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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg
):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }
}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 46 "Cartan Connection coefficients  f
rom the Frame" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "dim:=4;coo
rd := [r, theta, phi,t];LGUN:=array([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,
0,0,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dimG\"\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&coordG7&%\"rG%&thetaG%$phiG%\"tG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LGUNG-%'matrixG6#7&7&\"\"\"\"\"!F+F
+7&F+F*F+F+7&F+F+F*F+7&F+F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "Specify the Frame Matrtix" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "FF1:= array([[1,0,0,0],[0,r,0,0],[0,0,r*sin(theta),0]
,[0,0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FF1G-%'matrixG6#7&
7&\"\"\"\"\"!F+F+7&F+%\"rGF+F+7&F+F+*&F-F*-%$sinG6#%&thetaGF*F+7&F+F+F
+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "GG1:=evalm(simplify(
inverse(FF1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GG1G-%'matrixG6#
7&7&\"\"\"\"\"!F+F+7&F+*&\"\"\"F.%\"rG!\"\"F+F+7&F+F+*&F.F.*&F/\"\"\"-
%$sinG6#%&thetaG\"\"\"F0F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "GUN:=simplify(innerprod(transpose(FF1),LGUN,FF1));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GUNG-%'matrixG6#7&7&\"\"\"\"\"!F+F
+7&F+*$)%\"rG\"\"#\"\"\"F+F+7&F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!
\"\"F+7&F+F+F+F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "FF:=inn
erprod(FF1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG%$FF1G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "GG:=simplify(inverse(FF));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#GGG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+*&\"\"\"F.%\"r
G!\"\"F+F+7&F+F+*&F.F.*&F/\"\"\"-%$sinG6#%&thetaG\"\"\"F0F+7&F+F+F+F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "First \+
compute the differentials of the inverse matrix" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 119 "for i from 1 to dim do for j from 1 to dim do
 for k from 1 to dim do d1GG[i,j,k] := (diff(GG[i,j],coord[k])) od od \+
od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Compute the elements of t
he matrix product of - d[G][F]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 165 "for b from 1 to dim do for a from 1 to dim do for k from 1 to
 dim do  s:=0;for m from 1 to dim do s := s+(d1GG[a,m,k]*FF[m,b]); CC[
a,b,k]:=simplify(-s) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "for b from 1 to dim
 do for a from 1 to dim do for k from 1 to dim do if CC[a,b,k]=0 then \+
else print(`CCabk`(a,b,k)=CC[a,b,k]) fi od od od ;" }}{PARA 0 "" 0 "" 
{TEXT 257 44 "THE non zero CARTAN CONNECTION coefficients." }}{PARA 0 
"" 0 "" {TEXT 258 51 "                            CC(abk) index (1,-1,
-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#F'\"\"\"*&\"\"
\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$F'
\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCab
kG6%\"\"$F'\"\"#*&-%$cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "These results agree with matrix method." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Now compute the Shipov connecti
on." }}{PARA 0 "" 0 "" {TEXT -1 43 "NOTE SIGN CHANGE TO FIT RESULTS IN
 BOOK !!!" }}{PARA 0 "" 0 "" {TEXT -1 22 "Delta(i,j,k)= - F d[G]" }}
{PARA 0 "" 0 "" {TEXT -1 46 "This makes the formula Delta = Gamma + T \+
work." }}{PARA 0 "" 0 "" {TEXT -1 23 "DO YOU AGREE WITH THIS?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "for a from 1 to dim do for \+
j from 1 to dim do for k from 1 to dim do d1GG[a,j,k] := (-diff(GG[a,j
],coord[k])) od od od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Comput
e the elements of the matrix product of [F]d[G]" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 157 "for i from 1 to dim do for j from 1 to dim do
 for k from 1 to dim do  s:=0;for m to dim do s := s+FF[i,m]*(d1GG[m,j
,k]); DD[i,j,k]:=simplify(s) od od od od ;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "for i from 1 to dim do for j from \+
1 to dim do for k from 1 to dim do if DD[i,j,k]=0 then else print(`Del
ta`(i,j,k)=DD[i,j,k]) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 256 39 "N
ON-ZERO SHIPOV CONNECTION coefficients" }}{PARA 256 "" 0 "" {TEXT -1 
0 "" }{TEXT 256 26 "Delta(ijk) index (1,-1,-1)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"#F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"\"*&\"\"\"F*%\"rG!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$F'\"\"#*&-%$cos
G6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
41 "The anti-symmetric part of the Connection" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 126 "for j from 1 to dim do for i from 1 to dim do f
or k from 1 to dim do  s := (DD[i,j,k]-DD[i,k,j])/2; TTS[i,j,k]:=-s od
 od od  ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 149 "for i from 1 to dim do for j from 1 to dim do
 for k from 1 to dim do if TTS[i,j,k]=0 then else print(`ShipovTorsion
`(i,k,j)=TTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"#F'\"\"\",$*&\"\"\"F+%\"rG!\"\"
#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"#
\"\"\"F',$*&\"\"\"F+%\"rG!\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%.ShipovTorsionG6%\"\"$F'\"\"\",$*&\"\"\"F+%\"rG!\"\"#!\"\"\"\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$F'\"\"#,$*&-
%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#!\"\"F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"
#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$
\"\"#F',$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-!\"\"#\"\"\"F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 52 "Christoffel Connec
tion coefficients from the metric " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "metric:
= simplify((innerprod(transpose(FF),-LGUN,FF)));" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'met
ricG-%'matrixG6#7&7&!\"\"\"\"!F+F+7&F+,$*$)%\"rG\"\"#\"\"\"F*F+F+7&F+F
+,&F.F**&F/F2)-%$cosG6#%&thetaGF1F2\"\"\"F+7&F+F+F+F;" }}}{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 
0 145 "#for i from 1 to dim do for j from 1 to dim do for k from 1 to \+
dim do if d1gun[i,j,k]=0 then  else print(`dgun`(i,j,k)=d1gun[i,j,k]) \+
fi od od od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "for i from 1 to di
m 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; " }}{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; " }}{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 C
2S[i,j,k]=0 then  else print(`Gamma2`(i,j,k)=C2S[i,j,k]) fi od od od;
" }}{PARA 0 "" 0 "" {TEXT 256 61 "The non zero Christoffel Connection \+
coefficients 2nd kind    " }}{PARA 0 "" 0 "" {TEXT 257 50 "           \+
          Gamma2(i,j,k) index (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%'Gamma2G6%\"\"\"\"\"#F(,$%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"$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%\"\"$\"
\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamm
a2G6%\"\"$\"\"#F',$*&*&-%$cosG6#%&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 10 "Th
e T(ijk)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 144 "for i from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do s:=0; s := (DD[i,j,k]-C2S[i,j,k]); SHIPTR[i,j,k]:=sim
plify(s)  od od od   ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 257 "> " 0 "" 
{MPLTEXT 1 0 166 "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 and DD[i,j,k]=0 then else print(`T`(i
,j,k)=simplify(SHIPTR[i,j,k])) fi od od od ;" }{TEXT -1 0 "" }{TEXT 
256 22 "T(ijk) index (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"TG6%\"\"\"\"\"#F(%\"rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%
\"\"\"\"\"$F(,&%\"rGF'*&F*F')-%$cosG6#%&thetaG\"\"#\"\"\"!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"\"F',$*&\"\"\"F+%\"r
G!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F(*&-%$cos
G6#%&thetaG\"\"\"-%$sinGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"T
G6%\"\"$\"\"\"F',$*&\"\"\"F+%\"rG!\"\"!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#F'*&*&-%$cosG6#%&thetaG\"\"\"-%$sinG
F-F/\"\"\",&!\"\"F/*$)F+F(F2F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%\"TG6%\"\"$F'\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"T
G6%\"\"$F'\"\"#\"\"!" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 32 "Delta(ijk) = Gamma(ijk) + T(ijk)" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "but note sign change on delta." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{MARK "38 1 0" 30 }{VIEWOPTS 1 1 0 1 1 1803 }