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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 28 "D\351mo de tenssurf (version 6)
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "interface(showassumed=0):\nwith(ten
ssurf):" }}{PARA 7 "" 1 "" {TEXT -1 216 "Warning, these names have bee
n rebound: &**, &++, &--, &det, &inv, &t, &t1, &t2, &tt, RU, VR, antis
ym, base_pr, christoffel, composantes, def_base_unit, def_tenseur, dev
, sph, sym, vect_base, vect_base_dual, vect_pr\n" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 376 "Note 1 : la biblioth\350que tenssurf appelle la b
iblioth\350que tens3d. Un certain nombre d'op\351rateurs de tens3d son
t donc surcharg\351s par tenssurf. Il n'y a pas \340 s'inqui\351ter de
 cette surcharge. les op\351rateurs de tenssurf savent traiter des ten
seurs non surfaciques. Toutes les fonctionnalit\351s de tens3d sont do
nc toujours disponibles.\nNote 2 : les tableaux de composantes sont de
s " }{TEXT 256 5 "array" }{TEXT -1 2 ", " }{TEXT 257 6 "vector" }
{TEXT -1 4 " ou " }{TEXT 258 6 "matrix" }{TEXT -1 21 "  de la biblioth
\350que " }{TEXT 259 6 "linalg" }{TEXT -1 12 " et non des " }{TEXT 
260 5 "Array" }{TEXT -1 2 ", " }{TEXT 261 6 "Vector" }{TEXT -1 4 " ou \+
" }{TEXT 262 6 "Matrix" }{TEXT -1 20 " de la biblioth\350que " }{TEXT 
263 14 "Linear Algebra" }{TEXT -1 104 ". Les feullies de calcul d\351v
elopp\351es sous les versions pr\351c\351dentes sont donc compl\350tem
ent compatibles.  " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "D\351finit
ion d'une surface" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "La premi\350
re chose \340 faire est de d\351finir une surface sur laquelle seront \+
d\351finis les tenseurs ou les champs de tenseurs." }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "A titre d'exemple, on d\351fi
nit un cylindre de rayon " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 7 "
 d'axe " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 28 ", dont le point c
ourant est " }{XPPEDIT 18 0 "M;" "6#%\"MG" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 158 "assume(R>0):assume(theta,real):assume(z,real):\ncomp
_OM:=array(1..3,[R*cos(theta),R*sin(theta),z]);\ncoords:=theta,z;\nOM:
=def_tenseur(comp_OM,[cont],base_fond);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(comp_OMG-%'vectorG6#7%*&%#R|irG\"\"\"-%$cosG6#%'theta|irGF+*&
F*F+-%$sinGF.F+%#z|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'coordsG6$
%'theta|irG%#z|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OMG%5_tenseur
_3d_ordre_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "On d\351finit la
 surface et sa base naturelle tangente:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "surf:=def_surf(OM,coords);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%surfG%F_base_naturelle_tangente_orthogonale_G" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 69 "Les vecteurs de base de la base naturelle tangente
 de la surface sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a[coords[1]]
:=vect_base(surf,1);\na[coords[2]]:=vect_base(surf,2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"aG6#%'theta|irG%:_tenseur_tangent_ordre_1_G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%#z|irG%:_tenseur_tangent_o
rdre_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "La normale unitaire \+
\340 la surface est" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "n:=vect_base
(surf,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG%5_tenseur_3d_ordre
_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "On regarde les composante
s de ces vecteurs dans la base fondamentale" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 187 "evaln(a[coords[1]]) = composantes(a[coords[1]],[cont
],base_fond);\nevaln(a[coords[2]]) = composantes(a[coords[2]],[cont],b
ase_fond);\nevaln(n)            = composantes(n,[cont],base_fond);\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%'theta|irG-%'vectorG6#7%,$
*&%#R|irG\"\"\"-%$sinGF&F/!\"\"*&F.F/-%$cosGF&F/\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/&%\"aG6#%#z|irG-%'vectorG6#7%\"\"!F,\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"nG-%'vectorG6#7%-%$cosG6#%'theta|i
rG-%$sinGF+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Les vecteurs \+
de base duaux de la base naturelle tangente de la surface sont" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "ad[coords[1]]:=vect_base_dual(surf,
1);\nad[coords[2]]:=vect_base_dual(surf,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#adG6#%'theta|irG%:_tenseur_tangent_ordre_1_G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#adG6#%#z|irG%:_tenseur_tangent_ord
re_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Le dual de la normale u
nitaire est la normale unitaire elle-m\352me" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "composantes( vect_base_dual(surf,3) &-- n , [cont],su
rf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "On peut observer les coefficients \+
de connexion Riemannienne de la surface" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "christoffel(surf,XXX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q@To
us~les~coefficients~sont~nuls6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
34 "Le tenseur de courbure normale est" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "B:=tens_courb(surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG
%:_tenseur_tangent_ordre_2_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "S
es composantes cov-cov dans la base naturelle de la surface sont" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "composantes(B,[cov,cov],surf);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,$%#R|irG!\"\"\"\"!7$F
+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Le tenseur m\351trique de \+
surface est" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=tens_metr_surf(su
rf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%:_tenseur_tangent_ordre
_2_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "dont les composantes cov-
cov dans la base naturelle de la surface sont" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "composantes(A,[cov,cov],surf);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$*$)%#R|irG\"\"#\"\"\"\"\"!7$F-F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Le tenseur d'orientation de surfac
e est" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ES:=tens_orient_surf(surf)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ESG%:_tenseur_tangent_ordre_2_
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "dont les composantes cov-cov
 dans la base naturelle de la surface sont" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "composantes(ES,[cov,cov],surf);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!%#R|irG7$,$F)!\"\"F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "et les composantes cont-cont dans la base
 naturelle de la surface sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "co
mposantes(ES,[cont,cont],surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
matrixG6#7$7$\"\"!*&\"\"\"F*%#R|irG!\"\"7$,$F)F,F(" }}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 40 "D\351finition d'autres bases sur la surface" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Soient V1 et V2 deux vecteurs tang
ent d\351finis sur la base naturelle de la surface" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "V1:=a[coords[1]] &++ a[coords[2]];\nV2:=a[coords[1]
] &-- a[coords[2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G%:_tenseu
r_tangent_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G%:_tenseu
r_tangent_ordre_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "On aurait \+
pu aussi \351crire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "#V1:=def_tens
eur(vector([1,1]),[cont],surf):\n#V2:=def_tenseur(vector([1,-1]),[cont
],surf):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "On peut cr\351er une \+
nouvelle base tangente sur la surface" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "BB:=def_base_tang(V1,V2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
BBG%0_base_tangente_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Les comp
osantes cont-cov du tenseur courbure dans cette base sont" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "composantes(B,[cont,cov],BB);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,$*&\"\"\"F*%#R|irG!\"\"#F,\"\"#
F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "On peut construire une b
ase unitaire colin\351aire \340 une base donn\351e:\npar exemple la ba
se physique est la  base naturelle norm\351e :" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Bphy:=def_base_unit(surf);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%BphyG%<_base_tangente_orthonormee_G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "Les composantes cont-cov du tenseur courb
ure dans cette base sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "composa
ntes(B,[cont,cov],Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7$7$,$*&\"\"\"F*%#R|irG!\"\"F,\"\"!7$F-F-" }}}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 69 "D\351finition de tenseurs de surface par leurs composa
ntes dans une base" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "On a d\351j
\340 d\351fini deux tenseurs d'ordre 1 (les vecteurs V1 et V2) par leu
rs composantes sur la base naturelle." }}{PARA 0 "" 0 "" {TEXT -1 77 "
D\351finissons un tenseur du second ordre T par ses composantes sur la
 base Bphy" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "compT:=array(1..2,1..
2,[[T11,T12],[T21,T22]]);\nT:=def_tenseur(compT,[cont,cov],Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&compTG-%'matrixG6#7$7$%$T11G%$T12G7
$%$T21G%$T22G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG%:_tenseur_tang
ent_ordre_2_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "ses composantes \+
cont-cov dans la base naturelle sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "composantes(T,[cont,cov],surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7$7$%$T11G*&%$T12G\"\"\"%#R|irG!\"\"7$*&F,F+%$T21GF+%$T
22G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "ses composantes cont-cov d
ans la base BB sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "composantes(
T,[cont,cov],BB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,
*%$T11G#\"\"\"\"\"#*(F*F+%#R|irGF+%$T21GF+F+*&*&F*F+%$T12GF+F+F.!\"\"F
+*&F*F+%$T22GF+F+,*F)F**(F*F+F.F+F/F+F+*&#F+F,F+*&F2F+F.F3F+F3*&#F+F,F
+F5F+F37$,*F)F**&#F+F,F+*&F.F+F/F+F+F3*&*&F*F+F2F+F+F.F3F+*&#F+F,F+F5F
+F3,*F)F**&#F+F,F+FAF+F3*&#F+F,F+F:F+F3*&F*F+F5F+F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "ordre(T);ordre(V1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "On peut d\351finir un tenseur par \+
op\351rations tensorielles" }}{PARA 0 "" 0 "" {TEXT -1 87 "combinaison
 des composantes (ici contravariantes-covariantes) sur les tenseurs de
 base " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "Q :=   ( Q11 &** (a[coor
ds[1]] &t ad[coords[1]]) )\n  &++  ( Q12 &** (a[coords[1]] &t ad[coord
s[2]]) )\n  &++  ( Q21 &** (a[coords[2]] &t ad[coords[1]]) )\n  &++  (
 Q22 &** (a[coords[2]] &t ad[coords[2]]) );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"QG%:_tenseur_tangent_ordre_2_G" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 37 "On aurait pu \351crire plus simplement :" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "QQ := def_tenseur(matrix([[Q11,Q12],[Q21,
Q22]]),[cont,cov],surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QQG%:_t
enseur_tangent_ordre_2_G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"composantes(QQ &-- Q,[cov,cov],surf);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$\"\"!F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "composantes(Q,[cont,cov],surf);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$%$Q11G%$Q12G7$%$Q21G%$Q22G" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "composantes(Q,[cov,cov],BB);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,**&%$Q11G\"\"\")%#R|irG\"\"#F+F+%
$Q21GF+*&%$Q12GF+F,F+F+%$Q22GF+,*F)F+F/F+F0!\"\"F2F47$,*F)F+F/F4F0F+F2
F4,*F)F+F/F4F0F4F2F+" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Op\351r
ateurs alg\351briques " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 54 "additio
n, soustraction, multiplication par un scalaire" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "composantes( T &++ Q , [cont,cov], surf);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&%$T11G\"\"\"%$Q11GF*
,&*&%$T12GF*%#R|irG!\"\"F*%$Q12GF*7$,&*&F/F*%$T21GF*F*%$Q21GF*,&%$T22G
F*%$Q22GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "composantes( \+
T &-- Q , [cont,cov], surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat
rixG6#7$7$,&%$T11G\"\"\"%$Q11G!\"\",&*&%$T12GF*%#R|irGF,F*%$Q12GF,7$,&
*&F0F*%$T21GF*F*%$Q21GF,,&%$T22GF*%$Q22GF," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "composantes( (k+1) &** B , [cont,cov], Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&%\"kG\"\"\"%#R|irG
!\"\"F-*&F+F+F,F-F-\"\"!7$F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "composantes( B &** (k+1) , [cont,cov], Bphy);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&%\"kG\"\"\"%#R|irG!\"\"F-*&F+F
+F,F-F-\"\"!7$F/F/" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 "produits \+
tensoriels et produits tensoriels contract\351s" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 17 "produit tensoriel" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "V1V2:= V1 &t V2 ;\ncomposantes(V1V2,[cont,cont],surf); " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%V1V2G%:_tenseur_tangent_ordre_2_G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"!\"\"F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "T &t V1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%:_tenseur_tangent_ordre_3_G" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 67 "si les deux tenseurs ne sont pas tangents, on obtient u
n tenseur 3d" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "T &t n;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%5_tenseur_3d_ordre_3_G" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 36 "produit tensoriel contract\351 une fois" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "V1 &t1 V2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&!\"\"\"\"\"*$)%#R|irG\"\"#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "TV1:=T &t1 V1;\ncomposantes( TV1,[cont],Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$TV1G%:_tenseur_tangent_ordre_1_G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,&*&%$T11G\"\"\"%#R|irG
F*F*%$T12GF*,&*&F+F*%$T21GF*F*%$T22GF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "V1T:=V1 &t1 T;\ncomposantes( V1T,[cont],Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V1TG%:_tenseur_tangent_ordre_1_G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,&*&%$T11G\"\"\"%#R|irG
F*F*%$T21GF*,&*&F+F*%$T12GF*F*%$T22GF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "V1 &t1 n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "produit tensoriel contract\351 de
ux fois" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "TB:=T &t2 B;\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#TBG,$*&%$T11G\"\"\"%#R|irG!\"\"F*" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "op\351rations sur les tenseurs du
 second ordre" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "transposition" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "composantes( &tt(T) ,[cont,cov],Bph
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$%$T11G%$T21G7$%
$T12G%$T22G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "inverse" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "composantes( &inv(T) ,[cont,cov],Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,$*&%$T22G\"\"\",&*&%$
T11GF+F*F+!\"\"*&%$T12GF+%$T21GF+F+F/F/*&F1F+F,F/7$*&F2F+F,F/,$*&F.F+F
,F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "trace" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "&tr(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$T11G
\"\"\"%$T22GF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "d\351terminant
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "&det(T);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&%$T11G\"\"\"%$T22GF&F&*&%$T12GF&%$T21GF&!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "partie sym\351trique" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "composantes( sym(T) , [cont,cov] , Bphy);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$%$T11G,&%$T12G#\"\"\"
\"\"#*&F+F,%$T21GF,F,7$F)%$T22G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "partie antisym\351trique" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "com
posantes( antisym(T) , [cont,cov] , Bphy);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!,&%$T12G#\"\"\"\"\"#*&#F,F-F,%$T21
GF,!\"\"7$,&F0F+*&#F,F-F,F*F,F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
16 "partie sph\351rique" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "composan
tes( sph(T) , [cont,cov] , Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7$7$,&%$T11G#\"\"\"\"\"#*&F*F+%$T22GF+F+\"\"!7$F/F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "partie d\351viatorique" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 41 "composantes( dev(T) , [cont,cov] , Bphy);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&%$T11G#\"\"\"\"\"#
*&#F+F,F+%$T22GF+!\"\"%$T12G7$%$T21G,&F/F**&#F+F,F+F)F+F0" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "vecteurs propres et valeurs propres" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "vect_pr(T);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6$7%,(%$T22G#\"\"\"\"\"#*&F&F'%$T11GF'F'*&F&F'-%%sqrtG6#,
**$)F%F(F'F'*(F(F'F*F'F%F'!\"\"*$)F*F(F'F'*(\"\"%F'%$T12GF'%$T21GF'F'F
'F'F'<#%:_tenseur_tangent_ordre_1_G7%,(F%F&*&F&F'F*F'F'*&#F'F(F'*$F,F'
F'F3F'<#F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "base propre  d'un t
enseur du second ordre T2 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "T2:=
def_tenseur(array(1..2,1..2,[[1,2],[2,8]]),[cont,cov],Bphy);\nBprT2:=b
ase_pr(T2);\ncomposantes(T2,[cont,cov],BprT2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#T2G%:_tenseur_tangent_ordre_2_G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&BprT2G%<_base_tangente_orthogonale_G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*$-%%sqrtG6#\"#l\"\"\"#F.\"\"##
\"\"*F0F.\"\"!7$F3,&F)#!\"\"F0F1F." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 19 "produits vectoriels" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Prod
uit vectoriel entre deux vecteurs tangents (noter que le r\351sultat e
st un vecteur 3d )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "V1vV2:=V1 &v \+
V2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&V1vV2G%5_tenseur_3d_ordre_1_
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Ses composantes dans la base
 naturelle tangente sont" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "composa
ntes(V1vV2,[cont],surf);" }}{PARA 0 "" 0 "" {TEXT -1 147 "Remarquer qu
e les bases tangentes sont compl\351t\351es de mani\350re interne par \+
leur normale unitaire. On peut donc les utiliser comme des bases d'esp
ace." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F',$%#R|irG
!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Produit vectoriel de la \+
normale \340 la surface avec un vecteur tangent (le r\351sultat est un
 vecteur tangent)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "nvV1:=&nv(V1);
\ncomposantes(nvV1,[cont],Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%nvV1G%:_tenseur_tangent_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'vectorG6#7$!\"\"%#R|irG" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "o
p\351rations avec des tenseurs 3d" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
139 "Soient V3d et T3d un vecteur et un tenseur d\351finis par leurs c
omposantes sur la base Bphy (implicitement compl\351t\351e par sa norm
ale unitaire)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "V3d:=def_tenseur(
array(1..3,[V3d1,V3d2,V3d3]),[cont],Bphy);\nT3d:=def_tenseur(array(1..
3,1..3,[[T3d11,T3d12,T3d13],[T3d21,T3d22,T3d23],[T3d31,T3d32,T3d33]]),
[cont,cov],Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V3dG%5_tenseur
_3d_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T3dG%5_tenseur_3d_
ordre_2_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Les op\351rateurs al
g\351briques continuent \340 op\351rer sur les tenseurs 3d" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "W3d:=T3d &t1 V3d;\ncomposantes(W3d,[cont],B
phy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$W3dG%5_tenseur_3d_ordre_1_
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,(*&%&T3d11G\"\"\"%
%V3d1GF*F**&%&T3d12GF*%%V3d2GF*F**&%&T3d13GF*%%V3d3GF*F*,(*&%&T3d21GF*
F+F*F**&%&T3d22GF*F.F*F**&%&T3d23GF*F1F*F*,(*&%&T3d31GF*F+F*F**&%&T3d3
2GF*F.F*F**&%&T3d33GF*F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "V1 &++ V3d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5_tenseur_3d_ord
re_1_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "On extrait la partie ta
ngente d'un tenseur" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "tangV3d:=par
tie_tang(V3d,surf);\ncomposantes(tangV3d,[cont],Bphy);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(tangV3dG%:_tenseur_tangent_ordre_1_G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$%%V3d1G%%V3d2G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "tangT3d:=partie_tang(T3d,Bphy);\nco
mposantes(tangT3d,[cont,cov],Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%(tangT3dG%:_tenseur_tangent_ordre_2_G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$%&T3d11G%&T3d12G7$%&T3d21G%&T3d22G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 41 "On extrait la partie normale d'un tenseur
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "normV3d:=partie_norm(V3d,Bphy);
\ncomposantes(normV3d,[cont],surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%(normV3dG%5_tenseur_3d_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'vectorG6#7%\"\"!F'%%V3d3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "normT3d:=partie_norm(T3d,Bphy);\ncomposantes(normT3d,[cont,cov],
Bphy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(normT3dG%5_tenseur_3d_ord
re_2_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(%&T3d
13G7%F(F(%&T3d23G7%%&T3d31G%&T3d32G%&T3d33G" }}}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 24 "Op\351rateurs diff\351rentiels" }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 19 "gradient de surface" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 88 "Soit f une fonction scalaire des coordonn\351es sur la su
rface. Son gradient de surface est" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "alias(f=f(coords)):\nGSf:=GRADS(f,surf);\ncomposantes(GSf,[cov],su
rf);" }}{PARA 0 "" 0 "" {TEXT -1 157 "Remarquer qu'il est n\351cessair
e de pr\351ciser sur quelle surface on calcule le gradient (il peut y \+
avoir plusieurs surfaces d\351finies dans la feuille de calcul!)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GSfG%:_tenseur_tangent_ordre_1_G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$-%%diffG6$%\"fG%'theta|
irG-F(6$F*%#z|irG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Soit VS un c
hamp vectoriel tangent \340 la surface" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 160 "alias(VS1=VS1(coords),VS2=VS2(coords),VS3=VS3(coords)):\nVS:=
def_tenseur(array(1..2,[VS1,VS2]),[cont],surf);\nGSVS:=GRADS(VS);\ncom
posantes(GSVS,[cont,cov],Bphy); " }}{PARA 0 "" 0 "" {TEXT -1 88 "Remar
quer qu'il n'est pas n\351cessaire de pr\351ciser la surface, car VS e
st un champ tangent" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VSG%:_tenseu
r_tangent_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GSVSG%:_tens
eur_tangent_ordre_2_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
$7$-%%diffG6$%$VS1G%'theta|irG*&-F)6$F+%#z|irG\"\"\"%#R|irGF17$*&-F)6$
%$VS2GF,F1F2!\"\"-F)6$F7F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Soi
t TS un champ tensoriel du second ordre tangent \340 la surface" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "alias(TS11=TS11(coords),TS12=TS12(
coords),\n      TS21=TS21(coords),TS22=TS22(coords)):\nTS:=def_tenseur
(array(1..2,1..2,[[TS11,TS12],[TS21,TS22]]),[cont,cov],surf);\nGSTS:=G
RADS(TS);\ncomposantes(GSTS,[cont,cov,cov],surf);" }}{PARA 0 "" 0 "" 
{TEXT -1 88 "Remarquer qu'il n'est pas n\351cessaire de pr\351ciser la
 surface, car TS est un champ tangent" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#TSG%:_tenseur_tangent_ordre_2_G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%GSTSG%:_tenseur_tangent_ordre_3_G" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#-%&ARRAYG6$7%;\"\"\"\"\"#F'F'7*/6%F(F(F(-%%diffG6$%%TS11G%'theta
|irG/6%F(F(F)-F.6$F0%#z|irG/6%F(F)F(-F.6$%%TS12GF1/6%F(F)F)-F.6$F;F6/6
%F)F(F(-F.6$%%TS21GF1/6%F)F(F)-F.6$FDF6/6%F)F)F(-F.6$%%TS22GF1/6%F)F)F
)-F.6$FMF6" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "divergence de sur
face" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "La divergence de surface d
'un vecteur tangent est un scalaire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "DIVS(VS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%%diffG6$%$VS1G%'th
eta|irG\"\"\"-F%6$%$VS2G%#z|irGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
68 "La divergence de surface d'un tenseur tangent est un vecteur tange
nt" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "DSTS:=DIVS(TS);\ncomposantes(
DSTS,[cont],surf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DSTSG%:_tense
ur_tangent_ordre_1_G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$
,&*&-%%diffG6$%%TS11G%'theta|irG\"\"\"*$)%#R|irG\"\"#F.!\"\"F.-F*6$%%T
S12G%#z|irGF.,&*&-F*6$%%TS21GF-F.*$F0F.F3F.-F*6$%%TS22GF7F." }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "rotationnel de surface" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "Le rotationnel de surface d'un champ de v
ecteurs tangent est un scalaire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "R
OTS(VS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%diffG6$%$VS1G%#z|ir
G\"\"\"%#R|irGF*!\"\"*&-F&6$%$VS2G%'theta|irGF*F+F,F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 96 "Le rotationnel de surface d'un champ de t
enseurs du second ordre  tangent est un vecteur tangent" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "ROTS(TS);\ncomposantes(ROTS(TS),[cont],surf);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:_tenseur_tangent_ordre_1_G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,&*&-%%diffG6$%%TS12G%'t
heta|irG\"\"\"%#R|irG!\"\"F.*&-F*6$%%TS11G%#z|irGF.F/F0F0,&*&-F*6$%%TS
22GF-F.F/F0F.*&-F*6$%%TS21GF5F.F/F0F0" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 20 "laplacien de surface" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
76 "le laplacien d'un champ surfacique scalaire est un champ surfaciqu
e scalaire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "LAPS(f,surf);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%diffG6$%\"fG-%\"$G6$%'theta|irG
\"\"#\"\"\"*$)%#R|irGF-F.!\"\"F.-F&6$F(-F*6$%#z|irGF-F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 78 "le laplacien d'un champ surfacique vector
iel est un champ surfacique vectoriel" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "composantes( LAPS(VS) , [cont] , surf);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7$,&*&-%%diffG6$%$VS1G-%\"$G6$%'theta|irG\"
\"#\"\"\"*$)%#R|irGF1F2!\"\"F2-F*6$F,-F.6$%#z|irGF1F2,&*&-F*6$%$VS2GF-
F2*$F4F2F6F2-F*6$F@F9F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "le la
placien d'un champ surfacique tensoriel du second ordre est un champ s
urfacique tensoriel du second ordre" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "composantes( LAPS(TS) , [cont,cov] , surf);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$,&*&-%%diffG6$%%TS11G-%\"$G6$%'theta|ir
G\"\"#\"\"\"*$)%#R|irGF2F3!\"\"F3-F+6$F--F/6$%#z|irGF2F3,&*&-F+6$%%TS1
2GF.F3*$F5F3F7F3-F+6$FAF:F37$,&*&-F+6$%%TS21GF.F3*$F5F3F7F3-F+6$FJF:F3
,&*&-F+6$%%TS22GF.F3*$F5F3F7F3-F+6$FRF:F3" }}}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 21 "Quelques applications" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 70 "Construction du syst\350me de coordonn\351es spatiales induit p
ar la surface" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "assume(x3,real);\n
ON:=OM &++ ( x3 &** n );\nSC:=def_SC(ON,coords,x3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#ONG%5_tenseur_3d_ordre_1_G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#QTL'orientation~de~la~base~n'a~pas~pu~|eytre~d|dytermin
|dye6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SCG%6_base_3d_orthogonal
e_G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 118 "L'orientat
ion de la base SC n'a pas pu \352tre d\351termin\351e car le produit m
ixte des vecteurs de base n'est positif que si " }{XPPEDIT 18 0 "-R < \+
x3;" "6#2,$%\"RG!\"\"%#x3G" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "(vect_base(SC,1) &v vect_base(SC,2)) &t1 vect_base(SC
,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#R|irG\"\"\"%$x3|irGF%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "On observe les composantes d'un t
enseur tangent \340 la surface dans la base naturelle (3d) du syst\350
me de coordonn\351es 3d induit par la surface." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "composantes(TS,[cont,cov],SC);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%%%TS11G*&*&%%TS12G\"\"\"%#R|irGF,F,,&F-
F,%$x3|irGF,!\"\"\"\"!7%,&%%TS21GF,*&*&F4F,F/F,F,F-F0F,%%TS22GF17%F1F1
F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Soit V un prolongement spat
ial du champ surfacique tangent VS" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
71 "cV:=array(1..3,[VS1+x3*f,VS2+x3*f,x3*f]);\nV:=def_tenseur(cV,[cont
],SC);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cVG-%'vectorG6#7%,&%$VS1G
\"\"\"*&%$x3|irGF+%\"fGF+F+,&%$VS2GF+F,F+F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"VG%5_tenseur_3d_ordre_1_G" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 15 "On v\351rifie que " }{XPPEDIT 18 0 "ROTS(VS) = n &t1 RO
T(V) ,``[x3=0]" "6$/-%%ROTSG6#%#VSG-%$&t1G6$%\"nG-%$ROTG6#%\"VG&%!G6#/
%#x3G\"\"!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ROTS(VS) - subs(x3=0 \+
, (n &t1 ROT(V,SC)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "On v\351rifie que ROTS( &nv(VS)) =
 DIVS(VS)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " ROTS( &nv(VS)) - DIVS
(VS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "9 7 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }