{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 "Vessiot_Text" -1 256 "AGaramond" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou tput" -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 "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output " 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Ves s_Title2" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 2 0 0 2 0 0 0 }1 0 0 0 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_IO" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 }1 0 0 -1 -1 -1 3 30 0 0 0 0 -1 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 128 0 64 1 2 2 2 0 0 2 3 0 0 }2 1 0 0 10 10 3 6 3 30 0 0 -1 0 }{PSTYLE "" 256 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vessiot_T itle3" 0 260 1 {CSTYLE "" -1 -1 "" 0 0 255 128 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 261 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 256 262 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 256 263 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 256 264 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 256 265 1 {CSTYLE "" -1 -1 "" 1 9 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 258 "" 0 "" {TEXT -1 65 "Vessiot Tutorial: Tablea u, Cartan Characters & Spencer Cohomology" }}}{EXCHG {PARA 259 "" 0 " " {TEXT -1 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT 257 199 "to illustrat e the commands in the package Spencer, for creating the tableau for \+ a given system of PDE, for prolongation of tableau, for testing invol utivity, and for computing Spencer cohomology. " }{TEXT 256 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with(Vessiot):with(tensors): with(Spencer):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 256 "" 0 "" {TEXT 260 17 "Laplaces Equation" }}{PARA 257 "" 0 "" {TEXT 256 77 "We check that Laplace's equation is involu tive by 3 different calculations." }}{PARA 257 "" 0 "" {TEXT 256 259 " First we construct the tableau A and its first prolongation. We \+ then compute the restricted tableaus with respect to the standard b asis and note that differentiation with respect to x gives a sequen ce of surjectibe maps. See page 317 of BCGGG. " }}{PARA 257 "" 0 " " {TEXT 256 56 "Second, we check involutivity by Cartan's test. p 3 18" }}{PARA 257 "" 0 "" {TEXT 256 62 "Third, we check involutivity u sing Spencer cohomology, p325." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "coord_init([x,y,z],[u],euc): " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 41 "Delta:= [u[2,0,0] + u [0,2,0] + u[0,0,2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG7#,(& %\"uG6%\"\"#\"\"!F+\"\"\"&F(6%F+F*F+F,&F(6%F+F+F*F," }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 17 "B:=[D_x,D_y,D_z]:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 256 81 "The command symbol_to_pr_tableau constructs th e tableau and its prolongations." }}{PARA 0 "" 0 "" {TEXT 256 101 "The command tableau_sequence constructes the sequence of subtableaus d efined above eq 5, page 317." }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tableau(Delta,3):" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#A0G7$7%%(tableauG%$eucG7\"7'7#*&%\"xG\"\"\"%\"yGF.7#,&*$)F-\"\"#\" \"\"!\"\"*$)F/F4F5F.7#*&F-F5%\"zGF.7#*&F/F5F;F57#,&F2F6*$)F;F4F5F." }} }{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A3:=prA[4]:" }}}{EXCHG {PARA 260 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "METHOD 1" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 31 " A0_seq:=tableau_sequence(A0,B):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A0_0:=A0_seq[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%A0_0G7$7%%(tableauG%$eucG7\"7'7#*&%\"xG\"\"\"%\"yGF.7#,&*$)F-\"\"#\" \"\"!\"\"*$)F/F4F5F.7#*&F-F5%\"zGF.7#*&F/F5F;F57#,&F2F6*$)F;F4F5F." }} }{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A0_1:=A0_seq[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%A0_1G7$7%%(tableauG%$eucG7\"7$7#*&% \"yG\"\"\"%\"zGF.7#,&*$)F-\"\"#\"\"\"F.*$)F/F4F5!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A0_2:=A0_seq[3];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%A0_2G7$7%%(tableauG%$eucG7\"7#7#\"\"!" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 31 "A1_seq:=tableau_sequence (A1,B):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A1_0:=A1_seq [1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%A1_0G7$7%%(tableauG%$eucG7 \"7)7#*(%\"xG\"\"\"%\"yGF.%\"zGF.7#,&*$)F-\"\"$\"\"\"F.*&F-F6)F/\"\"#F 6!\"$7#,&F7!\"\"*&F-F6)F0F9F6F.7#,&*&)F-F9F6F/F6F:*$)F/F5F6F.7#,&*&FCF 6F0F6F=*&F8F6F0F6F.7#,&FBF=*&F/F6F?F6F.7#,&FHF:*$)F0F5F6F." }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A1_1:=A1_seq[2];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%A1_1G7$7%%(tableauG%$eucG7\"7$7#,&*$)%\"yG\" \"$\"\"\"\"\"\"*&F/F2)%\"zG\"\"#F1!\"$7#,&*&)F/F6F1F5F2F7*$)F5F0F1F2" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "A1_2:=A1_seq[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%A1_2G7$7%%(tableauG%$eucG7\"7#7#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 145 "For involutivity, differentiation with respect to x show carry A1_0 onto A0_0, and differentiation with respect to y sh ould carry A1_1 onto A0_1" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "C:=diff_tableau(A1_0,D_x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"CG7$7%%(tableauG%$eucG7\"7)7#*&%\"yG\"\"\"%\"zGF.7#,&*$)%\"xG\"\"#\" \"\"\"\"$*$)F-F5F6!\"$7#,&F8!\"\"*$)F/F5F6F.7#,$*&F4F.F-F6!\"'7#,$*&F4 F6F/F6!\"#7#,$FBFG7#,$FFFC" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "map(Spencer_linear_combo,C[2],A0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)-%'vectorG6#7'\"\"!F(F(\"\"\"F(-F%6#7'F(!\"$F(F(F(-F% 6#7'F(!\"\"F(F(F)-F%6#7'!\"'F(F(F(F(-F%6#7'F(F(!\"#F(F(-F%6#7'F9F(F(F( F(-F%6#7'F(F(F5F(F(" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 " C:=diff_tableau(A1_1,D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7$ 7%%(tableauG%$eucG7\"7$7#,&*$)%\"yG\"\"#\"\"\"\"\"$*$)%\"zGF0F1!\"$7#, $*&F/\"\"\"F5F:!\"'" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 36 " map(Spencer_linear_combo,C[2],A0_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7$-%'vectorG6#7$\"\"!\"\"$-F%6#7$!\"'F(" }}}{EXCHG {PARA 260 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "METHOD 2" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 15 " t:=nops(A1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG\"\"(" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_characters(A0_seq );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"$\"\"#\"\"!\"\"(" }}} {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "METHOD 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 124 "We check involutivity by the vanishing of all the Spencer c ohomology ( except H01 which is allowed under the definition)" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An1:=full_tableau(1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$An1G7$7%%(tableauG%$eucG7\"7%7#%\" xG7#%\"yG7#%\"zG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An2 :=full_tableau(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$An2G7$7%%(tab leauG%$eucG7\"7#7#\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 42 "H0:=Spencer_cohomology([A0,An1, An2],1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H0G7#7#,(**%\"xG\"\"\"&%#dxG6#%!GF*%#^~GF*&&%#duG6#7 %\"\"!F5F56#F.F*F***%\"yGF*&%#dyG6#F.F*F/F*&F16#F.F*F***%\"zGF*&%#dzG6 #F.F*F/F*&F16#F.F*F*" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 46 "H1:=Spencer_cohomology([A1, A0,An1, An2],1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7% \"\"!F4F46#F-F)7#*.F(\"\"\"&F+6#F-F)F.F)&%#dyG6#F-F)F.F)&F06#F-F)" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 50 "H2:=Spencer_cohomology([ A2, A1,A0, An1, An2],1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7% 7#**%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7%\"\"!F4F46#F-F)7#*.F(\" \"\"&F+6#F-F)F.F)&%#dyG6#F-F)F.F)&F06#F-F)7#*2F(F8&F+6#F-F)F.F)&F<6#F- F)F.F)&%#dzG6#F-F)F.F)&F06#F-F)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 49 "H3:=Spencer_cohomology([A3, A2,A1, A0, An1],1,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G7%7#**%#0~G\"\"\"&%#dxG6#%!GF)%# ^~GF)&&%#duG6#7%\"\"!F4F46#F-F)7#*.F(\"\"\"&F+6#F-F)F.F)&%#dyG6#F-F)F. F)&F06#F-F)7#*2F(F8&F+6#F-F)F.F)&F<6#F-F)F.F)&%#dzG6#F-F)F.F)&F06#F-F) " }}}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 12 "The gradient" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "c oord_init([x,y,z],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 39 "Delta:=[ u[1,0,0], u[0,1,0], u[0,0,1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG7%&%\"uG6%\"\"\"\"\"!F*&F'6%F*F)F*&F'6%F*F*F) " }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tab leau(Delta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$prAG7&7$7%%(table auG%$eucG7\"7#7#\"\"!F&F&F&" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "A0:=prA[1]: A1:=prA[2]: A2:=prA[3]:" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 21 "An1:=full_tableau(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$An1G7$7%%(tableauG%$eucG7\"7#7#\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 22 "An2:=full_tableau(-1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$An2G7$7%%(tableauG%$eucG7\"F)" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "H0:=Spencer_cohomology([ A0,An1,An2],1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H0G7#7%*(&%#dx G6#%!G\"\"\"%#^~GF,&&%#duG6#7%\"\"!F3F36#%!GF,*(&%#dzG6#%!GF,%#^~GF,&F /6#%!GF,*(&%#dyG6#%!GF,%#^~GF,&F/6#%!GF," }}}{EXCHG {PARA 0 "euc>" 0 " " {MPLTEXT 1 0 44 "H1:=Spencer_cohomology([A1,A0,An1,An2],1,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%# ^~GF)&%#dyG6#%!GF)7%*,&F+6#%!GF)%#^~GF)&F06#%!GF)%#^~GF)&&%#duG6#7%\" \"!FBFB6#%!GF)*,&F06#%!GF)%#^~GF)&%#dzG6#%!GF)%#^~GF)&F>6#%!GF)*,&F+6# %!GF)%#^~GF)&FK6#%!GF)%#^~GF)&F>6#%!GF)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 47 "H2:=Spencer_cohomology([A2,A1,A0,An1,An2],1,3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7%7#**%#0~G\"\"\"&%#dxG6#%!GF)% #^~GF)&%#dyG6#%!GF)7#*.F(\"\"\"&F+6#%!GF)%#^~GF)&F06#%!GF)%#^~GF)&%#dz G6#%!GF)7#*0&F+6#%!GF)%#^~GF)&F06#%!GF)%#^~GF)&F?6#%!GF)%#^~GF)&&%#duG 6#7%\"\"!FUFU6#%!GF)" }}}}{SECT 1 {PARA 262 "" 0 "" {TEXT -1 8 "The cu rl" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 28 "coord_init([x,y,z],[u,v,w]):" }}}{EXCHG {PARA 0 "eucl id>" 0 "" {MPLTEXT 1 0 59 "omega:=v_zip([u[0,0,0],v[0,0,0],w[0,0,0]],[ Dx,Dy,Dz],plus);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,(*&&%\"u G6%\"\"!F*F*\"\"\"&%#DxG6#%!GF+F+*&&%\"vGF)F+&%#DyG6#%!GF+F+*&&%\"wGF) F+&%#DzG6#%!GF+F+" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 33 "D elta:=coeff_list(dH(omega),all);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &DeltaG7%,&&%\"vG6%\"\"\"\"\"!F+F*&%\"uG6%F+F*F+!\"\",&&%\"wGF)F*&F-6% F+F+F*F/,&&F2F.F*&F(F4F/" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tableau(Delta,3):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G7$7%%(tableauG%'euclidG7\"7(7%\"\"!