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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 266 "" 0 "" {TEXT 282 21 "Geometr\355a Diferencial" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 46 "FORMAS \+
DIFERENCIALES Y EL TEOREMA DE FROBENIUS" }{TEXT 256 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 58 "Laboratorio confec
cionado por  Mirta Iriondo,  FaMAF, UNC." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 267 "E
n este laboratorio vamos a familiarizarnos con el c\341lculo de formas
 diferenciales. Lo vamos a aplicar centralmente a controlar si sistema
s de formas diferenciales son integrables o no. Para ello usaremos el \+
teorema de Frobenius en la versi\363n de formas diferenciales." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 22 "Teorema \+
de Frobenius: " }{TEXT -1 3 "Sea" }{TEXT 284 1 " " }{TEXT -1 2 "( " }
{XPPEDIT 18 0 "omega[p+1];" "6#&%&omegaG6#,&%\"pG\"\"\"F(F(" }{TEXT 
-1 7 ", ..., " }{XPPEDIT 18 0 "omega[n];" "6#&%&omegaG6#%\"nG" }{TEXT 
-1 18 ")  un sistema de  " }{XPPEDIT 18 0 "n-p;" "6#,&%\"nG\"\"\"%\"pG
!\"\"" }{TEXT -1 43 "  formas diferenciales  de grado 1 y clase " }
{XPPEDIT 18 0 "C^[1];" "6#)%\"CG7#\"\"\"" }{TEXT -1 20 " en un abierto
 U de " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 24 ", tal que \+
en cada punto " }{TEXT 259 1 "x" }{TEXT -1 29 " en U el rango del sist
ema ( " }{XPPEDIT 18 0 "omega[p+1];" "6#&%&omegaG6#,&%\"pG\"\"\"F(F(" 
}{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "omega[n];" "6#&%&omegaG6#%\"nG" 
}{TEXT -1 14 ") sea igual a " }{XPPEDIT 18 0 "n-p" "6#,&%\"nG\"\"\"%\"
pG!\"\"" }{TEXT -1 33 ". Entonces el sistema diferencial" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "omega[i] = 0;
" "6#/&%&omegaG6#%\"iG\"\"!" }{TEXT -1 10 "        ( " }{TEXT 260 11 "
p < i < n+1" }{TEXT -1 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 66 "es comple
tamente integrable si y s\363lo si las formas diferenciales " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 "d" }{XPPEDIT 
18 0 "`&^`(omega[i],omega[p+1]);" "6#-%#&^G6$&%&omegaG6#%\"iG&F'6#,&%
\"pG\"\"\"F.F." }{TEXT -1 8 " ...... " }{XPPEDIT 18 0 "`&^`(omega[n-1]
,omega[n]);" "6#-%#&^G6$&%&omegaG6#,&%\"nG\"\"\"F+!\"\"&F'6#F*" }
{TEXT -1 8 "      ( " }{TEXT 261 11 "p < i < n+1" }{TEXT -1 2 " )" }}
{PARA 0 "" 0 "" {TEXT -1 24 "son id\351nticamente nulas." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "Comience colocando el cursor en el signo " }{TEXT 277 
3 "(+)" }{TEXT -1 71 " en la secci\363n de abajo y cliquee con el mous
e, luego apriete la tecla " }{TEXT 276 5 "Enter" }{TEXT -1 242 " para \+
ejecutar cada comando que aparecer\341 en letras rojas. Si el comando \+
termina con punto y coma, Maple responder\341 con un eco de la terea r
ealizada y si el comando termina con dos puntos Maple ejecuta el coman
do pero no muestra el resultado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Los comandos  " }{TEXT 262 1 "d" 
}{TEXT 283 2 ", " }{TEXT 281 6 "deform" }{TEXT 285 5 ",    " }{TEXT 
263 2 "&^" }{TEXT 286 1 "," }{TEXT 270 9 " simpform" }{TEXT 268 1 " " 
}{TEXT -1 1 "y" }{TEXT 267 1 " " }{TEXT 288 1 " " }{TEXT 266 10 "scala
rpart" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 54 "Vamos a comenzar limpiando la memoria con el comando  " }
{HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 26 "  y cargando el pa
quete   " }{HYPERLNK 17 "difforms" 2 "difforms" "" }{TEXT -1 2 "  " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "with(difforms);" }}{PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Declaramos las formas d
iferenciales con el comando " }{HYPERLNK 17 "defform" 2 "defform" "" }
{TEXT -1 82 " , en este ejemplo son escalares, o sea de grado cero, y \+
definimos con el comando " }{HYPERLNK 17 "alias" 2 "alias" "" }{TEXT 
-1 18 "  las funciones P(" }{TEXT 278 5 "x,y,z" }{TEXT -1 1 ")" }
{TEXT 279 2 ", " }{TEXT -1 2 "Q(" }{TEXT 265 5 "x,y,z" }{TEXT -1 6 ") \+
y R(" }{TEXT 280 5 "x,y,z" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "defform(x=0,y=0,z=0,
P=0,Q=0,R=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "alias(P=P(
x,y,z),Q=Q(x,y,z),R=R(x,y,z));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 
"En la variable " }{TEXT 264 2 "w " }{TEXT -1 45 "guardamos una forma \+
diferencial de grado uno " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "w:=P*d(x)+Q*d(y)+R*d(z);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Para saber si la ecuaci\363n " }
{XPPEDIT 18 0 "w = 0;" "6#/%\"wG\"\"!" }{TEXT -1 59 " es completamente
 integrable haremos  el producto exterior " }{XPPEDIT 18 0 "`&^`(d(ome
ga),omega);" "6#-%#&^G6$-%\"dG6#%&omegaGF)" }{TEXT -1 34 " y simplific
aremos con el comando " }{HYPERLNK 17 "simpform" 2 "simpform" "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Omega:=simpform(d(w)&^w);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "La condici\363n para que la ecuaci\363n " }{XPPEDIT 18 0 
"w = 0;" "6#/%\"wG\"\"!" }{TEXT -1 32 " sea completamente integrable e
s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "collect(scalarpart(Ome
ga)=0,[P,Q,R]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Observe que he
mos usado el comando " }{HYPERLNK 17 "scalarpart" 2 "scalarpart" "" }
{TEXT -1 62 "  para obtener la parte escalar de esta 3-forma y  el com
ando " }{HYPERLNK 17 "collect" 2 "collect" "" }{TEXT -1 57 "  para jun
tar los t\351rminos con P, Q y R respectivamente. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Por lo tanto  si definimo
s " }{XPPEDIT 18 0 "X = [P, Q, R];" "6#/%\"XG7%%\"PG%\"QG%\"RG" }
{TEXT -1 78 ",  la condici\363n para que la ecuaci\363n sea completame
nte integrable es que  rot(" }{TEXT 269 1 "X" }{TEXT -1 19 ")  sea ort
ogonal a " }{TEXT 271 2 "X." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Ejerci
taci\363n" }}{PARA 0 "" 0 "" {TEXT -1 24 "1)  Considere la 2-forma" }}
{PARA 259 "" 0 "" {XPPEDIT 18 0 "omega = x*`&^`(d(y),d(z))-2*z*f(y)*`&
^`(d(x),d(y))+y*f(y)*`&^`(dz,d(x));" "6#/%&omegaG,(*&%\"xG\"\"\"-%#&^G
6$-%\"dG6#%\"yG-F-6#%\"zGF(F(**\"\"#F(F2F(-%\"fG6#F/F(-F*6$-F-6#F'-F-6
#F/F(!\"\"*(F/F(-F66#F/F(-F*6$%#dzG-F-6#F'F(F(" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "donde " }{TEXT 287 1 "f" }{TEXT -1 25 " es
 una aplicaci\363n clase " }{XPPEDIT 18 0 "C^[1]" "6#)%\"CG7#\"\"\"" }
{TEXT -1 4 " de " }{TEXT 272 1 "R" }{TEXT -1 5 "  en " }{TEXT 273 3 "R
  " }{TEXT -1 15 "tal que f(1)=1." }}{PARA 0 "" 0 "" {TEXT -1 14 "a) D
etermine  " }{TEXT 289 1 "f" }{TEXT -1 11 "  para que " }{XPPEDIT 18 
0 "d(omega) = `&^`(`&^`(d(x),d(y)),d(z));" "6#/-%\"dG6#%&omegaG-%#&^G6
$-F)6$-F%6#%\"xG-F%6#%\"yG-F%6#%\"zG" }{TEXT -1 7 ".\n     " }{TEXT 
274 8 "Ayuda : " }{TEXT -1 63 "el comando para resolver una ecuaci\363
n diferencial ordinaria es " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }}
{PARA 0 "" 0 "" {TEXT -1 14 "b) Determine  " }{TEXT 290 2 "f " }{TEXT 
-1 10 " para que " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 13 "
 sea cerrada." }}{PARA 0 "" 0 "" {TEXT -1 14 "c) Determine  " }{TEXT 
291 2 "f " }{TEXT -1 26 " para que exista una forma" }}{PARA 265 "" 0 
"" {TEXT -1 26 "                          " }{XPPEDIT 18 0 "omega[1] =
 P(x,y,z)*d(x)+Q(x,y,z)*d(y);" "6#/&%&omegaG6#\"\"\",&*&-%\"PG6%%\"xG%
\"yG%\"zGF'-%\"dG6#F-F'F'*&-%\"QG6%F-F.F/F'-F16#F.F'F'" }}{PARA 0 "" 
0 "" {TEXT -1 5 " con " }{XPPEDIT 18 0 "P(x,y,0) = 0;" "6#/-%\"PG6%%\"
xG%\"yG\"\"!F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Q(x,y,0) = 0;" "6#/-%
\"QG6%%\"xG%\"yG\"\"!F)" }{TEXT -1 3 " y " }{XPPEDIT 18 0 "d(omega[1])
 = omega;" "6#/-%\"dG6#&%&omegaG6#\"\"\"F(" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "2) Sea U un abier
to de " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 30 " constitu
\355do por los puntos ( " }{XPPEDIT 18 0 "x,y,z;" "6%%\"xG%\"yG%\"zG" 
}{TEXT -1 13 " ) tales que " }{XPPEDIT 18 0 "xyz;" "6#%$xyzG" }{TEXT 
-1 48 " sea distinto de 0; consideremos en U la 1-forma" }}{PARA 260 "
" 0 "" {XPPEDIT 18 0 "omega = d(x)/(y*z)+d(y)/(x*z)+d(z)/(x*y);" "6#/%
&omegaG,(*&-%\"dG6#%\"xG\"\"\"*&%\"yGF+%\"zGF+!\"\"F+*&-F(6#F-F+*&F*F+
F.F+F/F+*&-F(6#F.F+*&F*F+F-F+F/F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "a) Probar que la ecuaci\363n " }{XPPEDIT 18 0 "omega = 0;
" "6#/%&omegaG\"\"!" }{TEXT -1 29 " es completamente integrable." }}
{PARA 0 "" 0 "" {TEXT -1 38 "b) Determinar el factor integrante de " }
{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 24 ", es decir, una func
i\363n " }{TEXT 292 1 "f" }{TEXT -1 10 " de clase " }{XPPEDIT 18 0 "C^
[1]" "6#)%\"CG7#\"\"\"" }{TEXT -1 9 " tal que " }{XPPEDIT 18 0 "d(f*om
ega) = 0;" "6#/-%\"dG6#*&%\"fG\"\"\"%&omegaGF)\"\"!" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "3) Consi
dere la ecuaci\363n diferencial ordinaria de tercer orden" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 
"" {TEXT -1 26 "                          " }{XPPEDIT 18 0 "diff(y,`$`
(x,3)) = F(diff(y,`$`(x,2)),diff(y,x),x);" "6#/-%%diffG6$%\"yG-%\"$G6$
%\"xG\"\"$-%\"FG6%-F%6$F'-F)6$F+\"\"#-F%6$F'F+F+" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "y construya  un sistema d
e 1-formas equivalente a esta ecuaci\363n diferencial en  las variable
s   (" }{TEXT 275 1 "x" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "u,w,r;" "6%%
\"uG%\"wG%\"rG" }{TEXT -1 16 " ),  definiendo " }}{PARA 262 "" 0 "" 
{XPPEDIT 18 0 "u = y,w = diff(y,x),r = diff(y,`$`(x,2));" "6%/%\"uG%\"
yG/%\"wG-%%diffG6$F%%\"xG/%\"rG-F)6$F%-%\"$G6$F+\"\"#" }}{PARA 0 "" 0 
"" {TEXT -1 72 "El sistema de 1-formas  correspondiente a esta ecuaci
\363n diferencial ser\341" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
263 "" 0 "" {XPPEDIT 18 0 "sigma[1] = 0,sigma[2] = 0,sigma[3] = 0;" "6
%/&%&sigmaG6#\"\"\"\"\"!/&F%6#\"\"#F(/&F%6#\"\"$F(" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Determine
 la forma m\341s general de estas tres 1-formas y demuestre que este s
istema es integrable." }}}}{MARK "0 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }