{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "Helvetica" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Helvetica" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Helvetica" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Century Schoolbook" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Lucida Sans Typewriter " 1 10 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with (linal g):with(liesymm):with(difforms):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "setup(x,y,z,t):defform(x=0,y=0,z=0,t=0,Vx=0,Vy=0,Vz=0,D1=0,D2=0,D 3=0,Ax=0,Ay=0,Az=0,C=0,Phi=0,phi=0,theta=0,r=0,a=const,b=const,c=const ,Lx=0,Ly=0,Lz=0);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definit ion for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition f or trace" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for c lose" }}{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 258 "" 0 "" {TEXT 256 42 "Spheres in 3D space and Cartan \+ Connections" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 13 "R. M. Kiehn " }}{PARA 261 "" 0 "" {TEXT -1 30 " last upd ate February 25, 2000" }}{PARA 260 "" 0 "" {TEXT -1 18 "rkiehn2352@aol .com" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 13 "Introduction:" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 129 "The Cartan connection coefficients w ill be computed for both the map of cartesian coordinates into spheric al coordinates, (Part1)" }}{PARA 0 "" 0 "" {TEXT -1 73 "and the map fr om spherical coordinates into cartesian coordinates (Part2)" }}{PARA 263 "" 0 "" {TEXT 261 73 "Part 1 The map is from \{x,y,z\} into spheri cal coordinates \{r,theta,phi\}. " }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "r:=(x^2+y^2+z^2)^(1/2);cos(theta):=z/(r);c os(phi):=x/(x^2+y^2)^(1/2);sin(theta):=(1-cos(theta)^2)^(1/2);sin(phi) :=(1-cos(phi)^2)^(1/2);rho:=(x^2+y^2)^(1/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*$-%%sqrtG6# ,(*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF-F.F/*$)%\"zGF-F.F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%$cosG6#%&thetaG*&%\"zG\"\"\"*$-%%sqrtG6#,(*$ )%\"xG\"\"#F*\"\"\"*$)%\"yGF3F*F4*$)F)F3F*F4F*!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%$cosG6#%$phiG*&%\"xG\"\"\"*$-%%sqrtG6#,&*$)F)\"\"# F*\"\"\"*$)%\"yGF2F*F3F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%$s inG6#%&thetaG*$-%%sqrtG6#,&\"\"\"F-*&*$)%\"zG\"\"#\"\"\"F3,(*$)%\"xGF2 F3F-*$)%\"yGF2F3F-*$F0F3F-!\"\"!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%$sinG6#%$phiG*$-%%sqrtG6#,&\"\"\"F-*&*$)%\"xG\"\"#\"\"\"F3,&* $F0F3F-*$)%\"yGF2F3F-!\"\"!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$rhoG*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF-F.F/F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Compute the induced \"dribeins\" ( they are not exact differentials, but they are closed)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d(r);DZ:=solve(d(r)=0,d(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*&%\"xG\"\"\"-%\"dG6#F&F'\"\"\"*$- %%sqrtG6#,(*$)F&\"\"#F+F'*$)%\"yGF3F+F'*$)%\"zGF3F+F'F+!\"\"F'*&*&F6F' -F)6#F6F'F+*$-F.6#F0F+F:F'*&*&F9F'-F)6#F9F'F+*$-F.6#F0F+F:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DZG,$*&,&*&%\"xG\"\"\"-%\"dG6#F)F*F**&%\" yGF*-F,6#F/F*F*\"\"\"%\"zG!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "`dphi`:=factor(d(cos(phi))/sin(phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dphiG*&*&%\"yG\"\"\",&*&F'\"\"\"-%\"dG6#%\"xGF+ F+*&F/F+-F-6#F'F+!\"\"F+F(*&),&*$)F/\"\"#F(F+*$)F'F9F(F+#\"\"$F9F(-%%s qrtG6#*&*$F;F(F(F6!\"\"F(FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`dtheta`:=factor(subs(d(z)=DZ,(d(cos(theta))/sin(theta))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dthetaG,$*&,&*&%\"xG\"\"\"-%\"dG6#F )F*F**&%\"yGF*-F,6#F/F*F*\"\"\"*(-%%sqrtG6#,(*$)F)\"\"#F2F**$)F/F:F2F* *$)%\"zGF:F2F*F2F?\"\"\"-F56#*&,&F8F*F;F*F2F7!\"\"F2FE!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "The Frame matrix of functions on \+ x,y,z that causes the differential structures dr,dtheta,dphi to be c reated linearly in terms of dx,dy,dz is the Frame matrix [FF] below:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "FF:=(array([[x/r,y/r,z/r ],[-z*x/(rho*r^2),-z*y/(rho*r^2),(x^2+y^2)/(rho*r^2)],[-y/rho^2,x/rho^ 2,0]]));GG:=evalm(inverse(FF)):DETFF:=factor(det(FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%'matrixG6#7%7%*&%\"xG\"\"\"*$-%%sqrtG6#,(* $)F+\"\"#F,\"\"\"*$)%\"yGF4F,F5*$)%\"zGF4F,F5F,!\"\"*&F8F,*$-F/6#F1F,F <*&F;F,*$-F/6#F1F,F<7%,$*&*&F;F5F+F5F,*&-F/6#,&F2F5F6F5F,F1\"\"\"F%&DETFFG,$*& \"\"\"F'*&-%%sqrtG6#,&*$)%\"xG\"\"#F'\"\"\"*$)%\"yGF0F'F1F'-F*6#,(F-F1 F2F1*$)%\"zGF0F'F1F'!\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Note that the Frame matrix has a singularity along the z axis and at \+ the origin (excluded) points." }}{PARA 0 "" 0 "" {TEXT -1 112 "The nex t equation checks to see that the specified frame produces the desired differential structures: [FF]|dR>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "zz:=evalm(innerprod(FF,[d(x),d(y),d(z)]));zzb:=innerp rod(GG,zz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zzG-%'vectorG6#7%*&, (*&%\"xG\"\"\"-%\"dG6#F,F-F-*&%\"yGF--F/6#F2F-F-*&%\"zGF--F/6#F6F-F-\" \"\"*$-%%sqrtG6#,(*$)F,\"\"#F9F-*$)F2FAF9F-*$)F6FAF9F-F9!\"\",$*&,**(F 6F9F,F9F.F9F-*(F6F9F2F9F3F9F-*&F7F9F@F9!\"\"*&F7F9FCF9FMF9*&-F<6#,&F?F -FBF-F9F>\"\"\"FFFM,$*&,&*&F2F9F.F9F-*&F,F9F3F9FMF9FRFFFM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$zzbG-%'vectorG6#7%-%\"dG6#%\"xG-F*6#%\"yG-F* 6#%\"zG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 285 "Note that each compon ent (except the first) is not exact, but is closed. Each component in terms of xyz has an infinity at the origin and along the Z axis. The induced spherical coordinate \"differentials\" are not exact , but ar e closed and obey the Frobenius integrability theorem. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "d(zz[1]);d(zz[2]);d(zz[3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 231 "The metric on the domain xyz is the unit matrix o f constants, by assumption. As such the Christoffel symbols will be z ero. The pushed forward metric on the spherical coordinate range is ( but with arguments on the domain x,y,z) is" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 108 "pushedmetric:=simplify(innerprod(transpose(GG),GG) );inducedmetric:=innerprod(transpose(FF),pushedmetric,FF):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-pushedmetricG-%'matrixG6#7%7%\"\"\"\"\"!F +7%F+,(*$)%\"xG\"\"#\"\"\"F**$)%\"yGF1F2F**$)%\"zGF1F2F*F+7%F+F+,&F.F* F3F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Now check to see if the \+ Frame matrix is normal and find the Left and Right representations P2 R=[FF][ transpose FF]" }}{PARA 0 "" 0 "" {TEXT -1 171 "P2L= [transpose FF] [ F]. Conclusion: the Frame matrix is not normal!!! Note that \+ P2R is the pushedforward inverse metric on \{r,theta,phi\} but with ar guments on \{x,y,z\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "P2R :=simplify(innerprod(FF,transpose(FF)));P2L:=simplify(innerprod(transp ose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P2RG-%'matrixG6#7% 7%\"\"\"\"\"!F+7%F+*&\"\"\"F.,(*$)%\"xG\"\"#F.F**$)%\"yGF3F.F**$)%\"zG F3F.F*!\"\"F+7%F+F+*&F.F.,&F0F*F4F*F:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$P2LG-%'matrixG6#7%7%*&,>*$)%\"xG\"\")\"\"\"\"\"\"*&)F.\"\"'F0)% \"yG\"\"#F0\"\"$*&)F.\"\"%F0)F6F;F0F8*&)F.F7F0)F6F4F0F1*&F3F0)%\"zGF7F 0F1*(F:F0FAF0F5F0F7*(F>F0FAF0FF0F5F0F8*&F:F0F 5F0F1*&F>F0FF0F1*$F 5F0F1*$FAF0F1\"\"#F0),&FPF1FQF1\"\"#F0!\"\"*&*(F.F1F6F1,<*$F3F0F1FGF8F HF8FIF1FEF1FFF7FJF1*&FAF0F>F0!\"\"*&F5F0FAF0Fgn*$F:F0Fgn*&F>F0F5F0!\"# *$FF2F1F9F8F=F8*$)F6F/F0F1FCF1FDF7*&F?F0 FAF0F1FFF8FJF1FenF1FGF7FEF7FHF1*&F>F0FLF0F1F0*&)FO\"\"#F0)FU\"\"#F0FW* &*(FBF0F6F0FeoF0F0*$)FO\"\"#F0FW7%FcoFep*&,,FfnF1FhnF1F]oF1FPF1FQF1F0* $)FO\"\"#F0FW" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "P2L is complicat ed algebracially. But the bottom line is that it is NOT equal to P2R, " }}{PARA 0 "" 0 "" {TEXT -1 39 "hence the Jacobian matrix is NOT NOR MAL" }}{PARA 0 "" 0 "" {TEXT -1 107 "Now any matrix with an inverse ca n be composed as a product of a symmetric matrix and an orthogonal mat rix." }}{PARA 0 "" 0 "" {TEXT -1 375 "There are in general two ways to construct this representation, which will be denoted as the Lefthande d and the Right handed formulations. [SR] is the symmetric matrix of \+ the \"right handed\" formulation, and [OR] is the orthogonal matrix fo r the right handed formulation. (If the matrices are complex, the not ions symmetric and orthogonal translate to Hermitean and Unitary)" }} {PARA 0 "" 0 "" {TEXT -1 20 "The two formats are:" }}{PARA 0 "" 0 "" {TEXT -1 23 "[F]= [SR][OR]=[OL] [SL]" }}{PARA 0 "" 0 "" {TEXT -1 17 "I f [G][F]=1, then" }}{PARA 0 "" 0 "" {TEXT -1 32 "[SR]=\{[FF].transpose [FF]\}^(1/2) " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 22 "[OR]=[SR].transpose[G]" }}{PARA 0 "" 0 "" {TEXT -1 32 "Th e Left handed representations " }}{PARA 0 "" 0 "" {TEXT -1 32 "[SL]=\{ transpose[FF].[FF]\}^(1/2) " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 0 "" 0 "" {TEXT -1 22 "[OL]=transpose[G].[SL]" }}{PARA 0 "" 0 " " {TEXT -1 74 "The representations are distinct if the Frame [FF] is n ot a normal matrix." }}{PARA 0 "" 0 "" {TEXT -1 2 "**" }}{PARA 0 "" 0 "" {TEXT -1 82 "So the first step is to find the square roots of these matrices P2R and P2L above." }}{PARA 0 "" 0 "" {TEXT -1 65 "This is e asy to do for the Right Handed P2R, for it is diagonal." }}{PARA 0 " " 0 "" {TEXT -1 73 "The symmetric component of the right handed repres entation: F = [SR][OR]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " SR:=array([[1,0,0],[0,1/r,0],[0,0,1/rho]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SRG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*&\"\"\"F.*$-%%s qrtG6#,(*$)%\"xG\"\"#F.F**$)%\"yGF7F.F**$)%\"zGF7F.F*F.!\"\"F+7%F+F+*& F.F.*$-F16#,&F4F*F8F*F.F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Comp ute the orthogonal factor [OR] of the right handed representation." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "GS:=transpose(inverse(FF)) :OR:=innerprod(SR,GS);simplify(innerprod(transpose(OR),OR)):`Should_be _zero`:=evalm(innerprod(SR,OR)-FF):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#ORG-%'matrixG6#7%7%*&%\"xG\"\"\"*$-%%sqrtG6#,(*$)F+\"\"#F,\"\"\"*$ )%\"yGF4F,F5*$)%\"zGF4F,F5F,!\"\"*&F8F,*$-F/6#F1F,F<*&F;F,*$-F/6#F1F,F <7%,$*&*&F;F5F+F5F,*&-F/6#F1F,-F/6#,&F2F5F6F5F,F " 0 "" {MPLTEXT 1 0 11 " dOR:=d(OR):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Omega_RR:=si mplify(innerprod(transpose(OR),dOR));Delta_RL:=innerprod(d(SR),inverse (SR));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Omega_RRG-%'matrixG6#7%7% \"\"!*&,&*&%\"yG\"\"\"-%\"dG6#%\"xGF/!\"\"*&F3F/-F16#F.F/F/\"\"\",&*$) F3\"\"#F8F/*$)F.FF8F/F/F3F8F8*&F9\"\"\",(F:F/F=F/*$)FDF%)Delta_RLG-%'matrixG6#7%7%\"\"!F*F*7 %F*,$*&,(*&%\"xG\"\"\"-%\"dG6#F0F1F1*&%\"yGF1-F36#F6F1F1*&%\"zGF1-F36# F:F1F1\"\"\",(*$)F0\"\"#F=F1*$)F6FAF=F1*$)F:FAF=F1!\"\"!\"\"F*7%F*F*,$ *&,&F/F1F5F1F=,&F?F1FBF1FFFG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "O r one can find the Left Cartan matrix for [OR] and the right Cartan ma trix for [SR]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Omega_RL:= simplify(innerprod(dOR,transpose(OR)));Delta_RR:=innerprod(d(SR),inver se(SR));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Omega_RLG-%'matrixG6#7% 7%\"\"!*&,**(%\"zG\"\"\"%\"xGF/-%\"dG6#F0F/!\"\"*(F.\"\"\"%\"yGF/-F26# F7F/F4*&-F26#F.F/)F0\"\"#F6F/*&F;F6)F7F>F6F/F6*&-%%sqrtG6#,&*$F=F6F/*$ F@F6F/F6,(FFF/FGF/*$)F.F>F6F/\"\"\"!\"\"*&,&*&F7F6F1F6F4*&F0F6F8F6F/F6 *&-FC6#FEF6-FC6#FHF6FL7%,$F+F4F*,$*&*&FNF/F.F6F6*&FE\"\"\"-FC6#FHF6FLF 47%,$FMF4FYF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Delta_RRG-%'matrix G6#7%7%\"\"!F*F*7%F*,$*&,(*&%\"xG\"\"\"-%\"dG6#F0F1F1*&%\"yGF1-F36#F6F 1F1*&%\"zGF1-F36#F:F1F1\"\"\",(*$)F0\"\"#F=F1*$)F6FAF=F1*$)F:FAF=F1!\" \"!\"\"F*7%F*F*,$*&,&F/F1F5F1F=,&F?F1FBF1FFFG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Note that the matrix elements of Delta are perfect ex act differentials, and the matrix elements of Omega are closed but not exact differentials. " }}{PARA 0 "" 0 "" {TEXT -1 76 "It is possible to write the differential of the Frame field in several ways:" }} {PARA 0 "" 0 "" {TEXT -1 28 "d[FF] = [FF][CR] = [CL][FF] " }}{PARA 0 " " 0 "" {TEXT -1 73 " = \{[DeltaRL][FF] + [FF][OmegaRR]\} = \{ [OmegaL][FF]+[FF][DeltaL]\}" }}{PARA 0 "" 0 "" {TEXT -1 81 " = d\{[SR][OR]\} = d[SR][OR] + [SR]d[OR] = [SR]\{[DeltaRR] + [OmegaLR]\} [OR] " }}{PARA 0 "" 0 "" {TEXT -1 1 "-" }}{PARA 0 "" 0 "" {TEXT -1 79 "Note that appropriate linear combinations can be constructed to repre sent d[FF]" }}{PARA 0 "" 0 "" {TEXT -1 41 "such as is done in polariza tion of light." }}{PARA 0 "" 0 "" {TEXT -1 1 "*" }}{PARA 0 "" 0 "" {TEXT 262 40 "Now Compute the Right Cartan Matrix [CR]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cartan:=simplify(innerprod(inverse(FF),d(FF))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The matrix elements of the Right Cartan c onnection matrix using the matrix methods:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 93 "Gamma11:=wcollect(cartan[1,1]);Gamma12:=wcollect(ca rtan[1,2]);Gamma13:=wcollect(cartan[1,3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G,(*&*&,0*&)%\"yG\"\"%\"\"\")%\"xG\"\"$F-\"\" #*()F+F1F-F.F-)%\"zGF1F-F0*&)F+\"\"'F-F/\"\"\"F9*(F3F-F/F-)F5F,F-F9*&F 3F-)F/\"\"&F-F9*&F=F-F4F-F9*(F*F-F/F-F4F-F1F9-%\"dG6#F/F9F-*&),&*$)F/F 1F-F9*$F3F-F9\"\"#F-),(FGF9FIF9*$F4F-F9\"\"#F-!\"\"!\"\"*&*&,0*&)F+F0F -F;F-F9*$)F+\"\"(F-F9*&FUF-)F/F,F-F9*(F+F9FZF-F4F-F9*&)F+F>F-FHF-F1*(F HF-FUF-F4F-F0*&FgnF-F4F-F1F9-FB6#F+F9F-*&)FF\"\"#F-)FL\"\"#F-FOFP*&*&, &*&FZF-)F5F0F-F9*(FHF-FeoF-F3F-F9F9-FB6#F5F9F-*&)FF\"\"#F-)FL\"\"#F-FO FP" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G,(*&*&,0*(%\"yG\"\"\" )%\"xG\"\"%\"\"\")%\"zG\"\"#F/!\"\"*&)F*\"\"$F/F,F/F3*()F-F2F/F5F/F0F/ !\"$*&)F*\"\"&F/F8F/!\"#*&F5F/)F1F.F/F3*$)F*\"\"(F/F3*&F;F/F0F/F=F+-% \"dG6#F-F+F/*&),&*$F8F/F+*$)F*F2F/F+\"\"#F/),(FJF+FKF+*$F0F/F+\"\"#F/! \"\"F+*&*&,4*&)F-F%(Gamma13G,(*&*()%\"zG\"\"$\"\"\")%\"xG\"\" #F+-%\"dG6#F-\"\"\"F+*&,&*$F,F+F2*$)%\"yGF.F+F2\"\"\"),(F5F2F6F2*$)F)F .F+F2\"\"#F+!\"\"!\"\"*&**F8F2F(F+F-F2-F06#F8F2F+*&F4\"\"\")F;\"\"#F+F ?F@*&*(,,*$)F-\"\"%F+F2*&F,F+F7F+F.*&F=F+F,F+F.*&F7F+F=F+F.*$)F8FNF+F2 F2F-F+-F06#F)F2F+*&F4\"\"\")F;\"\"#F+F?F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 93 "Gamma21:=wcollect(cartan[2,1]);Gamma22:=wcollect(ca rtan[2,2]);Gamma23:=wcollect(cartan[2,3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G,(*&*&,4*&)%\"yG\"\"$\"\"\")%\"xG\"\"%F-!\"& *()F/\"\"#F-F*F-)%\"zGF4F-F1*(F+\"\"\")F6F0F-F3F-!\"#*&)F+\"\"&F-F3F-! \"%*(F+F-F.F-F5F-!\"$*&F+F-)F/\"\"'F-F:*&F*F-F9F-!\"\"*$)F+\"\"(F-FE*& F%(Gamma22G,(*&*&,0*&)%\"xG\"\"&\"\"\")%\"zG\"\"#F-F0*& )%\"yGF0F-F*F-F0*$)F+\"\"(F-\"\"\"*()F3\"\"%F-F+F7F.F-F7*&F9F-)F+\"\"$ F-F7*&F%(Gamma23G,(*&**%\"yG\"\"\")% \"zG\"\"$\"\"\"%\"xGF)-%\"dG6#F.F)F-*&,&*$)F.\"\"#F-F)*$)F(F6F-F)\"\" \"),(F4F)F7F)*$)F+F6F-F)\"\"#F-!\"\"!\"\"*&*(F8F-F*F--F06#F(F)F-*&F3\" \"\")F;\"\"#F-F?F@*&*(,,*$)F.\"\"%F-F)*&F5F-F8F-F6*&F=F-F5F-F6*&F8F-F= F-F6*$)F(FNF-F)F)F(F--F06#F+F)F-*&F3\"\"\")F;\"\"#F-F?F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Gamma31:=wcollect(cartan[3,1]);Gamm a32:=wcollect(cartan[3,2]);Gamma33:=wcollect(cartan[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G,(*&*(,(*&%\"zG\"\"\")%\"xG\"\"$ \"\"\"!\"#*(F*F/F-F+)%\"yG\"\"#F/F0*&)F*F.F/F-F/!\"\"F+F-F/-%\"dG6#F-F +F/*&,&*$)F-F4F/F+*$F2F/F+\"\"\"),(F=F+F?F+*$)F*F4F/F+\"\"#F/!\"\"F7*& *(,(*(F*F/F3F+F>F/F0*&F*F/)F3F.F/F0*&F3F/F6F/F7F+F-F/-F96#F3F+F/*&F<\" \"\")FB\"\"#F/FFF7*&*(,(*$)F-\"\"%F/F+*&F>F/F2F/F4*$)F3FYF/F+F+F-F/-F9 6#F*F+F/*&F<\"\"\")FB\"\"#F/FFF7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (Gamma32G,(*&*(,(*&%\"zG\"\"\")%\"xG\"\"$\"\"\"!\"#*(F*F/F-F+)%\"yG\" \"#F/F0*&)F*F.F/F-F/!\"\"F+F3F+-%\"dG6#F-F+F/*&,&*$)F-F4F/F+*$F2F/F+\" \"\"),(F=F+F?F+*$)F*F4F/F+\"\"#F/!\"\"F7*&*(,(*(F*F/F3F/F>F/F0*&F*F/)F 3F.F/F0*&F3F/F6F/F7F+F3F/-F96#F3F+F/*&F<\"\"\")FB\"\"#F/FFF7*&*(,(*$)F -\"\"%F/F+*&F>F/F2F/F4*$)F3FYF/F+F+F3F/-F96#F*F+F/*&F<\"\"\")FB\"\"#F/ FFF7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G,(*&*&,&*&%\"xG\"\" \")%\"yG\"\"#\"\"\"F+*$)F*\"\"$F/F+F+-%\"dG6#F*F+F/*$),(*$)F*F.F/F+*$F ,F/F+*$)%\"zGF.F/F+\"\"#F/!\"\"!\"\"*&*&,&*&F-F+F:F/F+*$)F-F2F/F+F+-F4 6#F-F+F/*$)F8\"\"#F/F@FA*&*&,&*&F>F+F,F/F+*&F:F/F>F/F+F+-F46#F>F+F/*$) F8\"\"#F/F@FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Now the componen ts of the right Cartan matrix will be computed by the tensor method, a s a check" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dim:=3;coord:= [x,y,z];GG:=inverse(FF);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dimG\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&coordG7%%\"xG%\"yG%\"zG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG-%'matrixG6#7%7%*&%\"xG\"\"\"*$- %%sqrtG6#,(*$)F+\"\"#F,\"\"\"*$)%\"yGF4F,F5*$)%\"zGF4F,F5F,!\"\",$*&*& F;F5F+F5F,*$-F/6#,&F2F5F6F5F,F " 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 the matrix produc t of - d[G][F]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "for b fr om 1 to dim do for a from 1 to dim do for k from 1 to dim do ss:=0;fo r m from 1 to dim do ss := ss+(d1GG[a,m,k]*FF[m,b]); CC[a,b,k]:=simpli fy(-ss) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "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(`CCabk`(a,b,k)=factor(CC[a,b,k])) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 257 50 "THE non zero CARTAN RIGHT CONNECTION coefficients." }} {PARA 0 "" 0 "" {TEXT 258 51 " CC(abk) inde x (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"F'F', $*&*&,0*&)%\"xG\"\"%\"\"\")%\"yG\"\"#F0F'*&)%\"zGF3F0F-F0F'*&)F.F3F0)F 2F/F0F3*(F5F0F8F0F1F0\"\"$*&F5F0F9F0F3*&F1F0)F6F/F0F'*$)F2\"\"'F0F'F'F .F'F0*&),&*$F8F0F'*$F1F0F'\"\"#F0),(FEF'FFF'*$F5F0F'\"\"#F0!\"\"!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"F'\"\"#,$*&*&%\" yGF',0*&)%\"xG\"\"%\"\"\")F,F(F2F'*&)%\"zGF(F2F/F2F'*&)F0F(F2)F,F1F2F( *(F5F2F8F2F3F2\"\"$*&F5F2F9F2F(*&F3F2)F6F1F2F'*$)F,\"\"'F2F'F'F2*&),&* $F8F2F'*$F3F2F'\"\"#F2),(FEF'FFF'*$F5F2F'\"\"#F2!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"F'\"\"$,$*&*&)%\"zGF(\"\" \")%\"xG\"\"#F.F.*&,&*$F/F.F'*$)%\"yGF1F.F'\"\"\"),(F4F'F5F'*$)F-F1F.F '\"\"#F.!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\" #\"\"\"F(*&*&%\"yGF(,4*$)%\"xG\"\"'\"\"\"F'*&)F/\"\"%F1)F+F'F1\"\"&*&) F/F'F1)F+F4F1F4*&)%\"zGF'F1F3F1\"\"$*(F;F1F8F1F5F1F6*&F8F1)FF1F(F(F/F(F1*&),&*$F;F1F(*$F5F1F(\"\"#F1),(FEF(FFF(*$F8F1F(\"\" #F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#\"\" \"\"\"$,$*&*()%\"zGF)\"\"\"%\"xGF(%\"yGF(F/*&),(*$)F0F'F/F(*$)F1F'F/F( *$)F.F'F/F(\"\"#F/,&F5F(F7F(\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$\"\"\"F(*&*()%\"xG\"\"#\"\"\"%\"zGF(,( *$F+F.F-*$)%\"yGF-F.F-*$)F/F-F.F(F(F.*&),(F1F(F2F(F5F(\"\"#F.,&F1F(F2F (\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$\"\" \"\"\"#*&**%\"zGF(%\"xGF(%\"yGF(,(*$)F-F)\"\"\"F)*$)F.F)F2F)*$)F,F)F2F (F(F2*&),(F0F(F3F(F5F(\"\"#F2,&F0F(F3F(\"\"\"!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%&CCabkG6%\"\"$\"\"\"F',$*&*&,&*$)%\"xG\"\"#\"\"\"F (*$)%\"yGF0F1F(F(F/F(F1*$),(F-F(F2F(*$)%\"zGF0F1F(\"\"#F1!\"\"!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"#F',$*&*&%\"yGF ',0*&)%\"xG\"\"%\"\"\")F,F(F2F'*&)%\"zGF(F2F/F2F'*&)F0F(F2)F,F1F2F(*(F 5F2F8F2F3F2\"\"$*&F5F2F9F2F(*&F3F2)F6F1F2F'*$)F,\"\"'F2F'F'F2*&),&*$F8 F2F'*$F3F2F'\"\"#F2),(FEF'FFF'*$F5F2F'\"\"#F2!\"\"!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"#F(*&*&%\"xGF',4*&)F+\"\"% \"\"\")%\"yGF(F0F/*&)F+F(F0)F2F/F0\"\"&*$)F2\"\"'F0F(*()%\"zGF(F0F4F0F 1F0F6*&F;F0F5F0\"\"$*&F1F0)FF1F(F(F/F(F1*&),&*$F;F1F(*$F5F1F(\"\"#F1),(FEF(FFF(*$F8F1F(\" \"#F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#F' F',$*&*&,0*$)%\"xG\"\"'\"\"\"\"\"\"*&)F.\"\"%F0)%\"yGF'F0F'*&)%\"zGF'F 0F3F0F'*(F8F0)F.F'F0F5F0\"\"$*&F;F0)F6F4F0F1*&F;F0)F9F4F0F1*&F8F0F>F0F 1F1F6F1F0*&),&*$F;F0F1*$F5F0F1\"\"#F0),(FEF1FFF1*$F8F0F1\"\"#F0!\"\"! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#F'\"\"$,$*&*& )%\"zGF(\"\"\")%\"yGF'F.F.*&),(*$)%\"xGF'F.\"\"\"*$F/F.F7*$)F-F'F.F7\" \"#F.,&F4F7F8F7\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% &CCabkG6%\"\"$\"\"#\"\"\"*&**%\"zGF)%\"xGF)%\"yGF),(*$)F-F(\"\"\"F(*$) F.F(F2F(*$)F,F(F2F)F)F2*&),(F0F)F3F)F5F)\"\"#F2,&F0F)F3F)\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$\"\"#F(*&*(%\"zG\" \"\")%\"yGF(\"\"\",(*$)%\"xGF(F/F(*$F-F/F(*$)F+F(F/F,F,F/*&),(F1F,F4F, F5F,\"\"#F/,&F1F,F4F,\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %&CCabkG6%\"\"$\"\"#F',$*&*&,&*$)%\"xGF(\"\"\"\"\"\"*$)%\"yGF(F0F1F1F4 F1F0*$),(F-F1F2F1*$)%\"zGF(F0F1\"\"#F0!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"$F',$*&*&)%\"zGF(\"\"\")%\"xG\"\" #F.F.*&,&*$F/F.F'*$)%\"yGF1F.F'\"\"\"),(F4F'F5F'*$)F-F1F.F'\"\"#F.!\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"$\"\" #,$*&*()%\"zGF(\"\"\"%\"xGF'%\"yGF'F/*&),(*$)F0F)F/F'*$)F1F)F/F'*$)F.F )F/F'\"\"#F/,&F5F'F7F'\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"$F(*&*&%\"xGF',(*$)%\"zG\"\"#\"\"\"F0*$)F+ F0F1F'*$)%\"yGF0F1F'F'F1*$),(F2F'F4F'F-F'\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#\"\"$\"\"\",$*&*()%\"zGF(\"\"\"% \"xGF)%\"yGF)F/*&),(*$)F0F'F/F)*$)F1F'F/F)*$)F.F'F/F)\"\"#F/,&F5F)F7F) \"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"# \"\"$F',$*&*&)%\"zGF(\"\"\")%\"yGF'F.F.*&),(*$)%\"xGF'F.\"\"\"*$F/F.F7 *$)F-F'F.F7\"\"#F.,&F4F7F8F7\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#\"\"$F(*&*&%\"yG\"\"\",(*$)%\"zGF'\"\" \"F'*$)%\"xGF'F1F,*$)F+F'F1F,F,F1*$),(F2F,F5F,F.F,\"\"#F1!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$F'\"\"\",$*&*&,&*$)% \"xG\"\"#\"\"\"F(*$)%\"yGF0F1F(F(F/F(F1*$),(F-F(F2F(*$)%\"zGF0F1F(\"\" #F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$F'\" \"#,$*&*&,&*$)%\"xGF(\"\"\"\"\"\"*$)%\"yGF(F0F1F1F4F1F0*$),(F-F1F2F1*$ )%\"zGF(F0F1\"\"#F0!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&C CabkG6%\"\"$F'F',$*&*&,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF/F0F1F1%\"zG F1F0*$),(F,F1F2F1*$)F5F/F0F1\"\"#F0!\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "These results agree with matrix method." }}{PARA 0 "" 0 "" {TEXT -1 55 "Next check for Affine Torsion using the tensor metho ds:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for j from 1 to dim do for i from 1 to dim do for k from 1 to dim do ss := (CC[i,j,k]-CC [i,k,j])/2; CCTTS[i,j,k]:=ss od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 t o dim do for j from 1 to dim do for k from 1 to dim do if CCTTS[i,j,k] =0 then else print(`RIGHT_AffineTorsion`(i,k,j)=CCTTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 54 "IF NO ENTRIES APPEAR ABOVE, THE AFFINE TORSION IS ZERO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 38 "Now compute the CARTAN LEFT CONNECTION " }}}{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 0 161 "for i from 1 to dim do for \+ j from 1 to dim do for k from 1 to dim do ss:=0;for m to dim do ss := ss+FF[i,m]*(d1GG[m,j,k]); DD[i,j,k]:=simplify(ss) od 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 DD[i,j,k]=0 then else pri nt(`Cartan_LEFT`(i,j,k)=DD[i,j,k]) fi od od od ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"#F'*&*&%\"zGF'%\"xGF'\"\"\" *&-%%sqrtG6#,(*$)F,F(F-F'*$)%\"yGF(F-F'*$)F+F(F-F'F--F06#,&F3F'F5F'F-! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"#F (*&*&%\"zGF'%\"yGF'\"\"\"*&-%%sqrtG6#,(*$)%\"xGF(F-F'*$)F,F(F-F'*$)F+F (F-F'F--F06#,&F3F'F6F'F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,C artan_LEFTG6%\"\"\"\"\"#\"\"$,$*&*$-%%sqrtG6#,&*$)%\"xGF(\"\"\"F'*$)% \"yGF(F4F'F4F4*$-F.6#,(F1F'F5F'*$)%\"zGF(F4F'F4!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"$F'*&%\"yG\"\"\"*$- %%sqrtG6#,(*$)%\"xG\"\"#F+F'*$)F*F4F+F'*$)%\"zGF4F+F'F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"$\"\"#,$*&%\"xG \"\"\"*$-%%sqrtG6#,(*$)F,F)F-F'*$)%\"yGF)F-F'*$)%\"zGF)F-F'F-!\"\"!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"\"F(, $*&*&%\"zGF(%\"xGF(\"\"\"*&),(*$)F-F'F.F(*$)%\"yGF'F.F(*$)F,F'F.F(#\" \"$F'F.-%%sqrtG6#,&F2F(F4F(F.!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"\"F',$*&*&%\"zGF(%\"yGF(\"\"\"*&),(*$ )%\"xGF'F.F(*$)F-F'F.F(*$)F,F'F.F(#\"\"$F'F.-%%sqrtG6#,&F2F(F5F(F.!\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\" \"\"\"$*&*$-%%sqrtG6#,&*$)%\"xGF'\"\"\"F(*$)%\"yGF'F3F(F3F3*$),(F0F(F4 F(*$)%\"zGF'F3F(#\"\"$F'F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% ,Cartan_LEFTG6%\"\"#F'\"\"\"*&%\"xG\"\"\",(*$)F*F'F+F(*$)%\"yGF'F+F(*$ )%\"zGF'F+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6 %\"\"#F'F'*&%\"yG\"\"\",(*$)%\"xGF'F*\"\"\"*$)F)F'F*F/*$)%\"zGF'F*F/! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#F'\"\"$ *&%\"zG\"\"\",(*$)%\"xGF'F+\"\"\"*$)%\"yGF'F+F0*$)F*F'F+F0!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"$\"\"\",$*& *&%\"zGF)%\"yGF)\"\"\"*&-%%sqrtG6#,&*$)%\"xGF'F/F)*$)F.F'F/F)F/,(F5F)F 8F)*$)F-F'F/F)\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%, Cartan_LEFTG6%\"\"#\"\"$F'*&*&%\"zG\"\"\"%\"xGF,\"\"\"*&-%%sqrtG6#,&*$ )F-F'F.F,*$)%\"yGF'F.F,F.,(F4F,F6F,*$)F+F'F.F,\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"$\"\"\"F(,$*&%\"yG\"\"\"* &-%%sqrtG6#,(*$)%\"xG\"\"#F,F(*$)F+F5F,F(*$)%\"zGF5F,F(F,,&F2F(F6F(\" \"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\" \"$\"\"\"\"\"#*&%\"xG\"\"\"*&-%%sqrtG6#,(*$)F+F)F,F(*$)%\"yGF)F,F(*$)% \"zGF)F,F(F,,&F2F(F4F(\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%,Cartan_LEFTG6%\"\"$\"\"#\"\"\"*&*&%\"zGF)%\"yGF)\"\"\"*$),&*$)%\"xG F(F.F)*$)F-F(F.F)#\"\"$F(F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %,Cartan_LEFTG6%\"\"$\"\"#F(,$*&*&%\"zG\"\"\"%\"xGF-\"\"\"*$),&*$)F.F( F/F-*$)%\"yGF(F/F-#\"\"$F(F/!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"$F'\"\"\"*&%\"xG\"\"\",&*$)F*\"\"#F+F(*$)% \"yGF/F+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6% \"\"$F'\"\"#*&%\"yG\"\"\",&*$)%\"xGF(F+\"\"\"*$)F*F(F+F0!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The anti-symmetric part of the LEF T CARTAN Connection appear above." }}{PARA 0 "" 0 "" {TEXT -1 34 "Chec k for assymetry (LEFT Torsion)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "for j from 1 to dim do for \+ i from 1 to dim do for k from 1 to dim do ss := (DD[i,j,k]-DD[i,k,j]) /2; TTS[i,j,k]:=simplify(ss) od od od ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "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(`LEFT_Torsion`(i,k,j)=TTS[i,k,j]) fi od od od ;" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"\"\"\"#F',$*&*&%\"zGF'%\"xGF'\" \"\"*&-%%sqrtG6#,(*$)F-F(F.F'*$)%\"yGF(F.F'*$)F,F(F.F'F.-F16#,&F4F'F6F 'F.!\"\"#F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\" \"\"\"\"$F',$*&%\"yG\"\"\"*$-%%sqrtG6#,(*$)%\"xG\"\"#F,F'*$)F+F5F,F'*$ )%\"zGF5F,F'F,!\"\"#F'F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_T orsionG6%\"\"\"F'\"\"#,$*&*&%\"zGF'%\"xGF'\"\"\"*&-%%sqrtG6#,(*$)F-F(F .F'*$)%\"yGF(F.F'*$)F,F(F.F'F.-F16#,&F4F'F6F'F.!\"\"#!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"\"\"\"$\"\"#,$*&,&*$- %%sqrtG6#,&*$)%\"xGF)\"\"\"F'*$)%\"yGF)F5F'F5!\"\"F4F'F5*$-F/6#,(F2F'F 6F'*$)%\"zGF)F5F'F5!\"\"#F9F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-L EFT_TorsionG6%\"\"\"F'\"\"$,$*&%\"yG\"\"\"*$-%%sqrtG6#,(*$)%\"xG\"\"#F ,F'*$)F+F5F,F'*$)%\"zGF5F,F'F,!\"\"#!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"\"\"\"#\"\"$,$*&,&*$-%%sqrtG6#,& *$)%\"xGF(\"\"\"F'*$)%\"yGF(F5F'F5!\"\"F4F'F5*$-F/6#,(F2F'F6F'*$)%\"zG F(F5F'F5!\"\"#F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_Torsion G6%\"\"#F'\"\"\",$*&,&*(%\"xGF(-%%sqrtG6#,(*$)F-F'\"\"\"F(*$)%\"yGF'F4 F(*$)%\"zGF'F4F(F4-F/6#,&F2F(F5F(F4F(*&F:F(F7F(F(F4*&-F/6#F=F4)F1#\"\" $F'F4!\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6% \"\"#\"\"$\"\"\",$*&,(*$)%\"xGF'\"\"\"F)*$)%\"yGF'F0F)*(%\"zGF)F3F)-%% sqrtG6#,(F-F)F1F)*$)F5F'F0F)F0F)F0*&-F76#,&F-F)F1F)F0)F9#\"\"$F'F0!\" \"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"# \"\"\"F',$*&,&*(%\"xGF(-%%sqrtG6#,(*$)F-F'\"\"\"F(*$)%\"yGF'F4F(*$)%\" zGF'F4F(F4-F/6#,&F2F(F5F(F4F(*&F:F(F7F(F(F4*&-F/6#F=F4)F1#\"\"$F'F4!\" \"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"# \"\"$F',$*&*&%\"zG\"\"\",&*$-%%sqrtG6#,&*$)%\"xGF'\"\"\"F-*$)%\"yGF'F7 F-F7!\"\"F6F-F-F7*&-F16#F3F7,(F4F-F8F-*$)F,F'F7F-\"\"\"!\"\"#F-F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"#\"\"\"\"\"$,$* &,(*$)%\"xGF'\"\"\"F(*$)%\"yGF'F0F(*(%\"zGF(F3F(-%%sqrtG6#,(F-F(F1F(*$ )F5F'F0F(F0F(F0*&-F76#,&F-F(F1F(F0)F9#\"\"$F'F0!\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"#F'\"\"$,$*&*&%\"zG\"\" \",&*$-%%sqrtG6#,&*$)%\"xGF'\"\"\"F-*$)%\"yGF'F7F-F7!\"\"F6F-F-F7*&-F1 6#F3F7,(F4F-F8F-*$)F,F'F7F-\"\"\"!\"\"#F;F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$\"\"#\"\"\",$*&,&*&%\"xGF)-%%sq rtG6#,&*$)F.F(\"\"\"F)*$)%\"yGF(F5F)F5!\"\"*(%\"zGF)F8F)-F06#,(F3F)F6F )*$)F;F(F5F)F5F)F5*&)F2#\"\"$F(F5-F06#F>F5!\"\"#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$F'\"\"\",$*&%\"xG\"\"\",&*$) F+\"\"#F,F(*$)%\"yGF0F,F(!\"\"#F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%-LEFT_TorsionG6%\"\"$\"\"\"\"\"#,$*&,&*&%\"xGF(-%%sqrtG6#,&*$)F.F) \"\"\"F(*$)%\"yGF)F5F(F5F(*(%\"zGF(F8F(-F06#,(F3F(F6F(*$)F:F)F5F(F5!\" \"F5*&)F2#\"\"$F)F5-F06#F=F5!\"\"#F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$F'\"\"#,$*&%\"yG\"\"\",&*$)%\"xGF(F,\"\" \"*$)F+F(F,F1!\"\"#F1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_To rsionG6%\"\"$\"\"\"F',$*&%\"xG\"\"\",&*$)F+\"\"#F,F(*$)%\"yGF0F,F(!\" \"#!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$ \"\"#F',$*&%\"yG\"\"\",&*$)%\"xGF(F,\"\"\"*$)F+F(F,F1!\"\"#!\"\"F(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "For this example from cartesian \+ to spherical coordinates, there is no assymetry for the [CR], but ther e is assymetry for [CL]" }}{PARA 0 "" 0 "" {TEXT -1 45 "(The physical \+ implication is not clear to me)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Next the Christoffel symbols will be computed for the metric on th e initial state." }}{PARA 0 "" 0 "" {TEXT -1 102 "As the metric on \{x ,y,z\} is presumed to be the unit matrix, all the Christoffel symbols \+ should be zero" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 60 "Christoffel Connection coefficients from the induced metric " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "metric:= array([[1 ,0,0],[0,1,0],[0,0,1]]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'metricG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7 %F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "metricinverse:=i nverse(metric):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "for i f rom 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 0 145 "#for i from 1 to dim do for j from 1 to dim d o 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 dim do for j from i to dim do for k from 1 to dim do C 1S[i,j,k] := 0 od od od; for i from 1 to dim do for j from 1 to dim d o 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 0 188 " for k from 1 to dim do for i from 1 to dim do for j from 1 to dim do ss := 0; for m to dim do ss := ss+metricinverse[k,m]*C1S[i,j,m] od; C 2S[k,i,j] := simplify(factor(ss),trig) od od od; " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 160 "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(`Christoffel_Gamma 2`(i-1,j-1,k-1)=C2S[i,j,k]) fi od od od;" }}{PARA 0 "" 0 "" {TEXT 256 89 "The non zero Christoffel Connection coefficients 2nd kind on the initial space (domain)" }}{PARA 0 "" 0 "" {TEXT 257 50 " \+ Gamma2(i,j,k) index (1,-1,-1)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "If no entries appear above the Christoffel symbols on the domain s pace vanish" }}{PARA 0 "" 0 "" {TEXT -1 110 "The Right Cartan matrix i s often defined as the sum of Christoffel Symbols and Rotation coeffic ients, T(i,j,k)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 49 "CartanRight(ijk) = Chr istoffelGamma(ijk) + T(ijk)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Compute the T(i,j,k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "for i from 1 to dim do for j from 1 to dim do f or k from 1 to dim do ss:=0; ss := (CC[i,j,k]-C2S[i,j,k]); SHIPTR[i,j, k]:=simplify(ss) 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 CC[i,j,k]=0 then else print(`T`(i ,j,k)=simplify(SHIPTR[i,j,k])) fi od od od ;" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "T(ijk) ind ex (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"F'F',$* &*&,0*&)%\"xG\"\"%\"\"\")%\"yG\"\"#F0F'*&)%\"zGF3F0F-F0F'*&)F.F3F0)F2F /F0F3*(F5F0F8F0F1F0\"\"$*&F5F0F9F0F3*&F1F0)F6F/F0F'*$)F2\"\"'F0F'F'F.F 'F0*&),&*$F8F0F'*$F1F0F'\"\"#F0),(FEF'FFF'*$F5F0F'\"\"#F0!\"\"!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"F'\"\"#,$*&*&%\"yGF',0 *&)%\"xG\"\"%\"\"\")F,F(F2F'*&)%\"zGF(F2F/F2F'*&)F0F(F2)F,F1F2F(*(F5F2 F8F2F3F2\"\"$*&F5F2F9F2F(*&F3F2)F6F1F2F'*$)F,\"\"'F2F'F'F2*&),&*$F8F2F '*$F3F2F'\"\"#F2),(FEF'FFF'*$F5F2F'\"\"#F2!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"F'\"\"$,$*&*&)%\"zGF(\"\"\")%\"xG\"\" #F.F.*&,&*$F/F.F'*$)%\"yGF1F.F'\"\"\"),(F4F'F5F'*$)F-F1F.F'\"\"#F.!\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"#F',$*&* &%\"yGF',0*&)%\"xG\"\"%\"\"\")F,F(F2F'*&)%\"zGF(F2F/F2F'*&)F0F(F2)F,F1 F2F(*(F5F2F8F2F3F2\"\"$*&F5F2F9F2F(*&F3F2)F6F1F2F'*$)F,\"\"'F2F'F'F2*& ),&*$F8F2F'*$F3F2F'\"\"#F2),(FEF'FFF'*$F5F2F'\"\"#F2!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"#F(*&*&%\"xGF',4*&) F+\"\"%\"\"\")%\"yGF(F0F/*&)F+F(F0)F2F/F0\"\"&*$)F2\"\"'F0F(*()%\"zGF( F0F4F0F1F0F6*&F;F0F5F0\"\"$*&F1F0)FF1F(F(F/F(F1*&),&*$F;F1F(*$F5F1F(\"\"#F1),(FEF(FFF(*$F8F1F(\"\"# F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"\"\" \"$,$*&*()%\"zGF)\"\"\"%\"xGF(%\"yGF(F/*&),(*$)F0F'F/F(*$)F1F'F/F(*$)F .F'F/F(\"\"#F/,&F5F(F7F(\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"\",$*&*&,0*$)%\"xG\"\"'\"\"\"F(*&)F/\"\"%F1) %\"yGF'F1F'*&)%\"zGF'F1F3F1F'*(F8F1)F/F'F1F5F1\"\"$*&F;F1)F6F4F1F(*&F; F1)F9F4F1F(*&F8F1F>F1F(F(F/F(F1*&),&*$F;F1F(*$F5F1F(\"\"#F1),(FEF(FFF( *$F8F1F(\"\"#F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6% \"\"#F'F',$*&*&,0*$)%\"xG\"\"'\"\"\"\"\"\"*&)F.\"\"%F0)%\"yGF'F0F'*&)% \"zGF'F0F3F0F'*(F8F0)F.F'F0F5F0\"\"$*&F;F0)F6F4F0F1*&F;F0)F9F4F0F1*&F8 F0F>F0F1F1F6F1F0*&),&*$F;F0F1*$F5F0F1\"\"#F0),(FEF1FFF1*$F8F0F1\"\"#F0 !\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"$,$* &*&)%\"zGF(\"\"\")%\"yGF'F.F.*&),(*$)%\"xGF'F.\"\"\"*$F/F.F7*$)F-F'F.F 7\"\"#F.,&F4F7F8F7\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"TG6%\"\"#\"\"$\"\"\",$*&*()%\"zGF(\"\"\"%\"xGF)%\"yGF)F/*&),(*$)F 0F'F/F)*$)F1F'F/F)*$)F.F'F/F)\"\"#F/,&F5F)F7F)\"\"\"!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F',$*&*&)%\"zGF(\" \"\")%\"yGF'F.F.*&),(*$)%\"xGF'F.\"\"\"*$F/F.F7*$)F-F'F.F7\"\"#F.,&F4F 7F8F7\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\" #\"\"$F(*&*&%\"yG\"\"\",(*$)%\"zGF'\"\"\"F'*$)%\"xGF'F1F,*$)F+F'F1F,F, F1*$),(F2F,F5F,F.F,\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"TG6%\"\"$\"\"\"F(*&*()%\"xG\"\"#\"\"\"%\"zGF(,(*$F+F.F-*$)%\"yGF-F.F -*$)F/F-F.F(F(F.*&),(F1F(F2F(F5F(\"\"#F.,&F1F(F2F(\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"\"\"\"#*&**%\"zGF(% \"xGF(%\"yGF(,(*$)F-F)\"\"\"F)*$)F.F)F2F)*$)F,F)F2F(F(F2*&),(F0F(F3F(F 5F(\"\"#F2,&F0F(F3F(\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"TG6%\"\"$\"\"\"F',$*&*&,&*$)%\"xG\"\"#\"\"\"F(*$)%\"yGF0F1F(F(F/F(F1 *$),(F-F(F2F(*$)%\"zGF0F1F(\"\"#F1!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#\"\"\"*&**%\"zGF)%\"xGF)%\"yGF),(*$) F-F(\"\"\"F(*$)F.F(F2F(*$)F,F(F2F)F)F2*&),(F0F)F3F)F5F)\"\"#F2,&F0F)F3 F)\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#F( *&*(%\"zG\"\"\")%\"yGF(\"\"\",(*$)%\"xGF(F/F(*$F-F/F(*$)F+F(F/F,F,F/*& ),(F1F,F4F,F5F,\"\"#F/,&F1F,F4F,\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#F',$*&*&,&*$)%\"xGF(\"\"\"\"\"\"*$)% \"yGF(F0F1F1F4F1F0*$),(F-F1F2F1*$)%\"zGF(F0F1\"\"#F0!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"\",$*&*&,&*$)%\"xG \"\"#\"\"\"F(*$)%\"yGF0F1F(F(F/F(F1*$),(F-F(F2F(*$)%\"zGF0F1F(\"\"#F1! \"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"#,$*& *&,&*$)%\"xGF(\"\"\"\"\"\"*$)%\"yGF(F0F1F1F4F1F0*$),(F-F1F2F1*$)%\"zGF (F0F1\"\"#F0!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\" \"$F'F',$*&*&,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF/F0F1F1%\"zGF1F0*$),( F,F1F2F1*$)F5F/F0F1\"\"#F0!\"\"!\"\"" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 263 0 "" }{TEXT 256 51 "Right Cartan(ijk) = Christoffel Gamma(ij k) + T(ijk)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "These computations agree with Shipov on page 217, except for t he T(3,3,2), T(3,3,1) and the T(2,2,1) terms given above." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "***************************************** **************************************************" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with (linalg):with(liesymm):with(d ifforms):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "setup(x,y,z,t):deffor m(x=0,y=0,z=0,t=0,Vx=0,Vy=0,Vz=0,D1=0,D2=0,D3=0,Ax=0,Ay=0,Az=0,C=0,Phi =0,phi=0,theta=0,r=0,a=const,b=const,c=const,Lx=0,Ly=0,Lz=0);" }} {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 263 "" 0 "" {TEXT 261 49 "Part 2: The map is from \{r,theta,phi \} into \{x,y,z" }{TEXT 264 2 "\} " }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{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 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG*(% \"rG\"\"\"-%$sinG6#%&thetaGF'-%$cosG6#%$phiGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG*(%\"rG\"\"\"-%$sinG6#%&thetaGF'-F)6#%$phiGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&%\"rG\"\"\"-%$cosG6#%&thetaGF' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "R:=[x,y,z];FF:=jacobian (R,[r,theta,phi]);DR:=d(R):`dx`:=DR[1];`dy`:=DR[2];`dz`:=DR[3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%*(%\"rG\"\"\"-%$sinG6#%&thetaG F(-%$cosG6#%$phiGF(*(F'\"\"\"F)F2-F*F/F(*&F'F2-F.F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%'matrixG6#7%7%*&-%$sinG6#%&thetaG\"\"\"-%$ cosG6#%$phiGF/*(%\"rGF/-F1F-F/F0\"\"\",$*(F5F7F+F7-F,F2F/!\"\"7%*&F+F7 F:F7*(F5F7F6F7F:F7*(F5F7F+F7F0F77%F6,$*&F5F7F+F7F;\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#dxG,(*(-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiG F+-%\"dG6#%\"rGF+F+**F3F+F,\"\"\"-F-F)F+-F1F)F+F+**F3F5F'F5-F(F.F+-F1F .F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dyG,(*(-%$sinG6#%&theta G\"\"\"-F(6#%$phiGF+-%\"dG6#%\"rGF+F+**F2F+F,\"\"\"-%$cosGF)F+-F0F)F+F +**F2F4F'F4-F6F-F+-F0F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dzG, &*&-%$cosG6#%&thetaG\"\"\"-%\"dG6#%\"rGF+F+*(F/F+-%$sinGF)F+-F-F)F+!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "GG:=evalm(inverse(FF) ):DETFF:=simplify(det(FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DETF FG*&-%$sinG6#%&thetaG\"\"\")%\"rG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 102 "Note that the Frame matrix has a singularity at values of theta equal to multiples of pi, and at r=0." }}{PARA 0 "" 0 "" {TEXT -1 112 "The next equation checks to see that the specified frame produces the desired differential structures: [FF]|dR>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "zz:=simplify(evalm(innerprod(FF,[d( r),d(theta),d(phi)])));zzb:=innerprod(GG,zz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#zzG-%'vectorG6#7%,(*(-%$sinG6#%&thetaG\"\"\"-%$cosG6 #%$phiGF/-%\"dG6#%\"rGF/F/**F7F/F0\"\"\"-F1F-F/-F5F-F/F/**F7F9F+F9-F,F 2F/-F5F2F/!\"\",(*(F+F9F=F9F4F9F/**F7F9F=F9F:F9F;F9F/**F7F9F+F9F0F9F>F 9F/,&*&F:F9F4F9F/*(F7F9F+F9F;F9F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$zzbG-%'vectorG6#7%-%\"dG6#%\"rG-F*6#%&thetaG-F*6#%$phiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Note that each component is exact (by co nstruction) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "d(zz[1]);d( zz[2]);d(zz[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "The metric on the target xyz is the unit matrix of constants, by assumption. As such the Christof fel symbols will be zero. The induced pulled back metric on the spher ical coordinate range is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "inducedmetric:=simplify(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.inducedmetricG-%'matrixG6#7%7%\"\"\"\"\"!F+7 %F+*$)%\"rG\"\"#\"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Now check to see if the Fram e matrix is normal and find the Left and Right representations P2R=[F F][ transpose FF]" }}{PARA 0 "" 0 "" {TEXT -1 171 "P2L= [transpose FF] [ F]. Conclusion: the Frame matrix is not normal!!! Note that P2R \+ is the pushedforward inverse metric on \{r,theta,phi\} but with argume nts on \{x,y,z\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "P2R:=si mplify(innerprod(FF,transpose(FF)));P2L:=simplify(innerprod(transpose( FF),FF));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$P2RG-%'matrixG6#7%7%,. *$)-%$cosG6#%$phiG\"\"#\"\"\"\"\"\"*&)-F.6#%&thetaGF1F2F,F2!\"\"*()%\" rGF1F2F5F2F,F2F1*$F;F2F3*&F,F2F;F2F9*&F;F2F5F2F9,**&F-F3-%$sinGF/F3F3* (F-F2FBF2F5F2F9**F;F2F5F2F-F2FBF2F1*(F;F2FBF2F-F2F9,&*(-FCF7F3F-F2F6F3 F3**F;F2F6F2F-F2FIF2F97%F@,0F3F3F+F9*$F5F2F9F4F3F?F3F:!\"#F>F3,&*(FIF2 FBF2F6F2F3**F;F2F6F2FBF2FIF2F97%FGFO,(FMF3F=F3F?F9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$P2LG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)%\"rG\"\"# \"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "P2R is complicated algebracially. But th e bottom line is that it is NOT equal to P2L, " }}{PARA 0 "" 0 "" {TEXT -1 39 "hence the Jacobian matrix is NOT NORMAL" }}{PARA 0 "" 0 " " {TEXT -1 107 "Now any matrix with an inverse can be composed as a pr oduct of a symmetric matrix and an orthogonal matrix." }}{PARA 0 "" 0 "" {TEXT -1 375 "There are in general two ways to construct this repre sentation, which will be denoted as the Lefthanded and the Right hande d formulations. [SR] is the symmetric matrix of the \"right handed\" \+ formulation, and [OR] is the orthogonal matrix for the right handed fo rmulation. (If the matrices are complex, the notions symmetric and or thogonal translate to Hermitean and Unitary)" }}{PARA 0 "" 0 "" {TEXT -1 20 "The two formats are:" }}{PARA 0 "" 0 "" {TEXT -1 23 "[F]= [SR][ OR]=[OL] [SL]" }}{PARA 0 "" 0 "" {TEXT -1 17 "If [G][F]=1, then" }} {PARA 0 "" 0 "" {TEXT -1 32 "[SR]=\{[FF].transpose[FF]\}^(1/2) " }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 22 "[OR]=[ SR].transpose[G]" }}{PARA 0 "" 0 "" {TEXT -1 32 "The Left handed repre sentations " }}{PARA 0 "" 0 "" {TEXT -1 32 "[SL]=\{transpose[FF].[FF] \}^(1/2) " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 22 "[OL]=transpose[G].[SL]" }}{PARA 0 "" 0 "" {TEXT -1 74 "The repr esentations are distinct if the Frame [FF] is not a normal matrix." }} {PARA 0 "" 0 "" {TEXT -1 2 "**" }}{PARA 0 "" 0 "" {TEXT -1 82 "So the \+ first step is to find the square roots of these matrices P2R and P2L a bove." }}{PARA 0 "" 0 "" {TEXT -1 65 "This is easy to do for the Right Handed P2R, for it is diagonal." }}{PARA 0 "" 0 "" {TEXT -1 73 "The \+ symmetric component of the Left handed representation: F = [OL] [SL] " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "SL:=array([[1,0,0],[0,r ,0],[0,0,(r*sin(theta))]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SLG- %'matrixG6#7%7%\"\"\"\"\"!F+7%F+%\"rGF+7%F+F+*&F-F*-%$sinG6#%&thetaGF* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Compute the orthogonal factor [OL] of the Left handed representation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "GS:=simplify(transpose(inverse(FF))):OL:=innerprod(G S,SL);simplify(innerprod(transpose(OL),OL)):`Should_be_zero`:=evalm(in nerprod(OL,SL)-FF):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OLG-%'matrix G6#7%7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG6#%$phiGF/*&F0\"\"\"-F1F-F/,$-F ,F2!\"\"7%*&F+F5F8F/*&F8F5F6F5F07%F6,$F+F9\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Find Omega, the right Cartan matrix of [OL], which s hould be an anti-symmetric matrix" }}{PARA 0 "" 0 "" {TEXT -1 41 "and \+ Delta the left Cartan matrix for [SL]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dOL:=d(OL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dOL G-%'matrixG6#7%7%,&*(-%$cosG6#%$phiG\"\"\"-F-6#%&thetaGF0-%\"dGF2F0F0* (-%$sinGF2F0-F8F.F0-F5F.F0!\"\",&*(F1\"\"\"F9F>F:F>F;*(F,F>F7F>F4F>F;, $*&F,F>F:F>F;7%,&*(F9F>F1F>F4F>F0*(F7F>F,F>F:F>F0,&*(F1F>F,F>F:F>F0*(F 9F>F7F>F4F>F;,$*&F9F>F:F>F;7%,$*&F7F>F4F>F;,$*&F1F>F4F>F;\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Omega_R:=simplify(innerprod( transpose(OL),dOL));Delta_R:=innerprod(d(SL),inverse(SL));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Omega_RG-%'matrixG6#7%7%\"\"!,$-%\"dG6#%& thetaG!\"\",$*&-%$sinGF.\"\"\"-F-6#%$phiGF5F07%F,F*,$*&-%$cosGF.F5F6\" \"\"F07%F2F;F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta_RG-%'matrix G6#7%7%\"\"!F*F*7%F**&-%\"dG6#%\"rG\"\"\"F0!\"\"F*7%F*F**&,&*&-%$sinG6 #%&thetaG\"\"\"F-F;F;*(F0F;-%$cosGF9F;-F.F9F;F;F1*&F0\"\"\"F7\"\"\"F2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "Omega_L:=simplify(innerprod(dOL,transpose(OL)) );Delta_L:=innerprod(d(SL),inverse(SL));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Omega_LG-%'matrixG6#7%7%\"\"!,$-%\"dG6#%$phiG!\"\"*&-F-6#%&th etaG\"\"\"-%$cosGF.F57%F,F**&F2\"\"\"-%$sinGF.F57%,$F1F0,$F9F0F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta_LG-%'matrixG6#7%7%\"\"!F*F*7% F**&-%\"dG6#%\"rG\"\"\"F0!\"\"F*7%F*F**&,&*&-%$sinG6#%&thetaG\"\"\"F-F ;F;*(F0F;-%$cosGF9F;-F.F9F;F;F1*&F0\"\"\"F7\"\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Note that the matrix elements of Delta are per fect exact differentials, and the matrix elements of Omega are closed \+ but not exact differentials. " }}{PARA 0 "" 0 "" {TEXT -1 76 "It is p ossible to write the differential of the Frame field in several ways: " }}{PARA 0 "" 0 "" {TEXT -1 88 "d[FF] = [FF][CR] =\{[DeltaR][FF] + [F F][OmegaR]\} = [CL][FF] = \{[OmegaL][FF]+[FF][DeltaL]\}" }}{PARA 0 "" 0 "" {TEXT -1 61 "Note that appropriate linear combinations can be con structed." }}{PARA 0 "" 0 "" {TEXT -1 1 "*" }}{PARA 0 "" 0 "" {TEXT 262 40 "Now Compute the Right Cartan Matrix [CR]" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cart an:=simplify(innerprod(inverse(FF),d(FF))):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 83 "The matrix elements of the Right Cartan connection matr ix using the matrix methods:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Gamma11:=wcollect(cartan[1,1]);Gamma12:=wcollect(cartan[1,2]);Ga mma13:=wcollect(cartan[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Ga mma11G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G,$*&%\"rG\" \"\"-%\"dG6#%&thetaGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamm a13G*&,&%\"rG!\"\"*&F'\"\"\")-%$cosG6#%&thetaG\"\"#\"\"\"F*F*-%\"dG6#% $phiGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Gamma21:=wcollec t(cartan[2,1]);Gamma22:=wcollect(cartan[2,2]);Gamma23:=wcollect(cartan [2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G*&-%\"dG6#%&thet aG\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G*&-% \"dG6#%\"rG\"\"\"F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma23 G,$*(-%$cosG6#%&thetaG\"\"\"-%\"dG6#%$phiGF+-%$sinGF)F+!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Gamma31:=wcollect(cartan[3,1 ]);Gamma32:=wcollect(cartan[3,2]);Gamma33:=wcollect(cartan[3,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G*&-%\"dG6#%$phiG\"\"\"%\"rG !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G*&*&-%$cosG6#%&the taG\"\"\"-%\"dG6#%$phiGF+\"\"\"-%$sinGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G,&*&*&-%$cosG6#%&thetaG\"\"\"-%\"dGF*F,\"\" \"-%$sinGF*!\"\"F,*&-F.6#%\"rGF/F6F2F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Now the components of the right Cartan matrix will be com puted by the tensor method, as a check" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "dim:=3;coord:=[r,theta,phi];GG:=simplify(inverse(FF)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dimG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&coordG7%%\"rG%&thetaG%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG-%'matrixG6#7%7%*&-%$sinG6#%&thetaG\"\"\"-%$cosG6 #%$phiGF/*&F+\"\"\"-F,F2F/-F1F-7%*&*&F0F5F7F/F5%\"rG!\"\"*&*&F6F5F7F5F 5F;F<,$*&F+F5F;F " 0 "" {MPLTEXT 1 0 119 "for i from 1 to dim do for j from 1 to dim do f or 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 the matrix product of - d[G][F]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "for b from 1 to dim do for a from 1 to dim do for k from 1 to d im do ss:=0;for m from 1 to dim do ss := ss+(d1GG[a,m,k]*FF[m,b]); CC [a,b,k]:=simplify(-ss) od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "for b from 1 to d im do for a from 1 to dim do for k from 1 to dim do if CC[a,b,k]=0 the n else print(`CCabk`(a,b,k)=factor(CC[a,b,k])) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 257 50 "THE non zero CARTAN RIGHT CONNECTION coefficie nts." }}{PARA 0 "" 0 "" {TEXT 258 51 " CC(a bk) index (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\" \"#\"\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% &CCabkG6%\"\"$\"\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"\"\"\"#F(,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$\"\"#F'*&-%$cosG6#%&t hetaG\"\"\"-%$sinGF,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabk G6%\"\"\"\"\"$F(*(%\"rGF',&-%$cosG6#%&thetaGF'!\"\"F'F',&F,F'F'F'F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"#\"\"$F(,$*&-%$cosG6# %&thetaG\"\"\"-%$sinGF-F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%& CCabkG6%\"\"$F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&CCabkG6%\"\"$F'\"\"#*&-%$cosG6#%&thetaG\"\"\"-%$sin GF,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "These results agree w ith matrix method." }}{PARA 0 "" 0 "" {TEXT -1 55 "Next check for Affi ne Torsion using the tensor methods:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for j from 1 to dim do for i from 1 to dim do for k \+ from 1 to dim do ss := (CC[i,j,k]-CC[i,k,j])/2; CCTTS[i,j,k]:=ss od o d od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 to dim do for j from 1 to dim do for k \+ from 1 to dim do if CCTTS[i,j,k]=0 then else print(`RIGHT_AffineTorsio n`(i,k,j)=CCTTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 54 "IF NO ENTRIES \+ APPEAR ABOVE, THE AFFINE TORSION IS ZERO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 38 "Now compute the CARTAN LEFT CONNECTION" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for a from 1 to dim do for j from 1 to dim do for k from 1 to dim do d1GG[a,j,k] := simplify(diff(GG[a,j],coord[k])) od od od: " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Compute the elements of the matrix product of [F]d[G]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "for i from 1 to dim do for j from 1 to dim do for k from 1 to dim do ss: =0;for m to dim do ss := ss+FF[i,m]*(d1GG[m,j,k]); DD[i,j,k]:=simplify (ss) od od od od ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for \+ i from 1 to dim do for j from 1 to dim do for k from 1 to dim do if DD [i,j,k]=0 then else print(`Cartan_LEFT`(i,j,k)=DD[i,j,k]) fi od od od \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"F'F',$*&, (*&)-%$cosG6#%&thetaG\"\"#\"\"\")-F.6#%$phiGF1F2F'F'F'*$F3F2!\"\"F2%\" rG!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"F '\"\"#,$*&*(-%$sinG6#%&thetaGF',&!\"\"F'*$)-%$cosG6#%$phiGF(\"\"\"F'F' -F5F.F'F8,&F1F'*$)F9F(F8F'!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%,Cartan_LEFTG6%\"\"\"\"\"#F',$*&*(-%$cosG6#%$phiGF'-%$sinGF.F',&!\" \"F'*$)-F-6#%&thetaGF(\"\"\"F'F'F9%\"rG!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"#F(,$*&**-%$sinG6#%&thetaGF '-F-6#%$phiGF'-%$cosGF1F'-F4F.F'\"\"\",&!\"\"F'*$)F5F(F6F'!\"\"F8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"#\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"$F'*&*(- %$cosG6#%&thetaGF'-F,6#%$phiGF'-%$sinGF-F'\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"\"\"\"$\"\"#,$-%$cosG6 #%$phiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\" #\"\"\"F(,$*&*(-%$cosG6#%$phiGF(-%$sinGF.F(,&!\"\"F(*$)-F-6#%&thetaGF' \"\"\"F(F(F9%\"rG!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan _LEFTG6%\"\"#\"\"\"F',$*&**-%$sinG6#%&thetaGF(-F-6#%$phiGF(-%$cosGF1F( -F4F.F(\"\"\",&!\"\"F(*$)F5F'F6F(!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"\"\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#F'\"\"\"*&,(*$)-%$cosG6#%&thetaG F'\"\"\"!\"\"*&F,F1)-F.6#%$phiGF'F1F(*$F4F1F2F1%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#F'F'*&*(-%$sinG6#%&theta G\"\"\")-%$cosG6#%$phiGF'\"\"\"-F1F,F.F4,&!\"\"F.*$)F5F'F4F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"$\"\"\"*&*( -%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F1F.F)\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"#\"\"$F',$-%$sinG6#%$p hiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"$\" \"\"F(*&*(-%$cosG6#%&thetaGF(-F,6#%$phiGF(-%$sinGF-F(\"\"\"%\"rG!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"$\"\"\"\"\"# -%$cosG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6% \"\"$\"\"#\"\"\"*&*(-%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F1F.F)\"\"\"% \"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEFTG6%\"\"$\" \"#F(-%$sinG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%,Cartan_LEF TG6%\"\"$F'\"\"\"*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F3%\"rG! \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The anti-symmetric part o f the LEFT CARTAN Connection appear above." }}{PARA 0 "" 0 "" {TEXT -1 34 "Check for assymetry (LEFT Torsion)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "for j from 1 to dim do for i from 1 to dim do for k from 1 to dim do ss := (DD[i,j,k]-DD [i,k,j])/2; TTS[i,j,k]:=simplify(ss) od od od ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "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(`LEFT_Torsion`(i,k,j)=TTS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"\"\"\"#F',$*&,,*(-%$sinG6#%&thet aGF'-%$cosGF/F'%\"rGF'!\"\"**F-\"\"\"F1F6F3F6)-F26#%$phiGF(F6F'*&F8F'- F.F9F'F4*(F8F6FF6F'\"\"\"!\"\"#F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_Tors ionG6%\"\"\"\"\"$F',$*&*(-%$cosG6#%&thetaGF'-F-6#%$phiGF'-%$sinGF.F'\" \"\"%\"rG!\"\"#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_Tor sionG6%\"\"\"F'\"\"#,$*&,,*(-%$sinG6#%&thetaGF'-%$cosGF/F'%\"rGF'!\"\" **F-\"\"\"F1F6F3F6)-F26#%$phiGF(F6F'*&F8F'-F.F9F'F4*(F8F6FF6F'\"\"\"!\"\"#F4F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"\"\"\"$\"\"#,&- %$cosG6#%$phiG#!\"\"F)F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT _TorsionG6%\"\"\"F'\"\"$,$*&*(-%$cosG6#%&thetaGF'-F-6#%$phiGF'-%$sinGF .F'\"\"\"%\"rG!\"\"#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%- LEFT_TorsionG6%\"\"\"\"\"#\"\"$,&#F'F(F'-%$cosG6#%$phiGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"#F'\"\"\",$*&,.*,-%$sinG 6#%&thetaGF(-%$cosG6#%$phiGF(-F.F3F(-F2F/F(%\"rGF(F(*$)F6F'\"\"\"F(*$) F6\"\"%F:!\"\"*&F9F:)F1F'F:!\"#*&FF(F 8F(\"\"\"!\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_Torsio nG6%\"\"#\"\"$\"\"\",$*&,&%\"rGF)*(-%$sinG6#%&thetaGF)-F06#%$phiGF)-%$ cosGF1F)F)\"\"\"F-!\"\"#F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LE FT_TorsionG6%\"\"#\"\"\"F',$*&,.*,-%$sinG6#%&thetaGF(-%$cosG6#%$phiGF( -F.F3F(-F2F/F(%\"rGF(F(*$)F6F'\"\"\"F(*$)F6\"\"%F:!\"\"*&F9F:)F1F'F:! \"#*&FF(F8F(\"\"\"!\"\"#F>F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"#\"\"$F',$-%$si nG6#%$phiG#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_Torsion G6%\"\"#\"\"\"\"\"$,$*&,&%\"rGF(*(-%$sinG6#%&thetaGF(-F06#%$phiGF(-%$c osGF1F(F(\"\"\"F-!\"\"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%- LEFT_TorsionG6%\"\"#F'\"\"$,$-%$sinG6#%$phiG#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$\"\"#\"\"\",$*&,&*&-%$cosG 6#%$phiGF)%\"rGF)F)*(-%$sinG6#%&thetaGF)-F5F0F)-F/F6F)!\"\"\"\"\"F2!\" \"#F:F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$F' \"\"\",$*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F4%\"rG!\"\"#F(F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_TorsionG6%\"\"$\"\"\"\"\" #,$*&,&*&-%$cosG6#%$phiGF(%\"rGF(F(*(-%$sinG6#%&thetaGF(-F5F0F(-F/F6F( !\"\"\"\"\"F2!\"\"#F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-LEFT_To rsionG6%\"\"$\"\"\"F',$*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F4% \"rG!\"\"#F,F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "For this examp le from cartesian to spherical coordinates, there is no assymetry for \+ the [CR], but there is assymetry for [CL]" }}{PARA 0 "" 0 "" {TEXT -1 45 "(The physical implication is not clear to me)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 82 "Next the Christoffel symbols will be computed for \+ the metric on the initial state." }}{PARA 0 "" 0 "" {TEXT -1 102 "As t he metric on \{x,y,z\} is presumed to be the unit matrix, all the Chri stoffel symbols should be zero" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 256 60 "Christoffel Connection coefficients from the induced \+ metric " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "metric: = evalm(inducedmetric);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'metricG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+*$)% \"rG\"\"#\"\"\"F+7%F+F+,&F-F**&F.F1)-%$cosG6#%&thetaGF0F1!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "metricinverse:=inverse(metri c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "for i from 1 to dim do for j from 1 to dim do for k from 1 to dim do d1gun[i,j,k] := (dif f(metric[i,j],coord[k])) od od od: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "#for i from 1 to dim do for j from 1 to dim do for k from 1 to dim do if d1gun[i,j,k]=0 then else print(`dgun`(i,j,k)=d1g un[i,j,k]) fi od od od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "for i f rom 1 to dim do for j from i to dim do for k from 1 to dim do C1S[i,j, k] := 0 od od od; for i from 1 to dim do for j from 1 to dim do for k from 1 to dim do C1S[i,j,k] := 1/2*d1gun[i,k,j]+1/2*d1gun[j,k,i]-1/2* d1gun[i,j,k] od od od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 " for k from 1 to dim do for i from 1 to dim do for j from 1 to dim do ss := \+ 0; for m to dim do ss := ss+metricinverse[k,m]*C1S[i,j,m] od; C2S[k,i, j] := simplify(factor(ss),trig) od od od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "for i from 1 to dim do for j from 1 to dim do for k \+ from 1 to dim do if C2S[i,j,k]=0 then else print(`Christoffel_Gamma2` (i-1,j-1,k-1)=C2S[i,j,k]) fi od od od;" }}{PARA 0 "" 0 "" {TEXT 256 89 "The non zero Christoffel Connection coefficients 2nd kind on the initial space (domain)" }}{PARA 0 "" 0 "" {TEXT 257 50 " \+ Gamma2(i,j,k) index (1,-1,-1)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %3Christoffel_Gamma2G6%\"\"!\"\"\"F(,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6%\"\"!\"\"#F(,&%\"rG!\"\"*&F*\" \"\")-%$cosG6#%&thetaGF(\"\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %3Christoffel_Gamma2G6%\"\"\"\"\"!F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6%\"\"\"F'\"\"!*&\"\"\"F* %\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6 %\"\"\"\"\"#F(,$*&-%$cosG6#%&thetaGF'-%$sinGF-F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6%\"\"#\"\"!F'*&\"\"\"F*%\" rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6%\" \"#\"\"\"F',$*&*&-%$cosG6#%&thetaGF(-%$sinGF.F(\"\"\",&!\"\"F(*$)F,F'F 2F(!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%3Christoffel_Gamma2G6 %\"\"#F'\"\"!*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%3Christoffel_Gamma2G6%\"\"#F'\"\"\",$*&*&-%$cosG6#%&thetaGF(-%$sinGF .F(\"\"\",&!\"\"F(*$)F,F'F2F(!\"\"F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "If no entries appear above the Christoffel symbols on the domai n space vanish" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "The Right Cartan matrix is often defined as the sum of C hristoffel Symbols and Rotation coefficients, T(i,j,k)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT 256 49 "CartanRight(ijk) = ChristoffelGamma(ijk) + T(ijk)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Compute t he T(i,j,k):" }}}{EXCHG {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 ss:=0; ss : = (CC[i,j,k]-C2S[i,j,k]); SHIPTR[i,j,k]:=simplify(ss) 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 CC[i,j,k]=0 then else print(`T`(i,j,k)=simplify(SHIPTR[i,j,k])) fi od od od ;" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT 256 22 "T(ijk) index (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"#F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"$F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"\"F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"\"F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"#\"\"!" }}}{EXCHG {PARA 258 "" 0 " " {TEXT -1 0 "" }{TEXT 256 39 "Right Cartan(ijk) = Gamma(ijk) + T(ijk) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "In this example the rotation coef ficients on the domain space vanish for the Cartan right matrix is exa ctly equal to the the Christoffel symbols." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Note the differences between th e map from cartesian space and the map to cartesian space." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "124 12" 0 }{VIEWOPTS 1 1 0 1 1 1803 }