{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 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 "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 "Maple Plot" 0 13 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 852 "dismantle:=proc(Phi ,vars::\{set,list\})\n local Psi,n,V,d,A,i,j,b;\n if not type(Phi,po lynom(anything,vars)) then\n ERROR(`Phi is expected to be a polynom with respect to `,vars,` but dismantle received `,Phi, vars)\n fi;\n Psi:=collect(Phi,vars);\n if not degree(Psi,vars)<=2 then\n ERR OR(`Phi is expected to have degree at most 2, but dismantle received ` ,Phi, vars)\n fi;\n n:=nops(vars);\n if type(vars,set) then \n V :=[op(vars)]\n else \n V:=[op(\{op(vars)\})]\n fi;\n Psi:=collec t(subs(seq(v=t*v,v=V),Psi),t);\n for d from 0 to 2 do\n Psi.d:=coe ff(Psi,t,d)\n od;\n A:=array(1..n,1..n,symmetric);\n for i to n do \n A[i,i]:=coeff(Psi2,V[i],2);\n for j from i+1 to n do \n \+ A[i,j]:=coeff(coeff(Psi2,V[i],1),V[j],1)/2\n od\n od;\n b:=array( 1..n,1..1);\n for i to n do\n b[i,1]:=coeff(Psi1,V[i],1)\n od;\n \+ eval(A),eval(b),Psi0,V\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "X:=array(1..2,1..1,[[x],[y]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"XG-%'MATRIXG6#7$7#%\"xG7#%\"yG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Phi:=7/8*x^2-3/8*x*y+sqrt(3)/8*x*z+23/32*y^2+3*sqrt(3 )/16*y*z+29/32*z^2-2*sqrt(3)*x+sqrt(3)*y-z+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG,6*$%\"xG\"\"##\"\"(\"\")*&F'\"\"\"%\"yGF-#!\"$F +*(\"\"$#F-F(F'F-%\"zGF-#F-F+*$F.F(#\"#B\"#K*(F2F3F.F-F4F-#F2\"#;*$F4F (#\"#HF9*&F2F3F'F-!\"#*&F2F3F.F-F-F4!\"\"F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Dec:=dismantle(Phi,[x,y,z]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$DecG6&-%'MATRIXG6#7%7%#\"\"(\"\")#!\"$\"#;,$*$\"\" $#\"\"\"\"\"##F5F07%F.#\"#B\"#K,$F2#F3F;7%F1F<#\"#HF;-F'6#7%7#,$F2!\"# 7#F27#!\"\"F37%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A:=Dec[1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "b:=De c[2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c:=Dec[3]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "linalg[jordan](A,P);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"!F)7%F)#F(\" \"#F)7%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(P);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,&*$\"\"$#\"\"\"\"\"# F+#F*\"\"%F,#F,F/,$F)F+7%#!\"$\"\")#F*F5\"\"!7%,&F,F,F)#F,F5,$F)#!\"\" F5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tPP:=evalm(transpos e(P) &* P);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$tPPG-%'MATRIXG6#7%7% ,(*$,&*$\"\"$#\"\"\"\"\"#F/#F.\"\"%F0F1F0#\"\"*\"#kF0*$,&F0F0F-#F0\"\" )F1F0,(F-F9#F.F6F0*&F8F0F.F/#!\"\"F:,(*&F,F0F.F/F/F0F0F-F97%F;#F0F3\" \"!7%F@FD#\"\"(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tPP:=m ap(radnormal,eval(tPP));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tPPG-%' MATRIXG6#7%7%,&#\"\"&\"\"#\"\"\"*$\"\"$#F.F-F.\"\"!,&#\"\"(\"\"%F.F/F1 7%F2#F.F6F27%F3F2F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for \+ j to 3 do \n f(j):=linalg[submatrix](P,1..3,j..j)\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "scalarproduct:=proc(v,w)\n local i ;\n add(v[i,1]*w[i,1],i=1..3)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e(1):=map(radnormal,evalm(f(1)/sqrt(scalarproduct(f(1 ),f(1)))),rationalized):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "for j from 2 to 3 do\n s:=evalm(scalarproduct(e(1),f(j))*e(1)); \n for k from 2 to j-1 do\n s:=evalm(s+scalarproduct(e(k),f(j))*e( k))\n od;\n e(j):=evalm(f(j)-s);\n e(j):=map(radnormal,evalm(e(j)/s qrt(scalarproduct(e(j),e(j)))),rationalized)\nod:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "P1:=linalg[augment](seq(e(j),j=1..3));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#P1G-%'MATRIXG6#7%7%,&*&,&\"$g\"\"\" \"*$\"\"$#F.\"\"#\"#kF1F0F1#F.\"#_*$F,F1#F0\"$3#F1,$*$,&#\"#X\"$/\"F.F /#!\"*F5F1!\"\"7%,&F6#!#:\"$;%F+F7#F0\"\"%,&F:#\"\"(\"\"'*&F;F1F0F1#F. F07%,&F6#\"# " 0 " " {MPLTEXT 1 0 42 "map(radnormal,evalm(transpose(P1) &* P1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"!F)7%F)F(F)7 %F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "newX:=array(1..3 ,1..1,[[newx],[newy],[newz]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%n ewXG-%'MATRIXG6#7%7#%%newxG7#%%newyG7#%%newzG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "XX:=evalm(P1&*newX):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "tosubs1:=\{x=XX[1,1],y=XX[2,1],z=XX[3,1]\}:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "newPhi:=map(radnormal,collec t(subs(tosubs1,Phi),\{newx,newy,newz\},distributed),rationalized);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%'newPhiG,.*$%%newxG\"\"#\"\"\"\"\"$F )*(,&\"$g\"F)*$F*#F)F(\"#kF/,&F.F)\"\"%F)F)F'F)#!\"\"\"#E*$%%newyGF(F/ *(,&#\"#X\"$/\"F)F.#!\"*\"#_F/F*F/%%newzGF)#\"\")F**$F@F(F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "center:=solve(\{diff(Phi,x), diff(Phi,y),diff(Phi,z)\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'centerG<%/%\"zG#\"\"\"\"\"#/%\"yG,$*$\"\"$F(#!\"\"F*/%\"xGF." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "newcenter:=solve(\{diff(new Phi,newx),diff(newPhi,newy),diff(newPhi,newz)\},\{newx,newy,newz\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*newcenterG<%/%%newzG,$*&,&\"$I\" \"\"\"*$\"\"$#F,\"\"#!#_F/F.F/#!\"\"\"#8/%%newxG,&*&,&\"#5F,F-\"\"%F/F .F/#F,F4*$F9F/#F;F4/%%newyG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "map(radnormal,subs(newcenter,tosubs1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%/%\"zG#\"\"\"\"\"#/%\"yG,$*$\"\"$F&#!\"\"F(/% \"xGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "tosubs2:=subs(ne wxx=newx,newyy=newy,newzz=newz,subs(newcenter,\{newxx=xi+newx,newyy=et a+newy,newzz=zeta+newz\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(tosu bs2G<%/%%newxG,(%#xiG\"\"\"*&,&\"#5F**$\"\"$#F*\"\"#\"\"%F0F/F0#F*\"#8 *$F,F0#F2F4/%%newzG,&%%zetaGF**&,&\"$I\"F*F.!#_F0F/F0#!\"\"F4/%%newyG% $etaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "newnewPhi:=map(rad normal,collect(subs(tosubs2,newPhi),\{xi,eta,zeta\},distributed));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*newnewPhiG,**$%$etaG\"\"##\"\"\"F(* $%#xiGF(F*!\"\"F**$%%zetaGF(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "tosubs:=map(radnormal,subs(tosubs2,tosubs1));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'tosubsG<%/%\"yG,,*&%#xiG\"\"\",&\"$I\"F+*$\"\"$ #F+\"\"#!#_F0#!\"$\"$/\"F.#!\"\"F1%$etaG#F/\"\"%*&F,F0%%zetaGF+#\"\"(F 5*(F,F0F/F0F " 0 "" {MPLTEXT 1 0 107 "surface:=subs(subs(xi=cos(p hi)*sin(theta),eta=sqrt(2)*sin(phi)*sin(theta),zeta=cos(theta),tosubs) ,[x,y,z]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(surfaceG7%,,**\"\"$# \"\"\"\"\"#-%$cosG6#%$phiGF*-%$sinG6#%&thetaGF*,&\"$I\"F**$F(F)!#_F)#F *\"#E*(F,F*F0F*F4F)#F(\"#_F6F**(F+F)-F1F.F*F0F*F)*&F4F)-F-F2F*#!\"$F<, ,F:#FB\"$/\"F6#!\"\"F+F=#F(\"\"%F?#\"\"(FE*(F4F)F(F)F@F*#F*F<,$*&,.F'! \")F:FBF6F7F=\"#yFL!\"'F?!\"*F*F(F)#FG\"$7$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot3d(surface,theta=0..Pi,phi=-Pi..Pi,scaling=con strained,labels=[x,y,z],axes=boxed);" }}{PARA 13 "" 1 "" {INLPLOT "6&- %%MESHG6#7;7;7%$\"3aiGM&pwuO\"!#<$!3tm\\(ox>?I#!#=$\"3#>!f(\\b/.=\"F+F (F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(7;7%$\"3R$[\"[$3h!o7F+$ !3!f,T\"**H[=@F.$\"3*Q`Xoz)H(4\"F+7%$\"3;t0%R\"pmZ7F+$!3zwc19'4\\[#F.$ \"3XDk<\"F+7%$\"3!)*3&>QxRR7F+$!3# y`Y&4)3kV$F.$\"3JPPv#f8^?\"F+7%$\"3ch`]Cn!\\D\"F+$!3sue;O'*4KOF.$\"3n, @Z3WrJ7F+7%$\"34>7x@(*Hy7F+$!3J2U,(oa3u$F.$\"3-X>X(4:WD\"F+7%$\"3$*\\x )e`#)zI\"F+$!35RC5&[i_v$F.$\"3^aRDX'o;F\"F+7%$\"34#3E@LK>M\"F+$!3(4(3! 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