;*&*$F4F+F+*$)F.#\"\"#F9F+F:F;7%F
;F;,$*&*&F4F+,&!\"\"F3*$)-%$cosGF1F6F+F3F3F+*$)F.#\"\"#F9F+F:FH" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "metricinverse:=inverse(metri
c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "for i from 1 to dim
 do for j from 1 to dim do for k from 1 to dim do d1gun[i,j,k] := (dif
f(metric[i,j],coord[k])) od od od: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 241 "for i from 1 to dim do for j from i to dim do for k \+
from 1 to dim do C1S[i,j,k] := 0  od od od; for i from 1 to dim do for
 j from 1 to dim do for k from 1 to dim do C1S[i,j,k] := 1/2*d1gun[i,k
,j]+1/2*d1gun[j,k,i]-1/2*d1gun[i,j,k]  od od od; " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 184 " for k from 1 to dim do for i from 1 to dim
 do for j from 1 to dim do s := 0; for m to dim do s := s+metricinvers
e[k,m]*C1S[i,j,m] od; C2S[k,i,j] := simplify(factor(s),trig) od od od;
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "for i from 1 to dim d
o for j from 1 to dim do for k from 1 to dim do if C2S[i,j,k]=0 then  \+
else print(`Gamma2`(i,j,k)=C2S[i,j,k]) fi od od od;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"F'F',$*&\"\"\"F*%\"rG!\"\"#!\"#\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"F'\"\"#,$*&-
%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#!\"\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"#F',$*&-%$cosG6#%&thetaG\"\"\"-%
$sinGF-!\"\"#!\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G
6%\"\"\"\"\"#F(,$%\"rG#!\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%'Gamma2G6%\"\"\"\"\"$F(,&%\"rG#!\"\"F(*&F*F')-%$cosG6#%&thetaG\"\"#
\"\"\"#F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"#\"\"
\"F(,$*&-%$cosG6#%&thetaG\"\"\"*&-%$sinGF-\"\"\")%\"rG\"\"#F/!\"\"#F(
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"#\"\"\"F',$*
&\"\"\"F+%\"rG!\"\"#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gam
ma2G6%\"\"#F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#F(\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"#F'F',$*&-%$cosG6#%&thetaG\"\"\"-%$sin
GF,!\"\"#!\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"
\"#\"\"$F(,$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-F/#!\"#F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"
#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"\"#F',$*&
-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#F(F'" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"#,$*&-%$cosG
6#%&thetaG\"\"\"-%$sinGF-!\"\"#F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 153 "For the Jacobian mapping, the Christoffel coefficientsGamma(a,
b,c) are the same as the right Cartan connection coefficients.  There \+
is no affine torsion." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The Left
 Cartan matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for a from 1 to dim do for \+
j from 1 to dim do for k from 1 to dim do d1GG[a,j,k] := simplify(diff
(GG[a,j],coord[k])) od od od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 
"Compute the elements of the matrix product of [F]d[G]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
256 35 "LEFT CARTAN CONNECTION coefficients" }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 26 "Delta(ijk) index (1,-1,-1)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "for i from 1 to dim do for j from \+
1 to dim do for k from 1 to dim do  s:=0;for m to dim do s := s+FF[i,m
]*(d1GG[m,j,k]); DD[i,j,k]:=simplify(factor(s),trig) od od od od ;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "for i from 1 to dim do for
 j from 1 to dim do for k from 1 to dim do if DD[i,j,k]=0 then else pr
int(`Delta`(i,j,k)=DD[i,j,k]) fi od od od ;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"\"F'F',$*&,(*&)-%$cosG6#%$phiG\"\"#\"\"
\")-F.6#%&thetaGF1F2\"\"$*$F,F2!\"$F'F'F2%\"rG!\"\"#!\"\"F7" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"F'\"\"#,$*&*&-%$cosG6#%&th
etaGF',&*$)-F-6#%$phiGF(\"\"\"\"\"$!\"#F'F'F6-%$sinGF.!\"\"#F'F7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F',$*&*(-%$cosG
6#%$phiGF',&!\"\"F'*$)-F-6#%&thetaGF(\"\"\"F'F'-%$sinGF.F'F7%\"rG!\"\"
F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F(*&*(-%$
cosG6#%$phiGF'-%$sinGF-F'-F,6#%&thetaGF'\"\"\"-F0F2!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#\"\"$F'" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"$F'*&*(-%$cosG6#%$phiGF'-F,6#%
&thetaGF'-%$sinGF0F'\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%&DeltaG6%\"\"\"\"\"$\"\"#,$-%$cosG6#%$phiG!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"\"F(,$*&*(-%$cosG6#%$phiGF(,&!
\"\"F(*$)-F-6#%&thetaGF'\"\"\"F(F(-%$sinGF.F(F7%\"rG!\"\"F1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"\"F'*&*(-%$cosG6#%$phiG
F(-%$sinGF-F(-F,6#%&thetaGF(\"\"\"-F0F2!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"\"\"\"$!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"#F'\"\"\",$*&,**$)-%$cosG6#%&thetaGF'\"
\"\"!\"$*&)-F/6#%$phiGF'F2F-F2\"\"$F'F(*$F5F2F3F2%\"rG!\"\"#F(F9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#F'F',$*&*&-%$cosG6#%&
thetaG\"\"\",&!\"\"F/*$)-F,6#%$phiGF'\"\"\"\"\"$F/F7-%$sinGF-!\"\"#F1F
8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"$\"\"\"*&*(-
%$sinG6#%$phiGF)-%$cosG6#%&thetaGF)-F-F2F)\"\"\"%\"rG!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"$F',$-%$sinG6#%$phiG!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"\"F(*&*(-%$
cosG6#%$phiGF(-F,6#%&thetaGF(-%$sinGF0F(\"\"\"%\"rG!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"\"\"\"#-%$cosG6#%$phiG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"#\"\"\"*&*(-%$sin
G6#%$phiGF)-%$cosG6#%&thetaGF)-F-F2F)\"\"\"%\"rG!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"\"#F(-%$sinG6#%$phiG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"\",$*&,&*$)-%$cosG6#%
&thetaG\"\"#\"\"\"F'!\"\"F(F3%\"rG!\"\"#F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"#,$*&-%$cosG6#%&thetaG\"\"\"-%$s
inGF-!\"\"#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The anti-s
ymmetric part of the Connection" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 135 "for j from 1 to dim do for i from 1 to dim do for k from 1 to
 dim do  s := (DD[i,j,k]-DD[i,k,j])/2; TTS[i,j,k]:=simplify(s) od od o
d  ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 147 "for i from 1 to dim do for j from 1 to dim do for \+
k from 1 to dim do if TTS[i,j,k]=0 then else print(`LeftTorsion`(i,k,j
)=TTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"\"\"#F',$*&,**(-%$cosG6#%&theta
GF'%\"rGF')-F.6#%$phiGF(\"\"\"\"\"$*&F1F6F-F6!\"#*(-%$sinGF4F'F3F'-F<F
/F'!\"$**F;F6F3F6F=F6)F-F(F6F7F6*&F=\"\"\"F1\"\"\"!\"\"#!\"\"\"\"'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"\"\"$F',$*&*(-
%$cosG6#%$phiGF'-F-6#%&thetaGF'-%$sinGF1F'\"\"\"%\"rG!\"\"#F'\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"F'\"\"#,$*&,**
(-%$cosG6#%&thetaGF'%\"rGF')-F.6#%$phiGF(\"\"\"\"\"$*&F1F6F-F6!\"#*(-%
$sinGF4F'F3F'-F<F/F'!\"$**F;F6F3F6F=F6)F-F(F6F7F6*&F=\"\"\"F1\"\"\"!\"
\"#F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"
\"\"$\"\"#,&-%$cosG6#%$phiG#!\"\"F)F/F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%,LeftTorsionG6%\"\"\"F'\"\"$,$*&*(-%$cosG6#%$phiGF'-F-6#%&the
taGF'-%$sinGF1F'\"\"\"%\"rG!\"\"#!\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"\"\"\"#\"\"$,&#F'F(F'-%$cosG6#%$p
hiGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#F'\"\"
\",$*&,,**-%$cosG6#%$phiGF(-%$sinGF/F(-F.6#%&thetaGF(%\"rGF(!\"$*&-F2F
4F()F3F'\"\"\"F7*(F9F;)F-F'F;F:F;\"\"$F9F'*&F9F;F=F;F7F;*&F9\"\"\"F6\"
\"\"!\"\"#F(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6
%\"\"#\"\"$\"\"\",$*&,&%\"rGF)*(-%$sinG6#%$phiGF)-%$cosG6#%&thetaGF)-F
0F5F)F)\"\"\"F-!\"\"#F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftT
orsionG6%\"\"#\"\"\"F',$*&,,**-%$cosG6#%$phiGF(-%$sinGF/F(-F.6#%&theta
GF(%\"rGF(!\"$*&-F2F4F()F3F'\"\"\"F7*(F9F;)F-F'F;F:F;\"\"$F9F'*&F9F;F=
F;F7F;*&F9\"\"\"F6\"\"\"!\"\"#!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%,LeftTorsionG6%\"\"#\"\"$F',$-%$sinG6#%$phiG#!\"\"F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#\"\"\"\"\"$,$*&
,&%\"rGF(*(-%$sinG6#%$phiGF(-%$cosG6#%&thetaGF(-F0F5F(F(\"\"\"F-!\"\"#
!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#F'\"
\"$,$-%$sinG6#%$phiG#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,L
eftTorsionG6%\"\"$\"\"#\"\"\",$*&,&*&%\"rGF)-%$cosG6#%$phiGF)!\"\"*(-%
$sinGF1F)-F06#%&thetaGF)-F6F8F)F)\"\"\"F.!\"\"#F)F(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$F'\"\"\",$*&,&*$)-%$cosG6#%&th
etaG\"\"#\"\"\"F'!\"\"F(F3%\"rG!\"\"#F(\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$\"\"\"\"\"#,$*&,&*&%\"rGF(-%$cos
G6#%$phiGF(!\"\"*(-%$sinGF1F(-F06#%&thetaGF(-F6F8F(F(\"\"\"F.!\"\"#F3F
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$F'\"\"#,$*
&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#\"\"\"\"\"'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$\"\"\"F',$*&,&*$)-%$cosG6#%&th
etaG\"\"#\"\"\"F'!\"\"F(F3%\"rG!\"\"#F4\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$\"\"#F',$*&-%$cosG6#%&thetaG\"\"
\"-%$sinGF-!\"\"#!\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 155 "For the Jacobian mapping, the Christoffel coefficients
Gamma(a,b,c) are the same as the right Cartan connection coefficients.
  There is no affine torsion.  " }{TEXT 256 149 "But for the normalize
d Jacobian matrix the Christoffel symbols are NOT = to the coefficient
s of the right Cartan matrix.  The Frame exhibits torsion." }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 268 "" 0 "" {TEXT 279 12 "Metri
c basis" }}{PARA 0 "" 0 "" {TEXT 298 62 "use the sq of the metric, = s
cale factor matrix,  as the frame" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "with(linalg):with(plots):with(
liesymm):with(difforms): setup(u,v,r):\ndefform(Z=0,u=0,v=0,r=0,theta=
0,phi=0,p=const,q=const,a=const,Y=0,X=0,E=0,ss=const):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 66 "X:=r*sin(theta)*cos(phi):Y:=r*sin(theta)*sin(p
hi):Z:=r*cos(theta):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dim:=3:coo
rd:=[r,theta,phi]:R:=[X,Y,Z]:`dR`=d(R):FFJ:=jacobian(R,coord):FFJINV:=
simplify(inverse(FFJ)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "GMN:=si
mplify(innerprod(transpose(FFJ),FFJ)):Scalefactor:=array([[1,0,0],[0,1
/r,0],[0,0,1/(r*sin(u))]]):OTH:=simplify(innerprod(FFJ,Scalefactor)):s
implify(innerprod(transpose(OTH),OTH)):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 100 "DETFFJ:=simplify(factor(det(FFJ))):E:=1/(DETFFJ)^(1/3):Gmn:=s
implify(innerprod(transpose(FFJ),FFJ)):" }}{PARA 7 "" 1 "" {TEXT -1 
32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "W
arning, new definition for trace" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warni
ng, new definition for close" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, \+
new definition for `&^`" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new d
efinition for d" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definitio
n for mixpar" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition f
or wdegree" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 0 "" }}{PARA 0 "" 0 "
" {TEXT 262 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(GM
N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%
F)*$)%\"rG\"\"#\"\"\"F)7%F)F),&F+F(*&F,F/)-%$cosG6#%&thetaGF.F/!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "FF:=array([[1,0,0],[0,r,0
],[0,0,r*sin(theta)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%'ma
trixG6#7%7%\"\"\"\"\"!F+7%F+%\"rGF+7%F+F+*&F-F*-%$sinG6#%&thetaGF*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "DETFRAME:=simplify(det(FF))
;FFINVD:=simplify(evalm(FF^(-1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)DETFRAMEG*&-%$sinG6#%&thetaG
\"\"\")%\"rG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'FFINVDG-
%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*&\"\"\"F.%\"rG!\"\"F+7%F+F+*&F.F.*&F/
\"\"\"-%$sinG6#%&thetaG\"\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Compute the vector of \+
induced 1-forms [F]|dr,dtheta,dphi>" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sigma:=(innerprod(FF,[d(r)
,d(theta),d(phi)])):sigma1:=sigma[1];sigma2:=sigma[2];sigma3:=sigma[3]
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G-%\"dG6#%\"rG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G*&%\"rG\"\"\"-%\"dG6#%&thetaGF'" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G*(%\"rG\"\"\"-%$sinG6#%&the
taGF'-%\"dG6#%$phiGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Compute \+
the vector of " }{TEXT 264 30 "Closure Affine torsion 2 forms" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "dsigma1:=wcollect(d(sigma1));dsigma
2:=wcollect(d(sigma2));dsigma3:=wcollect(d(sigma3));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(dsigma1G\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(dsigma2G-%#&^G6$-%\"dG6#%\"rG-F)6#%&thetaG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(dsigma3G,&*&-%$sinG6#%&thetaG\"\"\"-%#&^G
6$-%\"dG6#%\"rG-F06#%$phiGF+F+*(F2F+-%$cosGF)F+-F-6$-F0F)F3F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Compute the vector of topological \+
torsion 3-forms" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "TTsigma1:=wcollect(sigma1&^d(sigma1));TTsigma2:=wcol
lect(sigma2&^d(sigma2));TTsigma3:=wcollect(sigma3&^d(sigma3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)TTsigma1G\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)TTsigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
)TTsigma3G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now compute the Cartan matrix of c
onnection 1-forms from C=[F(inverse)] times d[F]" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 129 "dFF:=simplify(array([[d(FF[1,1]),d(FF[1,2]),
d(FF[1,3])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3])],[d(FF[3,1]),d(FF[3,2]),
d(FF[3,3])]])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rightcartanconnection:=simplify(eva
lm(FFINVD&*dFF),trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6rightcart
anconnectionG-%'matrixG6#7%7%\"\"!F*F*7%F**&-%\"dG6#%\"rG\"\"\"F0!\"\"
F*7%F*F**&,&*&-%$sinG6#%&thetaG\"\"\"F-F;F;*(F0F;-%$cosGF9F;-F.F9F;F;F
1*&F0\"\"\"F7\"\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 44 "leftcartanconnection:=(evalm(-dFF&^FFINVD));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%5leftcartanconnectionG-%'matrixG6#7%
7%\"\"!F*F*7%F*,$*&-%\"dG6#%\"rG\"\"\"F1!\"\"!\"\"F*7%F*F*,$*&,&*&-%$s
inG6#%&thetaG\"\"\"F.F>F>*(F1F>-%$cosGF<F>-F/F<F>F>F2*&F1\"\"\"F:\"\"
\"F3F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 116 "ZZ1:=simplify(innerprod(FFINVD,leftcartanconnection,FF));TESTSI
MILARITY:=simplify(evalm(ZZ1+rightcartanconnection));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$ZZ1G-%'matrixG6#7%7%\"\"!F*F*7%F*,$*&-%\"dG6#%
\"rG\"\"\"F1!\"\"!\"\"F*7%F*F*,$*&,&*&-%$sinG6#%&thetaG\"\"\"F.F>F>*(F
1F>-%$cosGF<F>-F/F<F>F>F2*&F1\"\"\"F:\"\"\"F3F4" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%/TESTSIMILARITYG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 172 "The above computation demonstrates the similarity transform be
tween the left and the right cartan connection matrices.  The matrix T
ESTSIMILARITY should be the null matrix." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 121 "The above 11 component of the Left Cartan connection dem
onstrates the complexity of the  matrix for the Jacobian mapping." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "NOW use tensor methods" }}{PARA 0 
"" 0 "" {TEXT -1 53 "First compute the differentials of the inverse ma
trix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GG:=inverse(FF);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#GGG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*&\"\"\"F.%\"rG!\"\"F+7%F+F+*&
F.F.*&F/\"\"\"-%$sinG6#%&thetaG\"\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "for i from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do d1GG[i,j,k] := (diff(GG[i,j],coord[k])) od od od: " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Compute the elements of the matr
ix product of - d[G][F] " }}{PARA 0 "" 0 "" {TEXT 259 32 "which is the
 right Cartan matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "fo
r b from 1 to dim do for a from 1 to dim do for k from 1 to dim do  s:
=0;for m from 1 to dim do s := s+(d1GG[a,m,k]*FF[m,b]); CC[a,b,k]:=sim
plify(-s) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "for b from 1 to dim do for \+
a from 1 to dim do for k from 1 to dim do if CC[a,b,k]=0 then else pri
nt(`Cabk`(a,b,k)=CC[a,b,k]) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 
257 44 "THE non zero CARTAN CONNECTION coefficients." }}{PARA 0 "" 0 "
" {TEXT 258 52 "                            C(a,b,c) index (1,-1,-1)" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#F'\"\"\"*&\"\"\"F*%
\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"\"*
&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"
$F'\"\"#*&-%$cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 45 "These results agree with matrix method above." }}
{PARA 0 "" 0 "" {TEXT -1 72 "Now compute the Anti symmetric [bk] compo
nents of the Cartan connection:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 127 "for j from 1 to dim do for i from 1 to dim do for k from 1 to
 dim do  s := (CC[i,j,k]-CC[i,k,j])/2; TTCCS[i,j,k]:=s od od od  ;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 to dim do for
 j from 1 to dim do for k from 1 to dim do if TTCCS[i,j,k]=0 then else
 print(`CartanaffineTorsion`(i,k,j)=TTCCS[i,k,j]) fi od od od ;" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 60 "If no ent
ries appear in the area, the affine torsion is ZERO" }}{PARA 0 "" 0 "
" {TEXT -1 68 "Next construct the induced metric on the initial state \+
(r,theta,phi)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTorsi
onG6%\"\"#F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#F(F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"#\"\"\"F',$*&\"\"\"F+%\"r
G!\"\"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTors
ionG6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#F(\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"$F'\"\"#,$*&-%$cosG6#%&th
etaG\"\"\"-%$sinGF-!\"\"#\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%4CartanaffineTorsionG6%\"\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"#!\"\"\"\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%4CartanaffineTorsionG6%\"\"$\"
\"#F',$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#!\"\"F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 60 "Christoffel Connection coefficients from the in
duced metric " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "metric:=si
mplify(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%'metricG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)%\"rG\"\"#\"\"\"F+7%F+
F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "metricinverse:=inverse(metric):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 124 "for i from 1 to dim do for j from 1 to dim
 do for k from 1 to dim do d1gun[i,j,k] := (diff(metric[i,j],coord[k])
) od od od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "for i from
 1 to dim do for j from i to dim do for k from 1 to dim do C1S[i,j,k] \+
:= 0  od od od; for i from 1 to dim do for j from 1 to dim do for k fr
om 1 to dim do C1S[i,j,k] := 1/2*d1gun[i,k,j]+1/2*d1gun[j,k,i]-1/2*d1g
un[i,j,k]  od od od; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 " \+
for k from 1 to dim do for i from 1 to dim do for j from 1 to dim do s
 := 0; for m to dim do s := s+metricinverse[k,m]*C1S[i,j,m] od; C2S[k,
i,j] := simplify(factor(s),trig) od od od; " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 142 "for i from 1 to dim do for j from 1 to dim do for
 k from 1 to dim do if C2S[i,j,k]=0 then  else print(`Gamma2`(i,j,k)=C
2S[i,j,k]) fi od od od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G
6%\"\"\"\"\"#F(,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gam
ma2G6%\"\"\"\"\"$F(,&%\"rG!\"\"*&F*F')-%$cosG6#%&thetaG\"\"#\"\"\"F'" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"#\"\"\"F'*&\"\"\"F*
%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"#F'\"\"
\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6
%\"\"#\"\"$F(,$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-F/!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"\"\"F'*&\"\"\"F*%\"rG!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"\"#F',$*&*&-%$co
sG6#%&thetaG\"\"\"-%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F3F0!\"\"F5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"\"*&\"\"\"F*%
\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"#
,$*&*&-%$cosG6#%&thetaG\"\"\"-%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F3F0!\"
\"F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "For the Jacobian mapping
, the Christoffel coefficientsGamma(a,b,c) are the same as the right C
artan connection coefficients.  There is no affine torsion." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The Left Cartan matrix:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 127 "for a from 1 to dim do for j from 1 to dim do for \+
k from 1 to dim do d1GG[a,j,k] := simplify(diff(GG[a,j],coord[k])) od \+
od od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Compute the elements o
f the matrix product of [F]d[G]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 35 "LEFT CARTAN CONNECTIO
N coefficients" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 26 "Delta
(ijk) index (1,-1,-1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "f
or i from 1 to dim do for j from 1 to dim do for k from 1 to dim do  s
:=0;for m to dim do s := s+FF[i,m]*(d1GG[m,j,k]); DD[i,j,k]:=simplify(
factor(s),trig) od od od od ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 139 "for i from 1 to dim do for j from 1 to dim do for k from 1 to d
im do if DD[i,j,k]=0 then else print(`Delta`(i,j,k)=DD[i,j,k]) fi od o
d od ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#F'\"\"\",$*
&\"\"\"F+%\"rG!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG
6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"#*&*&-%$cosG6#%&thetaG\"\"\"-%$s
inGF-F/\"\"\",&!\"\"F/*$)F+F(F2F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "The anti-symmetric part of the Connection" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "for j from 1 to dim do for i from \+
1 to dim do for k from 1 to dim do  s := (DD[i,j,k]-DD[i,k,j])/2; TTS[
i,j,k]:=simplify(s) od od od  ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "for i from 1 to dim do \+
for j from 1 to dim do for k from 1 to dim do if TTS[i,j,k]=0 then els
e print(`LeftTorsion`(i,k,j)=TTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"#F'\"\"\",$*
&\"\"\"F+%\"rG!\"\"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Lef
tTorsionG6%\"\"#\"\"\"F',$*&\"\"\"F+%\"rG!\"\"#F(F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"
#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$
F'\"\"#,$*&*&-%$cosG6#%&thetaG\"\"\"-%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F
3F0!\"\"#F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,LeftTorsionG6%\"
\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"#F(\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%,LeftTorsionG6%\"\"$\"\"#F',$*&*&-%$cosG6#%&thetaG\"
\"\"-%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F3F0!\"\"#F5F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 121 "For the diagonal scale factor Frame matrix, th
e Left and the Right Cartan connections are the same to within a minus
 sign" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "134 0 1" 30 
}{VIEWOPTS 1 1 0 1 1 1803 }