{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "_cstyle1" -1 201 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle2" -1 202 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 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 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "_pstyle1 " -1 200 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 201 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 201 397 " This is the Maple f ile that I used to compute the prolongations of the exterior different ial\nsystem whose integral manifolds model the surfaces in 3-space wit h the property that \nthe principal curvatures are contstant along the ir principal curves. You need to read the \ntreatment of this example in Lecture 6 to see what things mean.\n The first thing we do is lo ad in the difforms package. " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 15 "with(difforms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%#&^G%\" dG%(defformG%)formpartG%'parityG%+scalarpartG%)simpformG%(wdegreeG" }} }{EXCHG {PARA 200 "" 0 "" {TEXT 201 146 "Now I want to set up the stru cture equations and define the variables and forms \nthat I want to us e. Note that I have to tell it explicitly that " }{TEXT 202 1 "w" } {TEXT 201 22 " is skew-symmetric. \n" }{MPLTEXT 1 0 0 "" }{TEXT 201 176 "(The extra single quotes around things are just so that I can rep eat this assignment if\nI decide to change my notation. It's a Maple \+ thing, so don't worry about it right now.)" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 443 "omega = table(): kappa := table(): \nq := tabl e(): theta := table():\nassign('omega[1,2]'=-omega[2,1],\n 'ome ga[1,3]'=-omega[3,1],\n 'omega[2,3]'=-omega[3,2],\n ''omeg a[i,i]'=0'$'i'=1..3);\ndefform(\nomega=1,theta=1,kappa=0,q=0,\n''d(ome ga[i])'\n =sum('-&^(omega[i,j],omega[j])','j'=1..3)'$'i'=1..3,\n'd( omega[2,1])'=-omega[2,3] &^ omega[3,1],\n'd(omega[3,1])'=-omega[3,2] & ^ omega[2,1],\n'd(omega[3,2])'=-omega[3,1] &^ omega[1,2]);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "d(omega[3,2]-kappa[2]*omega [2]):\nsubs(omega[3,1]=kappa[1]*omega[1]+theta[1],omega[3,2]=kappa[2]* omega[2]+theta[2],\nd(kappa[2])=(kappa[1]-kappa[2])*q[2]*omega[1]+thet a[5],\nd(kappa[1])=(kappa[1]-kappa[2])*q[1]*omega[2]+theta[3],%):\nsim pform(%);\nd(omega[3,1]-kappa[1]*omega[1]):\nsubs(omega[3,1]=kappa[1]* omega[1]+theta[1],omega[3,2]=kappa[2]*omega[2]+theta[2],\nd(kappa[2])= (kappa[1]-kappa[2])*q[2]*omega[1]+theta[5],\nd(kappa[1])=(kappa[1]-kap pa[2])*q[1]*omega[2]+theta[3],%):\nsimpform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.-%#&^G6$&%&thetaG6#\"\"\"&%&omegaG6$\"\"#F*F**&&%&kap paG6#F.F*-F%6$&F(F2&F,6#\"\"$F*!\"\"*&,&&F1F)F*F0F9F*-F%6$&F,F)F+F*F** &)F0F.F*-F%6$&F,F2F6F*F9*(,&FF.F/*(,&F2F/F5F.F. &%\"qGF4F.-F%6$F8F=F.F.-F%6$&F(F?F=F/" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 97 "Set up the ideal on the space of integral elements. We \+ list the 1-forms that generate the ideal:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 288 "Ideal[1] := [\ntheta[0] = omega[3],\ntheta[1] = \+ omega[3,1]-kappa[1]*omega[1],\ntheta[2] = omega[3,2]-kappa[2]*omega[2] ,\ntheta[3] = d(kappa[1])-(kappa[1]-kappa[2])*q[1]*omega[2],\ntheta[4] = omega[2,1] - q[1]*omega[1]-q[2]*omega[2],\ntheta[5] = d(kappa[2])-( kappa[1]-kappa[2])*q[2]*omega[1] \n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&IdealG6#\"\"\"7(/&%&thetaG6#\"\"!&%&omegaG6#\"\"$/&F+F&,&&F/6$F 1F'F'*&&%&kappaGF&F'&F/F&F'!\"\"/&F+6#\"\"#,&&F/6$F1F?F'*&&F9F>F'&F/F> F'F;/&F+F0,&-%\"dG6#F8F'*(,&F8F'FDF;F'&%\"qGF&F'FEF'F;/&F+6#\"\"%,(&F/ 6$F?F'F'*&FNF'F:F'F;*&&FOF>F'FEF'F;/&F+6#\"\"&,&-FJ6#FDF'*(FMF'FYF'F:F 'F;" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 186 "Since I'm going to nee d to back substitute to do computations modulo\nthe theta[i], I go ahe ad and figure out how to express everything in terms\nof the theta[i] \+ and omega[1] and omega[2]:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 138 "[op(solve(\{op(Ideal[1])\},\n\{omega[3],omega[3,1],omega[3,2], omega[2,1],\n d(kappa[1]),d(kappa[2])\} ) ) ]:\nReverseIdeal[1] := map (simpform,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%-ReverseIdealG6#\" \"\"7(/&%&omegaG6$\"\"$F',&*&&%&kappaGF&F'&F+F&F'F'&%&thetaGF&F'/&F+6$ F-\"\"#,&*&&F16#F8F'&F+F " 0 "" {MPLTEXT 1 -1 110 "d(subs(Ideal[1],theta[3])):\nsu bs(ReverseIdeal[1],%): subs(['theta[i]=0'$'i'=0..5],%):\nTheta[3] := s impform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG6#\"\"$*&,&&% &kappaG6#\"\"\"!\"\"&F+6#\"\"#F-F--%#&^G6$-%\"dG6#&%\"qGF,&%&omegaGF0F -" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 110 "d(subs(Ideal[1],th eta[5])):\nsubs(ReverseIdeal[1],%): subs(['theta[i]=0'$'i'=0..5],%):\n Theta[5] := simpform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG 6#\"\"&*&,&&%&kappaG6#\"\"\"!\"\"&F+6#\"\"#F-F--%#&^G6$-%\"dG6#&%\"qGF 0&%&omegaGF,F-" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 110 "d(sub s(Ideal[1],theta[4])):\nsubs(ReverseIdeal[1],%): subs(['theta[i]=0'$'i '=0..5],%):\nTheta[4] := simpform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG6#\"\"%,(*&,(*&&%&kappaG6#\"\"#\"\"\"&F-6#F0F0F0*$)&%\"qG F2F/F0F0*$)&F6F.F/F0F0F0-%#&^G6$&%&omegaGF.&F>F2F0F0-F;6$-%\"dG6#F5F?! \"\"-F;6$-FC6#F9F=FE" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 358 "It fo llows that on an integral element d(q[1]) will be a multiple of omega[ 2] and d(q[2]) \nwill be a multiple of omega[1]. These two multiples \+ are related by a single equation\n(coming from annihlating Theta[4]), \+ so we get a solution in the form below (I have\nchosen q[3] so that th e formulas will be as symmetric as possible). Now we check\nthat the y work:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 156 "subs( \nd(q[ 1])=(q[3]+q[1]^2+kappa[1]*kappa[2]/2)*omega[2],\nd(q[2])=(q[3]-q[2]^2- kappa[1]*kappa[2]/2)*omega[1],\n[Theta[3],Theta[4],Theta[5]] ): \nsimp form(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 112 "Now we add the two 1-forms to the ide al with the extra parameter q[3] that\nparametrizes these integral ele ments:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 151 "Ideal[2] := [ op(Ideal[1]),\ntheta[6]=d(q[1])-(q[3]+q[1]^2+kappa[1]*kappa[2]/2)*omeg a[2],\ntheta[7]=d(q[2])-(q[3]-q[2]^2-kappa[1]*kappa[2]/2)*omega[1] \n] ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%&IdealG6#\"\"#7*/&%&thetaG6#\" \"!&%&omegaG6#\"\"$/&F+6#\"\"\",&&F/6$F1F5F5*&&%&kappaGF4F5&F/F4F5!\" \"/&F+F&,&&F/6$F1F'F5*&&F;F&F5&F/F&F5F=/&F+F0,&-%\"dG6#F:F5*(,&F:F5FDF =F5&%\"qGF4F5FEF5F=/&F+6#\"\"%,(&F/6$F'F5F5*&FNF5F " 0 " " {MPLTEXT 1 -1 150 "[op(solve(\{op(Ideal[2])\},\n\{omega[3],omega[3,1 ],omega[3,2],omega[2,1],\nd(kappa[1]),d(kappa[2]),d(q[1]),d(q[2])\}))] :\nReverseIdeal[2] := map(simpform,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%-ReverseIdealG6#\"\"#7*/&%&omegaG6$\"\"$\"\"\",&*&&%&kappaG6# F.F.&F+F3F.F.&%&thetaGF3F./&F+6$F-F',&*&&F2F&F.&F+F&F.F.&F6F&F./&F+6#F -&F66#\"\"!/&F+6$F'F.,(&F66#\"\"%F.*&&%\"qGF3F.F4F.F.*&&FNF&F.F=F.F./- %\"dG6#F1,&*&,&*&FMF.F1F.F.*&FMF.F " 0 "" {MPLTEXT 1 -1 110 "d(subs(Ideal[2],theta[6])):\nsubs(ReverseIdeal[2],%): subs(['thet a[i]=0'$'i'=0..7],%):\nTheta[6] := simpform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG6#\"\"',&*&,(*&&%\"qG6#\"\"#\"\"\"&F-6#\"\"$F 0F0*&F,F0)&F-6#F0F/F0F0*&#F0F/F0*&F,F0)&%&kappaGF7F/F0F0F0F0-%#&^G6$&% &omegaGF.&FBF7F0F0-F?6$-%\"dG6#F1FA!\"\"" }}}{EXCHG {PARA 201 "> " 0 " " {MPLTEXT 1 -1 110 "d(subs(Ideal[2],theta[7])):\nsubs(ReverseIdeal[2] ,%): subs(['theta[i]=0'$'i'=0..7],%):\nTheta[7] := simpform(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG6#\"\"(,&*&,(*&#\"\"\"\"\"#F -*&&%\"qG6#F-F-)&%&kappaG6#F.F.F-F-!\"\"*&F0F-&F16#\"\"$F-F-*&F0F-)&F1 F6F.F-F7F--%#&^G6$&%&omegaGF6&FCF2F-F--F@6$-%\"dG6#F9FDF7" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 228 "It follows that on an integral elemen t d(q[3]) is a linear combination of omega[1]\nand omega[2] and these \+ coefficients are determined by the requirement that it\nannihilate The ta[6] and Theta[7]. Here, we check, that this works:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 161 "subs(\nd(q[3]) \n= -(q[2]*q[3]+ q[2]*q[1]^2+1/2*kappa[1]^2*q[2])*omega[1]\n +(q[1]*q[3]-q[1]*q[2]^2-1 /2*kappa[2]^2*q[1])*omega[2],\n[Theta[6],Theta[7]]):simpform(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}}{EXCHG {PARA 200 "" 0 " " {TEXT 201 81 "Now we add this expression that vanishes on all integr al elements into the ideal:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 154 "Ideal[3] := [op(Ideal[2]),\ntheta[8]=d(q[3])\n+(q[2]*q[3]+q[2] *q[1]^2+1/2*kappa[1]^2*q[2])*omega[1]\n-(q[1]*q[3]-q[1]*q[2]^2-1/2*kap pa[2]^2*q[1])*omega[2]\n];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%&Idea lG6#\"\"$7+/&%&thetaG6#\"\"!&%&omegaGF&/&F+6#\"\"\",&&F/6$F'F3F3*&&%&k appaGF2F3&F/F2F3!\"\"/&F+6#\"\"#,&&F/6$F'F?F3*&&F9F>F3&F/F>F3F;/&F+F&, &-%\"dG6#F8F3*(,&F8F3FDF;F3&%\"qGF2F3FEF3F;/&F+6#\"\"%,(&F/6$F?F3F3*&F NF3F:F3F;*&&FOF>F3FEF3F;/&F+6#\"\"&,&-FJ6#FDF3*(FMF3FYF3F:F3F;/&F+6#\" \"',&-FJ6#FNF3*&,(&FOF&F3*$)FNF?F3F3*&#F3F?F3*&FDF3F8F3F3F3F3FEF3F;/&F +6#\"\"(,&-FJ6#FYF3*&,(FeoF3*$)FYF?F3F;*&#F3F?F3FjoF3F;F3F:F3F;/&F+6# \"\"),(-FJ6#FeoF3*&,(*&FYF3FeoF3F3*&FYF3FgoF3F3*&FioF3*&FYF3)F8F?F3F3F 3F3F:F3F3*&,(*&#F3F?F3*&FNF3)FDF?F3F3F;*&FNF3FeoF3F3*&FNF3FepF3F;F3FEF 3F;" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 158 "[op(solve(\{op(I deal[3])\},\n\{omega[3],omega[3,1],omega[3,2],omega[2,1],\nd(kappa[1]) ,d(kappa[2]),d(q[1]),d(q[2]),d(q[3])\}))]:\nReverseIdeal[3] := map(sim pform,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%-ReverseIdealG6#\"\"$7 +/&%&omegaG6$F'\"\"\",&*&&%&kappaG6#F-F-&F+F2F-F-&%&thetaGF2F-/&F+6$F' \"\"#,&*&&F16#F9F-&F+F=F-F-&F5F=F-/&F+F&&F56#\"\"!/&F+6$F9F-,(&F56#\" \"%F-*&&%\"qGF2F-F3F-F-*&&FNF=F-F>F-F-/-%\"dG6#F0,&*&,&*&FMF-F0F-F-*&F MF-FF-F-&F5F&F-/-FS6#F<,&*&,&*&FPF-F0F-F-*&FPF-FF-F-&F56#\"\"'F-/-FS6#FP,&*&,(FgoF-*$)FPF9F-FZ*&#F-F9F-F\\pF-FZF-F 3F-F-&F56#\"\"(F-/-FS6#Fgo,(*&,(*&#F-F9F-*&FMF-)FF-F-*&,(*&FPF-FgoF-FZ*&FPF-FioF-FZ*&#F-F9F-*&FPF-)F0 F9F-F-FZF-F3F-F-&F56#\"\")F-" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 227 "Now we check on the exterior derivatives of this 1-form theta[8], \nwhich has to be added to the ideal Ideal[3]. We know that d(theta [i]) is in\nIdeal[3] for i=0,1,2,3,4,5,6,7 for theoretical reasons, so we just need to compute" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 -1 110 "d(subs(Ideal[3],theta[8])):\nsubs(ReverseIdeal[3],%): subs(['thet a[i]=0'$'i'=0..8],%):\nTheta[8] := simpform(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&ThetaG6#\"\")\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 201 143 "It follow s that Ideal[3] is a Frobenius system, so that the underlying 11-manif old is\nfoliated by 2-dimensional integral manifolds of Ideal[3]." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "28 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }