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{TEXT -1 91 "Effectivement, dans une direction, on a un minimum local,
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%\"xG%\"yGQ!F/" 1 2 0 1 10 0 2 1 1 2 2 1.000000 55.000000 13.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Il semblerait qu'i
l y ait deux maxima (resp. minima) locaux qui sont des extrema globaux
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solve(\{diff(f(x,y),x)
=0,diff(f(x,y),y)=0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'<$/%\"yG\"
\"!/%\"xGF&<$/F(\"\"\"/F%F+<$F*/F%!\"\"<$/F(F/F,<$F1F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(u,u),f(u,-u);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$*&)%\"uG\"\"#\"\"\"-%$expG6#,$*$F$F'!\"\"F',$F#F-" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Il y a donc un \"point selle\" en
 (0,0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(f(r*cos(
theta),r*sin(theta)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**)%\"rG\"\"
#\"\"\"-%$cosG6#%&thetaGF'-%$sinGF*F'-%$expG6#,$*$F$F'#!\"\"F&F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(diff(r^2*exp(-r^2/2),r
)=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!*$-%%sqrtG6#\"\"#\"\"\",
$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(r^2*exp(-r
^2/2),r=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 134 134 {PLOTDATA 2 
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*****pfa<*F-$\"3zZX9v))QEbF-7$$\"38HLLeg`!)**F-$\"3*HS,:<yM0'F-7$$\"3w
****\\#G2A3\"F1$\"3/Z'3$=p%3_'F-7$$\"3;LLL$)G[k6F1$\"3IPX$*QH^$)oF-7$$
\"3#)****\\7yh]7F1$\"38\"y5j>!>brF-7$$\"3GLLe**o4#H\"F1$\"3@,(>G!fNXsF
-7$$\"3xmmm')fdL8F1$\"35KQnp!e*3tF-7$$\"3rmmTl^ya8F1$\"31kxe'3v7L(F-7$
$\"3mmm;WV*fP\"F1$\"3\"3%f4*p]nM(F-7$$\"3gmm\"H_.sR\"F1$\"3F4Gal;XbtF-
7$$\"3bmmm,FT=9F1$\"3(y97!=#fuN(F-7$$\"3\"****\\P\")z/Y\"F1$\"3\"y`g![
N-UtF-7$$\"3FLL$e#pa-:F1$\"3ZAZ:=d`,tF-7$$\"3emm\"HB-7a\"F1$\"3D%H#yum
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!\"\"$F*F*Fdcl-%+AXESLABELSG6$Q\"r6\"Q!Ficl-%%VIEWG6$;F(F[cl%(DEFAULTG
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 282 "On a donc un maximum global (do
nc local) en les points de coordonn\351es polaires (Pi/4,sqrt(2)) et (
3Pi/4,sqrt(2)), et un minimum global donc local en les points de coord
onn\351es polaires (-Pi/4,sqrt(2)) et (-3Pi/4,sqrt(2)). Il s'agit bien
 des quatre points critiques trouv\351s plus haut." }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 17 "3 Une circulation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the prote
cted names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "A:=[a/3,a/3,a/3];n:=vector(
[1,1,1])/sqrt(3);f1:=vector([1,-1,0])/sqrt(2);f2:=crossprod(n,f1);R:=a
*sqrt(2/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%,$%\"aG#\"\"\"
\"\"$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG,$*&-%'vectorG6#7%
\"\"\"F+F+F+-%%sqrtG6#\"\"$F+#F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#f1G,$*&-%'vectorG6#7%\"\"\"!\"\"\"\"!F+-%%sqrtG6#\"\"#F+#F+F1" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G-%'vectorG6#7%,$*&-%%sqrtG6#\"\"
$\"\"\"-F,6#\"\"#F/#F/\"\"'F),$F*#!\"\"F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG,$*&%\"aG\"\"\"-%%sqrtG6#\"\"'F(#F(\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M:=theta->evalm(A+R*(cos(the
ta)*f1+sin(theta)*f2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGf*6#%
&thetaG6\"6$%)operatorG%&arrowGF(-%&evalmG6#,&%\"AG\"\"\"*&%\"RGF1,&*&
-%$cosG6#9$F1%#f1GF1F1*&-%$sinGF8F1%#f2GF1F1F1F1F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "M(theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,&%\"aG#\"\"\"\"\"$**F)F*F(F*-%%sqrtG6#\"
\"'F*,&*&-%$cosG6#%&thetaGF*-F.6#\"\"#F*#F*F9**#F*F0F*-%$sinGF5F*-F.6#
F+F*F7F*F*F*F*,&F(F)**F)F*F(F*F-F*,&F2#!\"\"F9**F<F*F=F*F?F*F7F*F*F*F*
,&F(F)*&#F*\"\"*F**,F(F*F-F*F=F*F?F*F7F*F*FE" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 93 "x:=theta->M(theta)[1]:x(theta);y:=theta->M(theta
)[2]:y(theta);z:=theta->M(theta)[3]:z(theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%\"aG#\"\"\"\"\"$**F%F&F$F&-%%sqrtG6#\"\"'F&,&*&-%$co
sG6#%&thetaGF&-F*6#\"\"#F&#F&F5**#F&F,F&-%$sinGF1F&-F*6#F'F&F3F&F&F&F&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG#\"\"\"\"\"$**F%F&F$F&-%%sq
rtG6#\"\"'F&,&*&-%$cosG6#%&thetaGF&-F*6#\"\"#F&#!\"\"F5**#F&F,F&-%$sin
GF1F&-F*6#F'F&F3F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG#\"
\"\"\"\"$*&#F&\"\"*F&*,F$F&-%%sqrtG6#\"\"'F&-%$sinG6#%&thetaGF&-F-6#F'
F&-F-6#\"\"#F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "i
nt(x(theta)*diff(y(theta),theta)+y(theta)*diff(z(theta),theta)+z(theta
)*diff(x(theta),theta),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*()%\"aG\"\"#\"\"\"-%%sqrtG6#\"\"$F(%#PiGF(#F'F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 88 "Ce calcul peut \351galement ^etre fait en
 deux lignes \340 l'aide de la ``formule de Stokes''." }}}}}{MARK "6" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }