{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 0 1 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 "" 0 1 0 0 0 0 0 
1 0 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 }{CSTYLE "" -1 266 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 9 0 0 0 0 0 0 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 "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 0 
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 0 0 0 0 0 0 3 0 0 }2 1 0 0 10 10 3 
6 3 30 0 0 -1 0 }{PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 
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 "Vessiot_Title3" 256 260 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 
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 "" 0 261 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 }}
{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 35 "Vessiot Tutorial: Gettin
g Started I" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 
257 "" 0 "" {TEXT -1 58 " an  brief introduction  to the software pack
age Vessiot. " }}{PARA 257 "" 0 "" {TEXT -1 63 "see the  Vessiot helpf
iles for more  information and examples. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(
Vessiot):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 14 "with(te
nsors):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 13 "with(Kosz
ul):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 56 "Use the Maple helpfiles more  information and examples. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?Vessiot
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?pullback;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" 
{TEXT 257 20 "Input, Output and Ar" }{TEXT -1 0 "" }{TEXT 267 8 "ithme
tic" }{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 
"" {TEXT 256 190 "Every  Vessiot Session  begins with the coord_init c
ommand to initialize the names of the independent and dependent variab
les. Note that the prompt changes to the name of  coordinate system." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "coord_init([x,y,z],[u],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%0frame~name:~eucG" }}}{PARA 0 "" 0 "" {TEXT 256 118 "Once a coordinat
e system is initialized, there are many ways to create  differential f
orms, vector fields and tensors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 155 "The easiest  way is to use the command \+
evalV. This is the Vessiot syntax parser which allows the user to use \+
  + for addition, * for scalar multiplication," }}{PARA 0 "" 0 "" 
{TEXT 256 31 "&w for wedge and &t for tensor." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 255 "Outside the parser,  us
e  &plus for addition, &minus for subtraction, &mult for scalar multip
lication, &wedge for exterior product,  &tensor for tensor product.  I
t is important  to properly  group  operations with  parenthesis when \+
using these  commands." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 
0 "" {TEXT 256 84 "The  command  v_zip   zips together an list of coef
ficients and  forms, vectors....." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 90 "Output is  displayed in \+
the  Vessiot print   format. Use op  to view the internal format. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 34 "Create \+
some  simple vector fields." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "euc>" 0 "" {MPLTEXT 1 0 8 "X1:=D_x;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#X1G&%$D_xG6#%!G" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 7 "op(X1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%vectG
%$eucG7\"7#7$7#\"\"\"F+" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 
12 "X2:=vect(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X2G&%$D_xG6#%!G
" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 "X3:= (y&mult D_x) &
minus  (x &mult D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X3G,&*&%\"
yG\"\"\"&%$D_xG6#%!GF(F(*&%\"xGF(&%$D_yG6#F,F(!\"\"" }}}{EXCHG {PARA 
0 "euc > " 0 "" {MPLTEXT 1 0 26 "X3:=evalV( y*D_x - x*D_y);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#X3G,&*&%\"yG\"\"\"&%$D_xG6#%!GF(F(*&%\"xG
F(&%$D_yG6#F,F(!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 
"X4:= v_zip([a,b,c],[x,y,z], plus, vect);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#X4G,(*&%\"aG\"\"\"&%$D_xG6#%!GF(F(*&%\"bGF(&%$D_yG6#
F,F(F(*&%\"cGF(&%$D_zG6#F,F(F(" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT -1 30 "Create some differential forms" }}
{PARA 260 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 22 "omega1:= dx &wedge dy;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'omega1G*(&%#dxG6#%!G\"\"\"%#^~GF*&%#dyG6#F)F*" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 53 "omega2:= form([x,y]) &plus ((z^
2) &mult form([y,z]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'omega2G,&
*(&%#dxG6#%!G\"\"\"%#^~GF+&%#dyG6#F*F+F+**)%\"zG\"\"#\"\"\"&F.6#F*F+F,
F+&%#dzG6#F*F+F+" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 45 "ome
ga3:=v_zip([z^2,y^2,x^2],[dx,dy,dz],plus);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'omega3G,(*&)%\"zG\"\"#\"\"\"&%#dxG6#%!G\"\"\"F/*&)%
\"yGF)F*&%#dyG6#F.F/F/*&)%\"xGF)F*&%#dzG6#F.F/F/" }}}{PARA 260 "" 0 "
" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 26 "Special differential \+
forms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 15 "nu:=vol_form();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#nuG*,&%#dxG6#%!G\"\"\"%#^~GF*&%#dyG6#F)F*F+F*&%#dzG6#F)F*" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 15 "scalar_form(r);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%\"rG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 
0 6 "op(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%%formG%$eucG\"\"!7
#7$7\"%\"rG" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 19 "Create some tensors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 20 "T1:=tens(z,cov_bas);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G&%#dzG6#%!G" }}}{EXCHG {PARA 0 "
euc > " 0 "" {MPLTEXT 1 0 21 "T2:=tens(z,con_bas); " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#T2G&%$D_zG6#%!G" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "T3:
=dx &tensor dx &tensor D_x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G*
(&%#dxG6#%!G\"\"\"&F'6#F)F*&%$D_xG6#F)F*" }}}{EXCHG {PARA 0 "euc > " 
0 "" {MPLTEXT 1 0 28 "T4:=evalV( dx &t dx &t D_x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#T4G*(&%#dxG6#%!G\"\"\"&F'6#F)F*&%$D_xG6#F)F*" }}}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 38 "T5:=v_zip([1,1,1],[dx,dx
,D_x],tensor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T5G*(&%#dxG6#%!G
\"\"\"&F'6#F)F*&%$D_xG6#F)F*" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 37 "M:=matrix([[a,b,0],[c,d,0],[0,0,e]]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%%\"aG%\"bG\"\"!7%%\"cG%\"dGF
,7%F,F,%\"eG" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 43 "g:=arra
y_to_tens(M,[[cov_bas,cov_bas],[]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gG,,*(%\"dG\"\"\"&%#dyG6#%!GF(&F*6#F,F(F(*(%\"aGF(&%#dxG6#F,F(&F
26#F,F(F(*(%\"bGF(&F26#F,F(&F*6#F,F(F(*(%\"eGF(&%#dzG6#F,F(&F?6#F,F(F(
*(%\"cGF(&F*6#F,F(&F26#F,F(F(" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 43 "h:=array_to_tens(M,[[con_bas,con_bas],[]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hG,,*(%\"dG\"\"\"&%$D_yG6#%!GF(&F*6#F,F(
F(*(%\"aGF(&%$D_xG6#F,F(&F26#F,F(F(*(%\"bGF(&F26#F,F(&F*6#F,F(F(*(%\"e
GF(&%$D_zG6#F,F(&F?6#F,F(F(*(%\"cGF(&F*6#F,F(&F26#F,F(F(" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 260 "" 0 "" {TEXT -1 
0 "" }}{PARA 260 "" 0 "" {TEXT -1 16 "Special  Tensors" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "g:=
canonical_flat_metric(2,1,bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"gG,(*&&%#dyG6#%!G\"\"\"&F(6#F*F+F+*&&%#dxG6#F*F+&F06#F*F+F+*&&%#dzG6
#F*F+&F66#F*F+!\"\"" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 258 21 "Calcu
lus on Manifolds" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 92 "There are commands for all the  standard operations on d
ifferential forms and vector fields." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "coord_init([x,y,z],[],
euc):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
96 "Let's enter in a few  vector fields and forms, and  tensors  which
 we shall use in this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 26 "X:=  evalV(x^2*D_x + D_y);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,&*&)%\"xG\"\"#\"\"\"&%$D_xG
6#%!G\"\"\"F/&%$D_yG6#F.F/" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 26 "Y:=  evalV(D_x +y^2 *D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"YG,&&%$D_xG6#%!G\"\"\"*&)%\"yG\"\"#\"\"\"&%$D_yG6#F)F*F*" }}}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "alpha:= evalV(y^3 * dx +
 x^2*dy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG,&*&)%\"yG\"\"$
\"\"\"&%#dxG6#%!G\"\"\"F/*&)%\"xG\"\"#F*&%#dyG6#F.F/F/" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 58 "beta:=evalV(z^2*dx &w dy  + y^2
*dx &w dz  + x^2*dy &w dz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%beta
G,(**)%\"zG\"\"#\"\"\"&%#dxG6#%!G\"\"\"%#^~GF/&%#dyG6#F.F/F/**)%\"yGF)
F*&F,6#F.F/F0F/&%#dzG6#F.F/F/**)%\"xGF)F*&F26#F.F/F0F/&F:6#F.F/F/" }}}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 35 "g:= canonical_flat_metri
c(2,1,bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*&&%#dyG6#%!G\"
\"\"&F(6#F*F+F+*&&%#dxG6#F*F+&F06#F*F+F+*&&%#dzG6#F*F+&F66#F*F+!\"\"" 
}}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 16 "In
terior product" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc \+
> " 0 "" {MPLTEXT 1 0 20 "tau:=hook(D_x,beta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tauG,&*&)%\"zG\"\"#\"\"\"&%#dyG6#%!G\"\"\"F/*&)%\"yG
F)F*&%#dzG6#F.F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 
"" {TEXT -1 12 "Lie brackets" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 4 "X,Y;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&*&)%\"xG\"\"#\"\"\"&%$D_xG6#%!G\"\"\"F-&%$D_yG6#F,F-,
&&F*6#F,F-*&)%\"yGF'F(&F/6#F,F-F-" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 20 "Z:=Lie_bracket(X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"ZG,&*&%\"xG\"\"\"&%$D_xG6#%!GF(!\"#*&%\"yGF(&%$D_yG6#F,F(\"\"#
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 17 "Li
e  derivatives " }}{PARA 260 "" 0 "" {TEXT -1 16 "    A. Functions" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 27 "Lie_derivative(X, a*x+b*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&*&)%\"xG\"\"#\"\"\"%\"aG\"\"\"F*%\"bGF*" }}}{PARA 260 "" 0 "" 
{TEXT -1 20 "    B. Vector Fields" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 23 "W:=Lie_derivative(X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"WG,&*&%\"xG\"\"\"&%$D_xG6#%!GF(!\"#*&%\"yGF(&%$D_yG6#F,F(\"
\"#" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 260 "" 1 "" 
{TEXT -1 24 "   C. Differential Forms" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 9 "X, alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&)%\"x
G\"\"#\"\"\"&%$D_xG6#%!G\"\"\"F-&%$D_yG6#F,F-,&*&)%\"yG\"\"$F(&%#dxG6#
F,F-F-*&F%F(&%#dyG6#F,F-F-" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 31 "sigma:=Lie_derivative(X,alpha);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&sigmaG,&*&,&*&)%\"yG\"\"$\"\"\"%\"xG\"\"\"\"\"#*$)F*F/F,F+F.&
%#dxG6#%!GF.F.*&)F-F+F,&%#dyG6#F5F.F/" }}}{PARA 260 "" 0 "" {TEXT -1 
1 " " }}{PARA 260 "" 0 "" {TEXT -1 11 " D. Tensors" }}{PARA 260 "" 0 "
" {TEXT -1 1 " " }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 5 "X, g;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&)%\"xG\"\"#\"\"\"&%$D_xG6#%!G
\"\"\"F-&%$D_yG6#F,F-,(*&&%#dyG6#F,F-&F46#F,F-F-*&&%#dxG6#F,F-&F:6#F,F
-F-*&&%#dzG6#F,F-&F@6#F,F-!\"\"" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 23 "h:=Lie_derivative(X,g);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"hG,$*(%\"xG\"\"\"&%#dxG6#%!GF(&F*6#F,F(\"\"%" }}}{PARA 260 "
" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 19 "Exterior derivat
ive" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 6 "alpha;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"yG\"
\"$\"\"\"&%#dxG6#%!G\"\"\"F-*&)%\"xG\"\"#F(&%#dyG6#F,F-F-" }}}{EXCHG 
{PARA 0 "euc>" 0 "" {MPLTEXT 1 0 13 "ext_d(alpha);" }}{PARA 0 "euc > \+
" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"
\"#*$)%\"yGF&\"\"\"!\"$\"\"\"&%#dxG6#%!GF,%#^~GF,&%#dyG6#F0F," }}}
{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 82 "Check
 Cartan's formula for the Lie derivative in terms of the exterior deri
vative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "
" {MPLTEXT 1 0 29 "ans1:=Lie_derivative(Y,beta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ans1G,(*,)%\"zG\"\"#\"\"\"%\"yG\"\"\"&%#dxG6#%!GF,%#
^~GF,&%#dyG6#F0F,F)**)F+\"\"$F*&F.6#F0F,F1F,&%#dzG6#F0F,F)**,&*&)%\"xG
F)F*F+F*F)FAF)F,&F36#F0F,F1F,&F;6#F0F,F," }}}{EXCHG {PARA 0 "euc > " 
0 "" {MPLTEXT 1 0 53 "ans2:= ext_d(hook(Y,beta)) &plus hook(Y,ext_d(be
ta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans2G,(*,)%\"zG\"\"#\"\"\"
%\"yG\"\"\"&%#dxG6#%!GF,%#^~GF,&%#dyG6#F0F,F)**)F+\"\"$F*&F.6#F0F,F1F,
&%#dzG6#F0F,F)**,&*&)%\"xGF)F*F+F*F)FAF)F,&F36#F0F,F1F,&F;6#F0F,F," }}
}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "ans1 &minus ans2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**%#0~G\"\"\"&%#dxG6#%!GF%%#^~GF%&%#dy
G6#F)F%" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
1 {PARA 256 "" 0 "" {TEXT 259 17 "Transformations 1" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 50 "Maps  between  spaces \+
 can be defined in Vessiot. " }}{PARA 0 "" 0 "" {TEXT 256 60 "Maps  ca
n be  composed and inverted and Jacobians  computed." }}{PARA 0 "" 0 "
" {TEXT 256 74 "Differential forms can be pullbacked and vector fields
  pushed forwarded. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 
0 "euc > " 0 "" {MPLTEXT 1 0 26 "coord_init([x,y],[],euc1):" }}}
{EXCHG {PARA 0 "euc1>" 0 "" {MPLTEXT 1 0 26 "coord_init([u,v],[],euc2)
:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 43 "T
o define a map use the command  transform." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "euc2>" 0 "" {MPLTEXT 1 0 47 "phi:=transform(
euc1, euc2,[u=x^2-y^2, v= x*y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$phiG7$/%\"uG,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF,F-!\"\"/%\"vG*&F+F.F
1F." }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 
24 "Inverse  transformation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 78 "Inverse transform use the Maple solve  command
 and is therefore subject to the" }}{PARA 0 "" 0 "" {TEXT 256 54 "limi
tations and envirnoment variables of this command." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "euc1 > " 0 "" {MPLTEXT 1 0 23 "inverse
_transform(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Solution~~1~~:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG*&%\"vG\"\"\"-%'RootOfG6#,
(*&%\"uG\"\"\")%#_ZG\"\"#F(F/*$)F'F2F(!\"\"*$)F1\"\"%F(F/!\"\"/%\"yGF)
" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}}
{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 23 "inverse_transform(phi);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Solution~~1~~:G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<$/%\"yG,$*$-%%sqrtG6#,&%\"uG!\"#*$-F)6#,&*$)F,\"
\"#\"\"\"\"\"\"*$)%\"vGF4F5\"\"%F5F4F5#F6F4/%\"xG,$*&F9F5*$-F)6#F+F5!
\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Solution~~2~~:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG,$*$-%%sqrtG6#,&%\"uG!\"#*$-F)6#,&*$
)F,\"\"#\"\"\"\"\"\"*$)%\"vGF4F5\"\"%F5F4F5#!\"\"F4/%\"xG,$*&F9F5*$-F)
6#F+F5!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Solution~~3~~:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG,$*$-%%sqrtG6#,&%\"uG!\"#*$-F)
6#,&*$)F,\"\"#\"\"\"\"\"\"*$)%\"vGF4F5\"\"%F5F-F5#F6F4/%\"xG,$*&F9F5*$
-F)6#F+F5!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Solution~~4~~:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG,$*$-%%sqrtG6#,&%\"uG!\"#*
$-F)6#,&*$)F,\"\"#\"\"\"\"\"\"*$)%\"vGF4F5\"\"%F5F-F5#!\"\"F4/%\"xG,$*
&F9F5*$-F)6#F+F5!\"\"F-" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 
0 30 "psi:=inverse_transform(phi,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$psiG7$/%\"xG,$*&%\"vG\"\"\"*$-%%sqrtG6#,&%\"uG!\"#*$-F.6#,&*$)F1
\"\"#F+\"\"\"*$)F*F9F+\"\"%F+F9F+!\"\"F9/%\"yG,$*$-F.6#F0F+#F:F9" }}}
{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "Jacobi
an" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc2 > " 0 "" 
{MPLTEXT 1 0 21 "Jacobian_matrix(phi);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$,$%\"xG\"\"#,$%\"yG!\"#7$F,F)" }}}{PARA 260 "" 
0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "pullback" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc1 > " 0 "" {MPLTEXT 1 0 
27 "g:=pullback(phi, u^2 +v^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"gG,(*$)%\"xG\"\"%\"\"\"\"\"\"*&)F(\"\"#F*)%\"yGF.F*!\"\"*$)F0F)F*F+
" }}}{EXCHG {PARA 0 "euc1 > " 0 "" {MPLTEXT 1 0 32 "mu:=pullback(phi, \+
du &wedge dv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG**,&*$)%\"yG\"
\"#\"\"\"F**$)%\"xGF*F+F*\"\"\"&%#dxG6#%!GF/%#^~GF/&%#dyG6#F3F/" }}}
{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 11 "pushf
orward" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
79 "Recall that for an arbitrary map one can only pushforward a vector
 at a point. " }}{PARA 0 "" 0 "" {TEXT 256 79 "If the transformation i
s invertible then a vector field can be pushed forward. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc1 > " 0 "" {MPLTEXT 1 0 4 "p
hi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/%\"uG,&*$)%\"xG\"\"#\"\"\"\"
\"\"*$)%\"yGF*F+!\"\"/%\"vG*&F)F,F/F," }}}{EXCHG {PARA 0 "euc1 > " 0 "
" {MPLTEXT 1 0 28 "Z1:=pt_pushforward(phi,D_x):" }}}{EXCHG {PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"&%$D_uG6#%!GF&\"\"#*&%\"yGF&&%$D_
vG6#F*F&F&" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 29 "Z2:=push
forward(phi,psi,D_x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z2G,&*&&,$
*&%\"vG\"\"\"*$-%%sqrtG6#,&%\"uG!\"#*$-F.6#,&*$)F1\"\"#F+\"\"\"*$)F*F9
F+\"\"%F+F9F+!\"\"F=6#%!GF:&%$D_uG6#F@F:F:*&&,$*$-F.6#F0F+#F:F9F?F:&%$
D_vG6#F@F:F:" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 21 "transforming tensors." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 256 31 "Use the push_pull_tens  command" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc2 > " 0 "" 
{MPLTEXT 1 0 46 "T:=push_pull_tensor(phi,psi,  dx &tensor D_y);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"TG,**(&,$*&%\"vG\"\"\"*$-%%sqrtG6#
,&*$)%\"uG\"\"#F+\"\"\"*$)F*F4F+\"\"%F+!\"\"!\"\"6#%!GF5&%#duG6#F<F5&%
$D_uG6#F<F5F5*(&,$*&*$F7F+F+*&,&F3F5*$-F.6#F0F+F:\"\"\"-F.6#F0F+F9F:F;
F5&F>6#F<F5&%$D_vG6#F<F5F5*(&,$*&,(*&FKF+F3F5F:F1F5F6F4F+*&FI\"\"\"-F.
6#F0F+F9F4F;F5&%#dvG6#F<F5&FA6#F<F5F5*(&,$*&*&FYF5F*F5F+*&)FI\"\"#F+-F
.6#F0F+F9F4F;F5&Fjn6#F<F5&FS6#F<F5F5" }}}{EXCHG {PARA 0 "euc2 > " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 260 17 "Transf
ormations 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 125 "Vector  fields can be converted to transformations (the flow \+
of the vector field) and one parameter groups of transformations" }}
{PARA 0 "" 0 "" {TEXT 256 36 "can be converted to transformations." }}
{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 60 "The vec
t_to_tranform  command uses the Maple command Dsolve." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 25 "coor
d_init([x,y],[],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:
~eucG" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 28 "X:= evalV(-y *D_
x + x* D_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,&*&%\"yG\"\"\"&
%$D_xG6#%!GF(!\"\"*&%\"xGF(&%$D_yG6#F,F(F(" }}}{EXCHG {PARA 0 "euc > \+
" 0 "" {MPLTEXT 1 0 28 "phi:=vect_to_transform(X,t);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$phiG7$/%\"xG,&*&-%$cosG6#%\"tG\"\"\"F'F.F.*&-%$si
nGF,F.%\"yGF.!\"\"/F2,&*&F0\"\"\"F'F7F.*&F*F7F2F7F." }}}{EXCHG {PARA 
0 "euc > " 0 "" {MPLTEXT 1 0 80 "psi:=transform(euc,euc,[x=cosh(t)*x +
sinh(t)*y +a , y=sinh(t)*x +cosh(t)*y +b]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$psiG7$/%\"xG,(*&-%%coshG6#%\"tG\"\"\"F'F.F.*&-%%sinh
GF,F.%\"yGF.F.%\"aGF./F2,(*&F0\"\"\"F'F7F.*&F*F7F2F7F.%\"bGF." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 58 "The  com
mand  transform_to_vect returns a list of vectors." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 53 "Gamma:=t
ransform_to_vect(psi,[a,b,t],[t=0 ,a=0,b=0]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&GammaG7%&%$D_xG6#%!G&%$D_yG6#F),&*&%\"yG\"\"\"&F'6#F
)F0F0*&%\"xGF0&F+6#F)F0F0" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 261 15 
"Tensor Algebra " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 122 "The  tensor package contains the  standard set of comma
nds for  tensor algebra, covariant differentiation,  and curvature." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 25 "coord_init([x,y],[],euc);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%0frame~name:~eucG" }}}{EXCHG {PARA 0 "euc>" 0 "" 
{MPLTEXT 1 0 17 "T:=dx &tensor dy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"TG*&&%#dxG6#%!G\"\"\"&%#dyG6#F)F*" }}}{PARA 260 "" 0 "" {TEXT -1 
0 "" }}{PARA 260 "" 0 "" {TEXT -1 17 "Rearrange indices" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "T1:
= rearrange_indices(T, [2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T
1G*&&%#dyG6#%!G\"\"\"&%#dxG6#F)F*" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" 
}}{PARA 260 "" 0 "" {TEXT -1 18 "Symmetrize indices" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 24 "T2:=sy
mmetrize(T,[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G,&*(&#\"\"
\"\"\"#6#%!GF)&%#dxG6#F,F)&%#dyG6#F,F)F)*(F'\"\"\"&F16#F,F)&F.6#F,F)F)
" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 13 "
Raise indices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc >
 " 0 "" {MPLTEXT 1 0 34 "g:=canonical_flat_metric(2,0,bas):" }}}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%#dxG6#%!G\"\"\"&F&6#F(F)F
)*&&%#dyG6#F(F)&F.6#F(F)F)" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 21 "h:=inverse_metric(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"h
G,&*&&%$D_xG6#%!G\"\"\"&F(6#F*F+F+*&&%$D_yG6#F*F+&F06#F*F+F+" }}}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "T3:=raise_indices(h,T1,[
1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G*&&%$D_yG6#%!G\"\"\"&%#d
xG6#F)F*" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 16 "Contract indices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 34 "T4:=contract_indices(T3,
 [[1,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T4G\"\"!" }}}{PARA 
260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 12 "Form_to_ten
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 31 "T5:=form_to_tens(dx &wedge dy);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#T5G,&*(&#\"\"\"\"\"#6#%!GF)&%#dxG6#F,F)&%#dyG6#F,F)F
)*(&#!\"\"F*F+F)&F16#F,F)&F.6#F,F)F)" }}}}{SECT 1 {PARA 256 "" 0 "" 
{TEXT 262 15 "Tensor Analysis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 25 "coord_init([x,y],[],euc)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~eucG" }}}{EXCHG 
{PARA 0 "euc>" 0 "" {MPLTEXT 1 0 30 "M:=linalg[diag](1/y^2, 1/y^2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7$7$*&\"\"\"F+*$)%
\"yG\"\"#F+!\"\"\"\"!7$F1F*" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 
0 43 "g:=array_to_tens(M,[[cov_bas,cov_bas],[]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG,&*(&*&\"\"\"F)*$)%\"yG\"\"#F)!\"\"6#%!G\"\"\"&%#
dyG6#F0F1&F36#F0F1F1*(F'F)&%#dxG6#F0F1&F96#F0F1F1" }}}{PARA 260 "" 0 "
" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 18 "Christoffel symbol" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 13 "C:=offel2(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
CG,***&,$*&\"\"\"F*%\"yG!\"\"!\"\"6#%!G\"\"\"&%#dxG6#F/F0&%$D_xG6#F/F0
&%#dyG6#F/F0F0**&F)F.F0&F26#F/F0&%$D_yG6#F/F0&F26#F/F0F0**F'F*&F86#F/F
0&F56#F/F0&F26#F/F0F0**F'F*&F86#F/F0&F?6#F/F0&F86#F/F0F0" }}}{PARA 
260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 20 "Covariant d
erivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 
0 "" {MPLTEXT 1 0 63 "T1:=directional_covariant_derivative(D_x,vect_to
_tens(D_x), C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G*&&*&\"\"\"F(
%\"yG!\"\"6#%!G\"\"\"&%$D_yG6#F,F-" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 31 "  T
2:= x &mult tens(x,con_bas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G
*&%\"xG\"\"\"&%$D_xG6#%!GF'" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 30 "T3:=covariant_derivative(T,C);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#T3G,(**&,$*&\"\"\"F*%\"yG!\"\"!\"\"6#%!G\"\"\"&%#dxG6#F/F0&F2
6#F/F0&F26#F/F0F0**&,$F)\"\"#F.F0&F26#F/F0&%#dyG6#F/F0&F?6#F/F0F0**&F)
F.F0&F?6#F/F0&F?6#F/F0&F26#F/F0F0" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" 
}}{PARA 260 "" 0 "" {TEXT -1 43 "Riemann tensor, Ricci tensor, Ricci s
calar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "
" {MPLTEXT 1 0 26 "Riem:=curvature_tensor(C);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%RiemG,**,&,$*&\"\"\"F**$)%\"yG\"\"#F*!\"\"!\"\"6#%!G
\"\"\"&%#dxG6#F2F3&%$D_yG6#F2F3&F56#F2F3&%#dyG6#F2F3F3*,&F)F1F3&F56#F2
F3&F86#F2F3&F=6#F2F3&F56#F2F3F3*,F@F*&F=6#F2F3&%$D_xG6#F2F3&F56#F2F3&F
=6#F2F3F3*,F'F*&F=6#F2F3&FM6#F2F3&F=6#F2F3&F56#F2F3F3" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 26 "Ricci:=Ricci_tensor(Riem);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RicciG,&*(&,$*&\"\"\"F**$)%\"yG\"\"
#F*!\"\"!\"\"6#%!G\"\"\"&%#dxG6#F2F3&F56#F2F3F3*(F'F*&%#dyG6#F2F3&F;6#
F2F3F3" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 40 "S:=Ricci_scal
ar(inverse_metric(g),Riem);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG!
\"#" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT 263 29 "Lie Alge
bras of Vector Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 256 72 "Vessiot  has an extensive set of commands for working
 with Lie algebras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"euc > " 0 "" {MPLTEXT 1 0 8 "?Koszul;" }}}{EXCHG {PARA 0 "euc > " 0 "
" {MPLTEXT 1 0 7 "?Mubar;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 90 "For an introduct
ion to the Koszul and Mubar packages, see the tutorial getting_started
_III" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
105 "Here we show  a   Lie algebra of  vector fields on a space can be
 converted into an abstract Lie algebra." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl3R>" 0 "" {MPLTEXT 1 0 26 "coor
d_init([x,y], [],euc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name
:~eucG" }}}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 8 "X:= D_x:" }}}
{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 0 8 "Y:= D_y:" }}}{EXCHG {PARA 
0 "euc>" 0 "" {MPLTEXT 1 0 33 "R:= v_zip([-y,x],[D_x,D_y],plus):" }}}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 32 "S:=v_zip([x,y],[D_x,D_y]
, plus):" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "Gamma:=[X,Y,R,S];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&GammaG7&&%$D_xG6#%!G&%$D_yG6#F),&*&%\"yG\"\"
\"&F'6#F)F0!\"\"*&%\"xGF0&F+6#F)F0F0,&*&F5\"\"\"&F'6#F)F0F0*&F/F:&F+6#
F)F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
63 "The command vect_to_Lie_alg  checks that  the vector fields in " }
{XPPEDIT 256 0 "Gamma;" "6#%&GammaG" }{TEXT 256 55 " form a Lie algebr
a and  returns  a Vessiot Lie algebra" }}{PARA 0 "" 0 "" {TEXT 256 
122 "data structure. This data structure contains the structure consta
nt information  which is then used to  initialize the Lie" }}{PARA 0 "
" 0 "" {TEXT 256 7 "algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "euc > " 0 "" {MPLTEXT 1 0 30 "L:=vect_to_Lie_alg(Gamma, sa);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7$7%%(Lie_algG%#saG7#\"\"$7$
7$7%\"\"\"F*\"\"#F.7$7%F/F*F.F." }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
0Lie~algebra:~saG" }}}{EXCHG {PARA 0 "sa > " 0 "" {MPLTEXT 1 0 25 "Lie
_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#
7'7'%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(7'F(%$---G%%----GF5F57'&F+
6#F(F)\"\"!F9&F.6#F(7'&F.6#F(F)F9F9&F+6#F(7'&F16#F(F),$&F.6#F(!\"\",$&
F+6#F(FGF9" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 35 "A sampling of Lie algebra commands." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "sa > " 0 "" {MPLTEXT 1 0 19 "Lie_brack
et(e1,e3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%#e2G6#%!G" }}}{EXCHG 
{PARA 0 "sa > " 0 "" {MPLTEXT 1 0 21 " check_semi_simple();" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "sa > " 0 "" 
{MPLTEXT 1 0 17 "check_solvable();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%%trueG" }}}{EXCHG {PARA 0 "sa > " 0 "" {MPLTEXT 1 0 18 "check_nilpote
nt();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "
sa > " 0 "" {MPLTEXT 1 0 21 "check_ideal([e1,e2]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%%trueG" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT -1 90 "Compute the  infinitesimal istropy subalgebra
  and the linear isotropy representations of " }{XPPEDIT 18 0 "Gamma;
" "6#%&GammaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "sa > " 0 "" {MPLTEXT 1 0 37 "isotropy_subalgebra(Gamma
,[x=0,y=0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$7%\"\"!F&\"\"\"-%'
matrixG6#7$7$F&F'7$F'F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 1 "" {TEXT 256 107 "We see that at the origin the  infinitesimal ist
ropy subalgebra is generated by the 4rd and 3rd vectors in " }
{XPPEDIT 256 0 "Gamma;" "6#%&GammaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT 256 59 "This command also give the linear isotropy representat
ions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 264 9 "Utilities
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 33 "We  \+
list a few handy  utilities. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "coord_init([x,y,z],[],eu
c):" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 
9 "coeff_set" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 56 "list all the coefficients in a form, vector, tensor....." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc>" 0 "" {MPLTEXT 1 
0 56 "omega:= evalV(a*dx &w dy +   b*dx &w dz  +  c*dy &w dz);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,(**%\"aG\"\"\"&%#dxG6#%!GF(%
#^~GF(&%#dyG6#F,F(F(**%\"bGF(&F*6#F,F(F-F(&%#dzG6#F,F(F(**%\"cGF(&F/6#
F,F(F-F(&F66#F,F(F(" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 17 "
coeff_set(omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%%\"aG%\"cG%\"bG
" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 10 "
coeff_list" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 36 "list a select number of coefficients" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 33 "coeff_list(ome
ga,[[1,2], [1,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"aG%\"bG" }
}}{PARA 260 "" 0 "" {TEXT -1 13 "\nlinear_combo" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 85 "find the  components of \+
a form, vector, tensor ...  with respect to  a list of  such." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 
1 0 24 "alpha1:=evalV(dx &w dy):" }}}{EXCHG {PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 25 "alpha2:=evalV(dx &w dz): " }}{PARA 0 "euc > " 0 "" 
{MPLTEXT 1 0 24 "alpha3:=evalV(dy &w dz):" }}}{EXCHG {PARA 0 "euc > " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 
44 "linear_combo(omega,[alpha1, alpha2,alpha3]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%%\"aG%\"bG%\"cG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT -1 12 "Vessiot_diff" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 86 "differentiate the components o
f a form, vector, tensor ... with respect to a parameter" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 6 "om
ega;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**%\"aG\"\"\"&%#dxG6#%!GF&%#
^~GF&&%#dyG6#F*F&F&**%\"bGF&&F(6#F*F&F+F&&%#dzG6#F*F&F&**%\"cGF&&F-6#F
*F&F+F&&F46#F*F&F&" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 28 "e
ta1:=Vessiot_diff(omega,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eta1
G*(&%#dyG6#%!G\"\"\"%#^~GF*&%#dzG6#F)F*" }}}{PARA 260 "" 0 "" {TEXT 
-1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "helmsimp" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 32 "simplify a form, vector,
 tensor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euc > " 0 
"" {MPLTEXT 1 0 22 "eta2:=subs(a=0,omega);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eta2G,(**%#0~G\"\"\"&%#dxG6#%!GF(%#^~GF(&%#dyG6#F,F(
F(**%\"bGF(&F*6#F,F(F-F(&%#dzG6#F,F(F(**%\"cGF(&F/6#F,F(F-F(&F66#F,F(F
(" }}}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 21 "eta3:=helmsimp(eta
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eta3G,&**%\"bG\"\"\"&%#dxG6#
%!GF(%#^~GF(&%#dzG6#F,F(F(**%\"cGF(&%#dyG6#F,F(F-F(&F/6#F,F(F(" }}}}
{SECT 1 {PARA 256 "" 0 "" {TEXT 265 15 "Frame Mangement" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 122 "One of the most im
portant features of Vessiot is its ability to work with many coordinat
e systems (or spaces)  within one " }}{PARA 0 "" 0 "" {TEXT 256 8 "ses
sion." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 256 
82 "The  name of the current  frame is a  global variable _Vessiot_cur
rent_frame_name." }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 96 "There are a host of commands available for working with \+
 different coordinate systems or frames." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "euc > " 0 "" {MPLTEXT 1 0 27 "coord_init([x,y,z
],[u], A):" }}}{EXCHG {PARA 0 "A>" 0 "" {MPLTEXT 1 0 32 "coord_init([t
],[q1,q2,p1,p2],B):" }}}{EXCHG {PARA 0 "B>" 0 "" {MPLTEXT 1 0 34 "coor
d_init([r,theta],[phi,psi],C):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT -1 60 "List all the currently defined coordina
te systems or frames." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "C>" 0 "" {MPLTEXT 1 0 19 "listOfFrameNames();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7*%%euc2G%\"AG%%euc1G%#saG%\"CG%$eucG%'euclidG%\"BG" }}
}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 34 "Get \+
 information on a given frame." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "C>" 0 "" {MPLTEXT 1 0 20 "frameInformation(A);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%0frame~name:~~~AG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%8library~name:~~~vst_eucG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%5Frame~Jet~Variables:G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7)%\"xG%\"yG%\"zG&%\"uG6%\"\"!F*F*&F(6%\"\"\"F*F*&F(6%F*F-F*&F(6
%F*F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Frame~LabelsG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7)&%$D_xG6#%!G&%$D_yG6#F'&%$D_zG6#F'&&%$D_uG
6#7%\"\"!F3F36#F'&&F06#7%\"\"\"F3F36#F'&&F06#7%F3F9F36#F'&&F06#7%F3F3F
96#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/CoFrame~LabelsG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7)&%#dxG6#%!G&%#dyG6#F'&%#dzG6#F'&&%#duG6#7%
\"\"!F3F36#F'&&F06#7%\"\"\"F3F36#F'&&F06#7%F3F9F36#F'&&F06#7%F3F3F96#F
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Horizontal~Coframe~LabelsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#DxG6#%!G&%#DyG6#F'&%#DzG6#F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Vertical~Coframe~LabelsG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&&&%#CuG6#7%\"\"!F)F)6#%!G&&F&6#7%\"\"\"F)F
)6#F+&&F&6#7%F)F0F)6#F+&&F&6#7%F)F)F06#F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%/--------------G" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }
}{PARA 260 "" 0 "" {TEXT -1 14 "Remove a frame" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "C > " 0 "" {MPLTEXT 1 0 16 "remove_fra
me(C);" }}}{EXCHG {PARA 0 "C > " 0 "" {MPLTEXT 1 0 19 "listOfFrameName
s();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)%%euc2G%\"AG%%euc1G%#saG%$eu
cG%'euclidG%\"BG" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 
"" {TEXT -1 12 "Change frame" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "C > " 0 "" {MPLTEXT 1 0 19 "change_frame_to(B);" }}}
{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 260 "" 0 "" 
{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 18 "Obtain  frame data" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 62 "These co
mmands are very used frequently when writing programs." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 21 "frame
BaseDimension();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "B > " 0 "" {MPLTEXT 1 0 22 "frameFiberDimension();" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "B > " 0 "" 
{MPLTEXT 1 0 28 "frameIndependentVariables();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7#%\"tG" }}}{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 26 
"frameDependentVariables();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%#q1G
%#q2G%#p1G%#p2G" }}}{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 19 "frameB
aseVectors();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%$D_tG6#%!G" }}}
{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 18 "frameFiberForms();" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&&%$dq1G6#7#\"\"!6#%!G&&%$dq2GF'6#F+
&&%$dp1GF'6#F+&&%$dp2GF'6#F+" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 266 
6 "Frames" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 64 "One of the most common problems in Maple is expression blow-up
. " }}{PARA 0 "" 0 "" {TEXT 256 59 "The  use of moving frames is one w
ay to avoid this problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "B > " 0 "" {MPLTEXT 1 0 21 "coo
rd_init([x,y],[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~eu
clidG" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 17 "Define a coframe." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 23 "beta1:= (1/y) &mult dx:
" }}{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 24 "beta2:= (1/y) &mult  dy:" 
}}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "euclid > " 0 "" {MPLTEXT 1 0 21 "CoFr:=[beta1, beta2];" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%CoFrG7$*&&%#dxG6#%!G\"\"\"%\"yG!\"\"*&&%#dyG
6#F*F+F,F-" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 
25 "Construct the dual frame." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 30 "Fr:=dual_frame([beta1
,beta2]);" }}{PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#FrG7$*&%\"yG\"\"\"&%$D_xG6#%!GF(*&F'\"\"\"&%$D_
yG6#F,F(" }}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" 
{TEXT -1 48 "Construct the structure equations for the frame." }}
{PARA 260 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 35 "CoF_data:=coframe_data([x,y],CoFr);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%)CoF_dataG7%7'%-moving_frameG%'CartanG7#\"\"#%'euc
lidG7$%\"xG%\"yG7$7$7$\"\"\"F2F.7$7$F*F*F.7#7$7%F2F*F2!\"\"" }}}{PARA 
260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 21 "Initialize \+
the  frame" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid >
 " 0 "" {MPLTEXT 1 0 21 "frame_init(CoF_data);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.frame:~CartanG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT -1 80 "The  default  frame labels are E1, E2 a
nd the coframe labels are Theta1, Theta2." }}{PARA 260 "" 0 "" {TEXT 
-1 44 "Do a  few sample computations in this frame." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 26 "f:=
 y &mult scalar_form();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%\"yG
" }}}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 9 "ext_d(f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"yG\"\"\"&%'Theta2G6#%!GF%" }}}
{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 14 "ext_d(Theta1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%'Theta1G6#%!G\"\"\"%#^~GF(&%'Theta
2G6#F'F(" }}}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 19 "Lie_brac
ket(E1,E2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$&%#E1G6#%!G!\"\"" }}}
{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 18 "E1 &tensor Theta1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#E1G6#%!G\"\"\"&%'Theta1G6#F'F(" 
}}}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 26 "Lie_derivative(E1,
Theta1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%'Theta2G6#%!G" }}}
{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 39 "Lie_derivative(E1,eva
lV(E2 &t Theta1));" }}{PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%#E1G6#%!G\"\"\"&%'Theta1G6#F(F)!
\"\"*&&%#E2G6#F(F)&%'Theta2G6#F(F)F)" }}}{EXCHG {PARA 0 "Cartan > " 0 
"" {MPLTEXT 1 0 59 "g:=array_to_tens(linalg[diag](1,1),[[cov_bas,cov_b
as],[]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&*&&%'Theta2G6#%!G
\"\"\"&F(6#F*F+F+*&&%'Theta1G6#F*F+&F06#F*F+F+" }}}{EXCHG {PARA 11 "" 
1 "" {XPPMATH 20 "6#,&*&&%'Theta1G6#%!G\"\"\"&F&6#F(F)F)*&&%'Theta2G6#
F(F)&F.6#F(F)F)" }}}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 13 "C
:=offel2(g):" }}}{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 23 "R:=c
urvature_tensor(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,***&%'Th
eta1G6#%!G\"\"\"&%#E2G6#F*F+&F(6#F*F+&%'Theta2G6#F*F+!\"\"**&F(6#F*F+&
F-6#F*F+&F26#F*F+&F(6#F*F+F+**&F26#F*F+&%#E1G6#F*F+&F(6#F*F+&F26#F*F+F
+**&F26#F*F+&FB6#F*F+&F26#F*F+&F(6#F*F+F4" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 60 "Convert from the moving frame
  back to the coordinate frame." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "Cartan > " 0 "" {MPLTEXT 1 0 36 "oldR:=change_frame_ba
sis(R,Fr,CoFr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oldRG,**,&,$*&\"
\"\"F**$)%\"yG\"\"#F*!\"\"!\"\"6#%!G\"\"\"&%#dxG6#F2F3&%$D_yG6#F2F3&F5
6#F2F3&%#dyG6#F2F3F3*,&F)F1F3&F56#F2F3&F86#F2F3&F=6#F2F3&F56#F2F3F3*,F
@F*&F=6#F2F3&%$D_xG6#F2F3&F56#F2F3&F=6#F2F3F3*,F'F*&F=6#F2F3&FM6#F2F3&
F=6#F2F3&F56#F2F3F3" }}}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT 268 19 "updated 01/22/03:IA" }}}{MARK "10 0 0" 0 }
{VIEWOPTS 1 1 0 3 4 1802 }