%\"yGF,7%F-%\"xGF,7%F,% \"zGF-7%F1F,F/7%F/F,F,7%F,F,F1" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]:" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3]:" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A3:=prA[4]:" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 21 "An1:=full_tableau(0):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 22 "An2:=full_tableau(-1):" }}}{EXCHG {PARA 0 "eucli d > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 17 "B:=[D_x,D_y,D_z]:" }}}{EXCHG {PARA 0 "euclid > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 30 "A_seq:=tableau_sequence(A0,B):" }}}{EXCHG {PARA 0 "euclid > " 0 " " {MPLTEXT 1 0 15 "t:=nops(A1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"tG\"#5" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 25 "Cartan _characters(A_seq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"$\"\"#\" \"\"\"#5" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 41 "H0:=Spen cer_cohomology([A0,An1,An2],1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#H0G7#7%,&*(&%#dxG6#%!G\"\"\"%#^~GF-&&%#dwG6#7%\"\"!F4F46#%!GF-F-*(&% #dzG6#%!GF-%#^~GF-&&%#duGF26#%!GF-!\"\",&*(&%#dyG6#%!GF-%#^~GF-&F06#%! GF-F-*(&F96#%!GF-%#^~GF-&&%#dvGF26#%!GF-FB,&*(&F*6#%!GF-%#^~GF-&FS6#%! GF-FB*(&FF6#%!GF-%#^~GF-&F>6#%!GF-F-" }}}{EXCHG {PARA 0 "euclid>" 0 " " {MPLTEXT 1 0 44 "H1:=Spencer_cohomology([A1,A0,An1,An2],1,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%# ^~GF)&&%#duG6#7%\"\"!F4F46#%!GF)7#,(*,&F+6#%!GF)%#^~GF)&%#dyG6#%!GF)%# ^~GF)&&%#dwGF26#%!GF)F)*,&F+6#%!GF)%#^~GF)&%#dzG6#%!GF)%#^~GF)&&%#dvGF 26#%!GF)!\"\"*,&F?6#%!GF)%#^~GF)&FN6#%!GF)%#^~GF)&F06#%!GF)F)" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 47 "H2:=Spencer_cohomology( [A2,A1,A0,An1,An2],1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7%7# **%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7%\"\"!F4F46#%!GF)7#*.F(\"\" \"&F+6#%!GF)%#^~GF)&%#dyG6#%!GF)%#^~GF)&F06#%!GF)7#*2F(F9&F+6#%!GF)%#^ ~GF)&F?6#%!GF)%#^~GF)&%#dzG6#%!GF)%#^~GF)&F06#%!GF)" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 261 14 "The divergence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 32 "coord _init([x,y,z],[u,v,w],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame ~name:~eucG" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 37 "Delta:=[u[ 1,0,0]+v[0,1,0]+ w[0,0,1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Delt aG7#,(&%\"uG6%\"\"\"\"\"!F+F*&%\"vG6%F+F*F+F*&%\"wG6%F+F+F*F*" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tableau( Delta,3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G7$7%%(tableauG%$eucG7\"7*7% %\"yG\"\"!F-7%%\"zGF-F-7%F-%\"xGF-7%F1F-,$F/!\"\"7%F-F-F,7%F-F/F-7%F-F ,F37%F-F-F1" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[ 2]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3]:" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A3:=prA[4]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An1:=full_tableau(0):" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 22 "An2:=full_tableau(-1):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 17 "B:=[D_x,D_y,D_z]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "A_seq:=tableau_sequence(A0,B):" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 15 "t:=nops(A1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"tG\"#:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Cartan_ch aracters(A_seq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"$F$\"\"#\"#: " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "H0:=Spencer_cohomol ogy([A0,An1,An2],1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H0G7#7#,( *(&%#dxG6#%!G\"\"\"%#^~GF-&&%#duG6#7%\"\"!F4F46#%!GF-F-*(&%#dyG6#%!GF- %#^~GF-&&%#dvGF26#%!GF-F-*(&%#dzG6#%!GF-%#^~GF-&&%#dwGF26#%!GF-F-" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 44 "H1:=Spencer_cohomology([A1 ,A0,An1,An2],1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~ G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7%\"\"!F4F46#%!GF)7#*.F(\"\"\"&F+6 #%!GF)%#^~GF)&%#dyG6#%!GF)%#^~GF)&F06#%!GF)" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 47 "H2:=Spencer_cohomology([A2,A1,A0,An1,An2],1,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G7%7#**%#0~G\"\"\"&%#dxG6#%!G F)%#^~GF)&&%#duG6#7%\"\"!F4F46#%!GF)7#*.F(\"\"\"&F+6#%!GF)%#^~GF)&%#dy G6#%!GF)%#^~GF)&F06#%!GF)7#*2F(F9&F+6#%!GF)%#^~GF)&F?6#%!GF)%#^~GF)&%# dzG6#%!GF)%#^~GF)&F06#%!GF)" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 262 14 "Pommaret p. 94" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 116 "This is a nice example of how the poor choice of basis B can lead to the w rong conclusion regarding involutivity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "coord_frame([x1,x2,x3,x 4,x5],[u],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~eucG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Delta:=[u[0,0,0,1,1] - \+ u[1,0,1,0,0],u[0,0,1,0,1] - u[1,1,0,0,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u[0,0,2,0,0] - u[0,1,0,1,0], u[0,1,0,0,1] - u[2,0,0,0 ,0], u[0,1,1,0,0] -" }}{PARA 0 "euc>" 0 "" {MPLTEXT 1 0 42 "u[1,0,0,1, 0], u[0,2,0,0,0] -u[1,0,1,0,0]];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% &DeltaG7(,&&%\"uG6'\"\"!F*F*\"\"\"F+F+&F(6'F+F*F+F*F*!\"\",&&F(6'F*F*F +F*F+F+&F(6'F+F+F*F*F*F.,&&F(6'F*F*\"\"#F*F*F+&F(6'F*F+F*F+F*F.,&&F(6' F*F+F*F*F+F+&F(6'F7F*F*F*F*F.,&&F(6'F*F+F+F*F*F+&F(6'F+F*F*F+F*F.,&&F( 6'F*F7F*F*F*F+F,F." }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA :=symbol_to_pr_tableau(Delta,1):" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1]:" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "nops(prA[2][2]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B1:=[vect(x1), vect(x2), vect(x3), vect(x4), vect(x5)]:" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B2:=[vect(x1), vect(x3), vect(x2), vect(x4), vect(x5)]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B3:=[vect(x5), vect(x4), vect(x3), vect(x2), vect(x1) ]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "A_seq1:=tableau_s equence(A0,B1):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cart an_characters(A_seq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\" \"F%F%F%\"#>" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "A_seq2:=tableau_sequence (A0,B2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_char acters(A_seq2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\"#\"\"!\" \"\"F'\"#=" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "A_seq3:=tableau_sequence (A0,B3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_char acters(A_seq3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\"%\"\"!F& F&\"#8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 54 "symbol_to_symbol_matrix(Delta, [xi1,xi2,x i3,xi4,xi5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7#,&*&% $xi1G\"\"\"%$xi3GF+!\"\"*&%$xi4GF+%$xi5GF+F+7#,&*&F*\"\"\"%$xi2GF+F-*& F,F4F0F4F+7#,&*&F5F4F/F4F-*$)F,\"\"#F4F+7#,&*$)F*F " 0 "" {MPLTEXT 1 0 38 "coord_frame([x1,x2,x3,x4 ,x5],[u],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 172 "Delta:=[u[0,0,0,1,1] - u[ 2,0,0,0,0], u[0,0,1,0,1] -u[1,0,0,1,0],u[0,1,0,0,1] -u[1,0,1,0,0],u[0, 0,0,2,0] -u[0,1,0,0,1],u[0,0,1,1,0]-u[1,1,0,0,0], u[0,0,2,0,0]-u[0,1,0 ,1,0]];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&DeltaG7(,&&%\"uG6'\"\"!F *F*\"\"\"F+F+&F(6'\"\"#F*F*F*F*!\"\",&&F(6'F*F*F+F*F+F+&F(6'F+F*F*F+F* F/,&&F(6'F*F+F*F*F+F+&F(6'F+F*F+F*F*F/,&&F(6'F*F*F*F.F*F+F6F/,&&F(6'F* F*F+F+F*F+&F(6'F+F+F*F*F*F/,&&F(6'F*F*F.F*F*F+&F(6'F*F+F*F+F*F/" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tableau( Delta,1):" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1]:" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 16 "nops(prA[2][2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#8" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B1:=[vect(x1), \+ vect(x2), vect(x3), vect(x4), vect(x5)]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B2:=[vect(x1), vect(x3), vect(x2), vect(x4), vec t(x5)]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 55 "B3:=[vect(x5 ), vect(x4), vect(x3), vect(x2), vect(x1)]:" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 32 "A_seq1:=tableau_sequence(A0,B1):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_characters(A_seq1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\"$\"\"!F&\"\"\"\"#;" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 32 "A_seq2:=tableau_sequence(A0,B2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_characters(A_seq2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\"#\"\"\"\"\"!F&\"#<" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 32 "A_seq3:=tableau_sequence(A0,B3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Cartan_characters(A_seq3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7'\"\"&\"\"#\"\"\"F&\"\"!\"#;" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "euc > " 0 " " {MPLTEXT 1 0 54 "symbol_to_symbol_matrix(Delta, [xi1,xi2,xi3,xi4,xi5 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7#,&*$)%$xi1G\"\" #\"\"\"!\"\"*&%$xi4G\"\"\"%$xi5GF1F17#,&*&F+F1F0F-F.*&%$xi3GF1F2F-F17# ,&*&F+F-F7F-F.*&%$xi2GF1F2F-F17#,&F;F.*$)F0F,F-F17#,&*&F+F-F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 263 30 "A Simple System of Finite Type" }}{PARA 0 "" 0 "" {TEXT 256 61 "A system is of finite type if some prolonged tableau is 0. " }}{PARA 0 "" 0 "" {TEXT 256 103 "In this example we see that one of the Spencer groups \+ is non-zero in degree where it must vanish for " }}{PARA 0 "" 0 "" {TEXT 256 29 "the tableau to be involutive." }}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 22 "coord_init([x,y],[u]):" }}}{EXCHG {PARA 0 "euc lid>" 0 "" {MPLTEXT 1 0 24 "Delta:=[u[4,0], u[0,4]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG7$&%\"uG6$\"\"%\"\"!&F'6$F*F)" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tableau(Delt a,4):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G7$7%%(tableauG%'euclidG7\"7% 7#*&)%\"xG\"\"$\"\"\"%\"yG\"\"\"7#*&F.F2)F1F/F07#*&)F.\"\"#F0)F1F9F0" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G7$7%%(tableauG%'euclidG7\"7$7#*& )%\"xG\"\"#\"\"\")%\"yG\"\"$F07#*&)F.F3F0)F2F/F0" }}}{EXCHG {PARA 0 "e uclid > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G7$7%%(tableauG%'euclidG7\"7#7#*&)%\"xG\"\"$\"\"\" )%\"yGF/F0" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A3:=pr A[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G7$7%%(tableauG%'euclidG 7\"7#7#\"\"!" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 11 "A4:= prA[5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G7$7%%(tableauG%'eucli dG7\"7#7#\"\"!" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 21 "An 1:=full_tableau(3):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 21 "An2:=full_tableau(2):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 44 " H1:=Spencer_cohomology([A1,A0,An1,An2],1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7$ \"\"!F46#%!GF)7#*.F(\"\"\"&F+6#%!GF)%#^~GF)&%#dyG6#%!GF)%#^~GF)&F06#%! GF)" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 43 "H2:=Spencer_coh omology([A2,A1,A0,An1],1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G 7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)&&%#duG6#7$\"\"!F46#%!GF)7#*.F(\" \"\"&F+6#%!GF)%#^~GF)&%#dyG6#%!GF)%#^~GF)&F06#%!GF)" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 42 "H3:=Spencer_cohomology([A3,A2,A1,A0] ,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G7$7#**%#0~G\"\"\"&%#dx G6#%!GF)%#^~GF)&&%#duG6#7$\"\"!F46#%!GF)7#*.F(\"\"\"&F+6#%!GF)%#^~GF)& %#dyG6#%!GF)%#^~GF)&F06#%!GF)" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 42 "H4:=Spencer_cohomology([A4,A3,A2,A1],1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H4G7$7#**%#0~G\"\"\"&%#dxG6#%!GF)%#^~GF)& %#dyG6#%!GF)7#*0)%\"xG\"\"$\"\"\")%\"yGF7F8&F+6#%!GF)%#^~GF)&F06#%!GF) %#^~GF)&&%#duG6#7$\"\"!FH6#%!GF)" }}}}{SECT 1 {PARA 264 "" 0 "" {TEXT -1 15 "Killing Tensors" }}{PARA 0 "" 0 "" {TEXT 256 103 "We check tha t the symbols of the differential equations for rank 2 and rank 3 Killing tensors k." }}{PARA 0 "" 0 "" {TEXT 256 35 "tensors on R^2 \+ are of finite type." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 256 13 "Rank 2 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 23 "k_11 = \+ k_12=b, k_22=c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "coord_init([x,y], [a,b,c],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 66 "Delta:= [ a[1,0] , a[0,1] + 2 *b[ 1,0], b[0,1] + 2*c[1,0], c[0,1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&DeltaG7&&%\"aG6$\"\"\"\"\"!,&&F'6$F*F)F)&%\"bGF(\"\"#,&&F/F-F)&%\"cG F(F0&F4F-" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "prA:=symbol_ to_pr_tableau(Delta,2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G7$7%%(tableau G%$eucG7\"7$7%\"\"!,$%\"yG!\"#%\"xG7%F-F0F," }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#A1G7$7%%(tableauG%$eucG7\"7#7%,$*$)%\"yG\"\"#\"\"\"\"\"%,$*&%\"xG \"\"\"F/F6!\"%*$)F5F0F1" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G7$7%%(tableau G%$eucG7\"7#7%\"\"!F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 256 7 "Rank 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 34 "k_111=a, k_112=b, k_122=c, k_222=d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 " coord_init([x,y], [a,b,c,d],euc):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 78 "Delta:=[a[1,0], a[0,1] +3*b[1,0] , b[0,1] +c[1,0], 3* c[0,1] +d[1,0] , d[0,1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG 7'&%\"aG6$\"\"\"\"\"!,&&F'6$F*F)F)&%\"bGF(\"\"$,&&F/F-F)&%\"cGF(F),&&F 4F-F0&%\"dGF(F)&F8F-" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "p rA:=symbol_to_pr_tableau(Delta,3):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G 7$7%%(tableauG%$eucG7\"7%7&\"\"!,$%\"yG!\"\"%\"xGF,7&F,F,F.,$F0!\"$7&, $F.F3F0F,F," }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[ 2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G7$7%%(tableauG%$eucG7\"7$ 7&,$*$)%\"yG\"\"#\"\"\"\"\"$,$*&%\"xG\"\"\"F/F6!\"#*$)F5F0F1\"\"!7&F:F -F3,$F8F2" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G7$7%%(tableauG%$eucG7\"7#7& *$)%\"yG\"\"$\"\"\",$*&%\"xG\"\"\")F.\"\"#F0!\"\"*&)F3F6F0F.F4,$*$)F3F /F0F7" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A3:=prA[4];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G7$7%%(tableauG%$eucG7\"7#7&\"\" !F,F,F," }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 44 "M:=symbol_to _symbol_matrix(Delta,[xi1,xi2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"MG-%'matrixG6#7'7&%$xi1G\"\"!F+F+7&%$xi2G,$F*\"\"$F+F+7&F+F-F*F+7&F+ F+,$F-F/F*7&F+F+F+F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT 264 22 "Maxwell's Equations V1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 54 "First we construc t the symbol for Maxwell's equation." }}{PARA 256 "" 0 "" {TEXT 256 46 "Then we apply the Cartan test for involutivity" }}{PARA 256 "" 0 " " {TEXT 256 36 "We compute the Spencer cohomology ." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 53 "coo rd_frame([x1,x2,x3,x4],[psi1,psi2,psi3,psi4],euc);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%0frame~name:~eucG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 123 "The following program creates the \+ symbol matrix for Maxwell's equations for a given background (contra variant) metric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Maxwell_symbol_matrix:=proc(m)" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 34 "local fr,n,i,Z,V,S, s1,s2,s3,s4,g;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fr:=_Vessiot_current_frame_name;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n:=frameBaseDimension();" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 32 " V:=frameIndependentVariables();" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 25 " S:=[seq(zeta.i,i=1..n)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Z:=form_to_tens(v_zip(S,V,plus,form));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "g:=array_to_tens(m,[[con_bas,con_bas],[]]);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s1:=g &tensor g;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s2:=skewtrize(s1,[2,4]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "s3:=contract_indices(s2&tensor Z,[[4,5]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "s4:=contract_indices(s3 &tensor Z, [[3,4] ]); " }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 18 "tens_to_array(s4);" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "eta :=evalm(linalg[diag](-1,1,1,1)):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "M:=Maxwell_symbol_matrix(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7&7&,(*$)%&zeta4G\"\"#\"\"\"#!\"\"F.* $)%&zeta3GF.F/F0*$)%&zeta2GF.F/F0,$*&F7\"\"\"%&zeta1GF:#F:F.,$*&F4F:F; F/F<,$*&F-F:F;F/F<7&F8,(F+F " 0 "" {MPLTEXT 1 0 60 "Sigma:=symbol_matrix_to_symbol( M,[zeta1,zeta2,zeta3,zeta4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Si gmaG7&,.&%%psi1G6&\"\"!F*F*\"\"##!\"\"F+&F(6&F*F*F+F*F,&F(6&F*F+F*F*F, &%%psi2G6&\"\"\"F5F*F*#F5F+&%%psi3G6&F5F*F5F*F6&%%psi4G6&F5F*F*F5F6,.& F(F4F6&F3F)F6&F3F/F6&F36&F+F*F*F*F,&F86&F*F5F5F*F,&F;6&F*F5F*F5F,,.&F( F9F6&F3FDF,&F8F)F6&F8F1F6&F8FBF,&F;6&F*F*F5F5F,,.&F(F " 0 "" {MPLTEXT 1 0 35 "prA:=symbol_to_pr_tablea u(Sigma,2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An1:=ful l_tableau(1):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An2:=f ull_tableau(0):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A0:= prA[1]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]: " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3]:" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc \+ > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "B:=[D_x1,D_x2,D_x3,D_x4]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "t:=nops(prA[2][2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"tG\"#l" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "A_seq:= tableau_sequence(prA[1],B):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Cartan_characters(A_seq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7 &\"#;\"#7\"\"(\"\"\"\"#l" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "16+ 2*12 +3*7 +4*1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#l" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "H0:=Spencer_cohomology([ A0,An1,An2],1,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#H0G7#7&,4**%#x 4G\"\"\"&%$dx1G6#%!GF*%#^~GF*&&%&dpsi1G6#7&\"\"!F5F5F56#%!GF*!\"\"**%# x1GF*&F,6#%!GF*%#^~GF*&&%&dpsi4GF36#%!GF*\"\"#**F)\"\"\"&%$dx2G6#%!GF* %#^~GF*&&%&dpsi2GF36#%!GF*F***%#x2GF*&FH6#%!GF*%#^~GF*&F@6#%!GF*!\"#** F)FF&%$dx3G6#%!GF*%#^~GF*&&%&dpsi3GF36#%!GF*F***%#x3GF*&Fgn6#%!GF*%#^~ GF*&F@6#%!GF*FZ**F:FF&%$dx4G6#%!GF*%#^~GF*&F16#%!GF*F8**FRFF&F[p6#%!GF *%#^~GF*&FM6#%!GF*F***FaoFF&F[p6#%!GF*%#^~GF*&F\\o6#%!GF*F*,4**FRFF&F, 6#%!GF*%#^~GF*&F16#%!GF*F8**F:FF&F,6#%!GF*%#^~GF*&FM6#%!GF*FD**F:FF&FH 6#%!GF*%#^~GF*&F16#%!GF*F8**FaoFF&FH6#%!GF*%#^~GF*&F\\o6#%!GF*F***F)FF &FH6#%!GF*%#^~GF*&F@6#%!GF*F***FaoFF&Fgn6#%!GF*%#^~GF*&FM6#%!GF*FZ**FR FF&Fgn6#%!GF*%#^~GF*&F\\o6#%!GF*F***F)FF&F[p6#%!GF*%#^~GF*&FM6#%!GF*FZ **FRFF&F[p6#%!GF*%#^~GF*&F@6#%!GF*F*,4**FRFF&F,6#%!GF*%#^~GF*&FM6#%!GF *F***FaoFF&F,6#%!GF*%#^~GF*&F\\o6#%!GF*F***F)FF&F,6#%!GF*%#^~GF*&F@6#% !GF*F***FRFF&FH6#%!GF*%#^~GF*&F16#%!GF*FZ**F:FF&FH6#%!GF*%#^~GF*&FM6#% !GF*F***FaoFF&Fgn6#%!GF*%#^~GF*&F16#%!GF*FZ**F:FF&Fgn6#%!GF*%#^~GF*&F \\o6#%!GF*F***F)FF&F[p6#%!GF*%#^~GF*&F16#%!GF*FZ**F:FF&F[p6#%!GF*%#^~G F*&F@6#%!GF*F*,4**FaoFF&F,6#%!GF*%#^~GF*&F16#%!GF*F***F:FF&F,6#%!GF*%# ^~GF*&F\\o6#%!GF*FZ**FaoFF&FH6#%!GF*%#^~GF*&FM6#%!GF*F8**FRFF&FH6#%!GF *%#^~GF*&F\\o6#%!GF*FD**F:FF&Fgn6#%!GF*%#^~GF*&F16#%!GF*F***FRFF&Fgn6# %!GF*%#^~GF*&FM6#%!GF*F8**F)FF&Fgn6#%!GF*%#^~GF*&F@6#%!GF*F8**F)FF&F[p 6#%!GF*%#^~GF*&F\\o6#%!GF*FD**FaoFF&F[p6#%!GF*%#^~GF*&F@6#%!GF*F8" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 12 "nops(H0[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 44 "H1:=Spencer_cohomology([A1,A0,An1,An2],1,2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%$dx1G6#%!GF)% #^~GF)&&%&dpsi1G6#7&\"\"!F4F4F46#%!GF)7#,:*.%#x2GF)&F+6#%!GF)%#^~GF)&% $dx2G6#%!GF)%#^~GF)&F06#%!GF)!\"\"*.%#x1GF)&F+6#%!GF)%#^~GF)&F@6#%!GF) %#^~GF)&&%&dpsi2GF26#%!GF)F)*.%#x3GF)&F+6#%!GF)%#^~GF)&%$dx3G6#%!GF)%# ^~GF)&F06#%!GF)FG*.FI\"\"\"&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&&%&dpsi 3GF26#%!GF)F)*.%#x4GF)&F+6#%!GF)%#^~GF)&%$dx4G6#%!GF)%#^~GF)&F06#%!GF) FG*.FIF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&&%&dpsi4GF26#%!GF)F)*.FXF `o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FS6#%!GF)F)*.F:F`o&F@6#%!GF)%#^~ GF)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)FG*.F_pF`o&F@6#%!GF)%#^~GF)&Fep6#%!GF) %#^~GF)&FS6#%!GF)F)*.F:F`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!G F)FG*.F_pF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Fjo6#%!GF)F)*.FXF`o&F hn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)FG" }}}{EXCHG {PARA 0 "eu c > " 0 "" {MPLTEXT 1 0 6 "H1[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 #**%#0~G\"\"\"&%$dx1G6#%!GF&%#^~GF&&&%&dpsi1G6#7&\"\"!F1F1F16#%!GF&" } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 12 "nops(H1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 43 "H2_3:=Spencer_cohomology([A0,An1,An2],3,3);" }}{PARA 8 "" 1 "" {TEXT -1 55 "Error, (in collect/series) too many levels of r ecursion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 62 "We need a more efficient algorithm for computing cohom ology!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 265 41 "Maxwel l's Equations V2 (In Coulomb Gauge)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 61 "This ti me we fix a gauge by adding equations to the symbol." }}{PARA 0 "" 0 "" {TEXT 256 87 "Run Maxwell' Equation V1 to initialize the coordi nates and compute the symbol Sigma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 68 "Gauge:= psi1[1,0,0,0] \+ + psi2[0,1,0,0] +psi3[0,0,1,0] +psi4[0,0,0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GaugeG,*&%%psi1G6&\"\"\"\"\"!F*F*F)&%%psi2G6&F*F)F*F *F)&%%psi3G6&F*F*F)F*F)&%%psi4G6&F*F*F*F)F)" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 49 "Gauge_Sigma:=[op(Sigma) ,op(pr_symbol([Gauge]) )];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,Gauge_SigmaG7*,.&%%psi1G6&\" \"!F*F*\"\"##!\"\"F+&F(6&F*F*F+F*F,&F(6&F*F+F*F*F,&%%psi2G6&\"\"\"F5F* F*#F5F+&%%psi3G6&F5F*F5F*F6&%%psi4G6&F5F*F*F5F6,.&F(F4F6&F3F)F6&F3F/F6 &F36&F+F*F*F*F,&F86&F*F5F5F*F,&F;6&F*F5F*F5F,,.&F(F9F6&F3FDF,&F8F)F6&F 8F1F6&F8FBF,&F;6&F*F*F5F5F,,.&F(FF5,*FMF5&F8F/F5FIF5FHF5,*&F;F )F5FRF5FQF5FPF5" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "prA: =symbol_to_pr_tableau(Gauge_Sigma,2):" }}}{EXCHG {PARA 0 "euc > " 0 " " {MPLTEXT 1 0 21 "An1:=full_tableau(1):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "An2:=full_tableau(0):" }}}{EXCHG {PARA 0 "euc > \+ " 0 "" {MPLTEXT 1 0 11 "A0:=prA[1]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A1:=prA[2]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 11 "A2:=prA[3]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "B:= [D_x1,D_x2,D_x3,D_x4]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "t:=nops(prA[2][2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG\"#b " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "A_seq:=tableau_sequ ence(prA[1],B):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Cart an_characters(A_seq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7&\"#:\"#6\" \"'\"\"!\"#b" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 41 "H0:=Spencer_cohomology([ A0,An1,An2],1,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#H0G7#7*,.**%#x 3G\"\"\"&%$dx1G6#%!GF*%#^~GF*&&%&dpsi1G6#7&\"\"!F5F5F56#%!GF*F***%#x1G F*&F,6#%!GF*%#^~GF*&&%&dpsi3GF36#%!GF*!\"\"**%#x2GF*&%$dx2G6#%!GF*%#^~ GF*&F?6#%!GF*F***F9\"\"\"&%$dx3G6#%!GF*%#^~GF*&F16#%!GF*F***F)FO&FQ6#% !GF*%#^~GF*&F?6#%!GF*F***%#x4GF*&%$dx4G6#%!GF*%#^~GF*&F?6#%!GF*F*,.**F EFO&F,6#%!GF*%#^~GF*&F16#%!GF*F***F9FO&F,6#%!GF*%#^~GF*&&%&dpsi2GF36#% !GF*FC**F9FO&FG6#%!GF*%#^~GF*&F16#%!GF*F***FEFO&FG6#%!GF*%#^~GF*&Fcp6# %!GF*F***F)FO&FQ6#%!GF*%#^~GF*&Fcp6#%!GF*F***F[oFO&F]o6#%!GF*%#^~GF*&F cp6#%!GF*F*,0**F9FO&F,6#%!GF*%#^~GF*&F16#%!GF*\"\"#**FEFO&F,6#%!GF*%#^ ~GF*&Fcp6#%!GF*F***F)FO&F,6#%!GF*%#^~GF*&F?6#%!GF*F***F[oFO&F,6#%!GF*% #^~GF*&&%&dpsi4GF36#%!GF*F***F9FO&FG6#%!GF*%#^~GF*&Fcp6#%!GF*F***F9FO& FQ6#%!GF*%#^~GF*&F?6#%!GF*F***F9FO&F]o6#%!GF*%#^~GF*&Fgt6#%!GF*F*,***F 9FO&F,6#%!GF*%#^~GF*&F16#%!GF*F***FEFO&FG6#%!GF*%#^~GF*&F16#%!GF*F***F )FO&FQ6#%!GF*%#^~GF*&F16#%!GF*F***F[oFO&F]o6#%!GF*%#^~GF*&F16#%!GF*F*, 2**F9FO&F,6#%!GF*%#^~GF*&F?6#%!GF*F***F)FO&FG6#%!GF*%#^~GF*&Fcp6#%!GF* F***FEFO&FG6#%!GF*%#^~GF*&F?6#%!GF*FC**FEFO&FQ6#%!GF*%#^~GF*&Fcp6#%!GF *F***F)FO&FQ6#%!GF*%#^~GF*&F?6#%!GF*F***F[oFO&FQ6#%!GF*%#^~GF*&Fgt6#%! GF*F***F[oFO&F]o6#%!GF*%#^~GF*&F?6#%!GF*FC**F)FO&F]o6#%!GF*%#^~GF*&Fgt 6#%!GF*F*,0**F[oFO&F,6#%!GF*%#^~GF*&F16#%!GF*F***F[oFO&FG6#%!GF*%#^~GF *&Fcp6#%!GF*F***F[oFO&FQ6#%!GF*%#^~GF*&F?6#%!GF*F***F9FO&F]o6#%!GF*%#^ ~GF*&F16#%!GF*F***FEFO&F]o6#%!GF*%#^~GF*&Fcp6#%!GF*F***F)FO&F]o6#%!GF* %#^~GF*&F?6#%!GF*F***F[oFO&F]o6#%!GF*%#^~GF*&Fgt6#%!GF*F`s,2**F9FO&F,6 #%!GF*%#^~GF*&Fcp6#%!GF*F***FEFO&FG6#%!GF*%#^~GF*&Fcp6#%!GF*F***F)FO&F G6#%!GF*%#^~GF*&F?6#%!GF*F***F[oFO&FG6#%!GF*%#^~GF*&Fgt6#%!GF*F***F)FO &FQ6#%!GF*%#^~GF*&Fcp6#%!GF*FC**FEFO&FQ6#%!GF*%#^~GF*&F?6#%!GF*F***F[o FO&F]o6#%!GF*%#^~GF*&Fcp6#%!GF*FC**FEFO&F]o6#%!GF*%#^~GF*&Fgt6#%!GF*F* ,.**F[oFO&F,6#%!GF*%#^~GF*&F16#%!GF*F***F9FO&F,6#%!GF*%#^~GF*&Fgt6#%!G F*FC**FEFO&FG6#%!GF*%#^~GF*&Fgt6#%!GF*F***F)FO&FQ6#%!GF*%#^~GF*&Fgt6#% !GF*F***F9FO&F]o6#%!GF*%#^~GF*&F16#%!GF*F***F[oFO&F]o6#%!GF*%#^~GF*&Fg t6#%!GF*F*" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 12 "nops(H0[1 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 44 "H1:=Spencer_cohomology([A1,A0,An1,An2],1,2); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#H1G7$7#**%#0~G\"\"\"&%$dx1G6#%! GF)%#^~GF)&&%&dpsi1G6#7&\"\"!F4F4F46#%!GF)7),:*.%#x2GF)&F+6#%!GF)%#^~G F)&%$dx2G6#%!GF)%#^~GF)&F06#%!GF)F)*.%#x1GF)&F+6#%!GF)%#^~GF)&F@6#%!GF )%#^~GF)&&%&dpsi2GF26#%!GF)!\"\"*.%#x3GF)&F+6#%!GF)%#^~GF)&%$dx3G6#%!G F)%#^~GF)&F06#%!GF)F)*.FH\"\"\"&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&&%& dpsi3GF26#%!GF)FV*.%#x4GF)&F+6#%!GF)%#^~GF)&%$dx4G6#%!GF)%#^~GF)&F06#% !GF)F)*.FHF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&&%&dpsi4GF26#%!GF)FV* .FXF`o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FR6#%!GF)FV*.F:F`o&F@6#%!GF) %#^~GF)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)F)*.F_pF`o&F@6#%!GF)%#^~GF)&Fep6#% !GF)%#^~GF)&FR6#%!GF)FV*.F:F`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6 #%!GF)F)*.F_pF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Fjo6#%!GF)FV*.FXF `o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)F),2*.FXF`o&F+6#%!GF) %#^~GF)&F@6#%!GF)%#^~GF)&F06#%!GF)F)*.F:F`o&F+6#%!GF)%#^~GF)&Fhn6#%!GF )%#^~GF)&F06#%!GF)FV*.FHF`o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&F06#%!G F)!\"#*.F:F`o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FR6#%!GF)!\"$*.FXF`o& F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)Fdy*.F_pF`o&F@6#%!GF)%#^~ GF)&Fhn6#%!GF)%#^~GF)&Ffq6#%!GF)Fgx*.FXF`o&F@6#%!GF)%#^~GF)&Fep6#%!GF) %#^~GF)&Ffq6#%!GF)FV*.F:F`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#% !GF)F),2*.F_pF`o&F+6#%!GF)%#^~GF)&F@6#%!GF)%#^~GF)&F06#%!GF)F)*.F:F`o& F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&F06#%!GF)FV*.F_pF`o&F@6#%!GF)%#^~GF )&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)FV*.FHF`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^ ~GF)&F06#%!GF)Fgx*.F:F`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&FR6#%!GF)F dy*.FXF`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Fjo6#%!GF)Fgx*.F_pF`o&F@6 #%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)Fdy*.F:F`o&Fhn6#%!GF)%#^~GF) &Fep6#%!GF)%#^~GF)&Fjo6#%!GF)FV,2*.FXF`o&F+6#%!GF)%#^~GF)&F@6#%!GF)%#^ ~GF)&FR6#%!GF)F)*.FHF`o&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&F06#%!GF)\" \"$*.F:F`o&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FR6#%!GF)\"\"#*.FXF`o&F+ 6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)F_dl*.F_pF`o&F+6#%!GF)%#^~G F)&Fhn6#%!GF)%#^~GF)&Ffq6#%!GF)F\\el*.FXF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF )%#^~GF)&Ffq6#%!GF)F)*.FHF`o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FR6#%! GF)F)*.FHF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)FV,2*.F_pF` o&F+6#%!GF)%#^~GF)&F@6#%!GF)%#^~GF)&FR6#%!GF)F)*.F_pF`o&F+6#%!GF)%#^~G F)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)F)*.FHF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%# ^~GF)&F06#%!GF)F_dl*.F:F`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&FR6#%!GF )F\\el*.FXF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Fjo6#%!GF)F\\el*.F_pF `o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)F_dl*.FHF`o&F@6#%!GF)% #^~GF)&Fep6#%!GF)%#^~GF)&FR6#%!GF)F)*.FHF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!G F)%#^~GF)&Fjo6#%!GF)F),2*.F_pF`o&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&F0 6#%!GF)F)*.FXF`o&F+6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&F06#%!GF)FV*.F_pF` o&F@6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&FR6#%!GF)F)*.FXF`o&F@6#%!GF)%#^~G F)&Fep6#%!GF)%#^~GF)&FR6#%!GF)FV*.FHF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%# ^~GF)&F06#%!GF)Fgx*.F:F`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&FR6#%!GF )Fgx*.FXF`o&Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Fjo6#%!GF)Fdy*.F_pF`o& Fhn6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)Fdy,2*.FHF`o&F+6#%!GF)%# ^~GF)&F@6#%!GF)%#^~GF)&F06#%!GF)F_dl*.F:F`o&F+6#%!GF)%#^~GF)&F@6#%!GF) %#^~GF)&FR6#%!GF)F_dl*.FXF`o&F+6#%!GF)%#^~GF)&F@6#%!GF)%#^~GF)&Fjo6#%! GF)F\\el*.F_pF`o&F+6#%!GF)%#^~GF)&F@6#%!GF)%#^~GF)&Ffq6#%!GF)F\\el*.F: F`o&F+6#%!GF)%#^~GF)&Fhn6#%!GF)%#^~GF)&Fjo6#%!GF)F)*.F:F`o&F+6#%!GF)%# ^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#%!GF)F)*.FHF`o&F@6#%!GF)%#^~GF)&Fhn6#%!GF )%#^~GF)&Fjo6#%!GF)FV*.FHF`o&F@6#%!GF)%#^~GF)&Fep6#%!GF)%#^~GF)&Ffq6#% !GF)FV" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "e uc > " 0 "" {MPLTEXT 1 0 6 "H1[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7#**%#0~G\"\"\"&%$dx1G6#%!GF&%#^~GF&&&%&dpsi1G6#7&\"\"!F1F1F16#%!GF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 12 "nops(H1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"( " }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 47 "H2:=Spencer_cohomol ogy([A2,A1,A0,An1,An2],1,3):" }}{PARA 8 "" 1 "" {TEXT -1 55 "Error, (i n collect/series) too many levels of recursion" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 258 18 "Einstein Equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 124 "The following program \+ creates the symbol matrix for Einstein's equations for a given back ground (contravariant) metric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 32 "Einstein_symbol_matrix:=proc( m) " }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 65 "local fr,n,X,Y,Z, S,V,g,s1,s2 ,s3,s4,s5,s6, index_list, i,j,t,A; " }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 32 "fr:=_Vessiot_current_frame_name;" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 24 "n:=frameBaseDimension();" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 24 "S:=[ seq(zeta.i,i=1..n)];" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 31 "V:=frameInde pendentVariables():" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 81 "Z:=form_to_te ns(v_zip(S,V,plus,form));g:=array_to_tens(m,[[con_bas,con_bas],[]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s1:=g &tensor g &tensor g;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s2:=skewtrize(s1,[2,4,6]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s3:=symmetrize(s2,[1,2]); " }} {PARA 0 ">" 0 "" {MPLTEXT 1 0 25 "s4:=symmetrize(s3,[3,4]);" }}{PARA 0 ">" 0 "" {MPLTEXT 1 0 38 " s5:=contract_indices2(s4, Z,[[6,1]]);" }} {PARA 0 ">" 0 "" {MPLTEXT 1 0 37 "s6:=contract_indices2(s5, Z,[[5,1]]) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "index_list:=[[1,1],[1,2],[1,3] ,[1,4],[2,2],[2,3],[2,4],[3,3],[3,4],[4,4]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(10,10);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1 to 10 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A[i,j]:=coeff_l ist(s6,[[op(index_list[i]), op(index_list[j])]]);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "od; od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm( 12*A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "e uc>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 74 "coord_frame([x1,x2,x3,x4],[g11,g12, g13,g14,g22,g23,g24,g33,g34, g44],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~eucG" }}} {EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 35 "eta:=evalm(linalg[diag](-1 ,1,1,1)):" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 31 "M:=Einstein_ symbol_matrix(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrix G6#7,7,\"\"!F*F*F*,&*$)%&zeta4G\"\"#\"\"\"!\"#*$)%&zeta3GF/F0F1,$*&%&z eta2G\"\"\"F4F8F/,$*&F7F0F.F8F/,&F,F1*$)F7F/F0F1,$*&F4F0F.F0F/,&F2F1F< F17,F*,&F,F8F2F8,$F6!\"\",$F:FDF*,$*&F4F0%&zeta1GF8FD,$*&F.F0FHF0FD,$* &F7F0FHF0F/F*FK7,F*FC,&F,F8FF*FIFFFYF:F6F*,&F " 0 "" {MPLTEXT 1 0 16 "linal g[rank](M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 60 "Sigma:=symbol_matrix_to_symbol(M,[zet a1,zeta2,zeta3,zeta4]):" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "prA: =symbol_to_pr_tableau(Sigma,2):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "B:= [D_x1,D_x2,D_x3,D_x4]:" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 19 "t:=nops(prA[2][2 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG\"$k\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "A_seq:=tableau_sequence(prA[1],B):" } }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "Cartan_characters(A_seq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7&\"#S\"#I\"#;\"\"%\"$k\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "40 +30*2 +16*3 +4*4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$k\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 19 "updated 01/23/03:IA" }{MPLTEXT 1 0 1 " " }{TEXT -1 0 "" }}}{MARK "12 \+ 0 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }