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{SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 291 1 " " }{TEXT 292 27 "An\341lisis II  - Matem\341tica 3" }
{TEXT -1 1 "\n" }}{PARA 259 "" 0 "" {TEXT 293 15 "TALLER DE MAPLE" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 14 " LABORA
TORIO 3" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "En este laboratorio representaremos gr\341ficamente " }
{TEXT 258 26 "superficies parametrizadas" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Superfici
es parametrizadas: gr\341fico" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(plots):\nwith(p
lottools):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Phi(u,v)=(x(u,v)
,y(u,v),z(u,v))" }{TEXT -1 39 " es una superficie parametrizada donde \+
" }{TEXT 260 1 "u" }{TEXT -1 1 " " }{TEXT 261 8 "= u1..u2" }{TEXT -1 
4 " y  " }{TEXT 258 3 "v =" }{TEXT 256 7 " v1..v2" }{TEXT 259 1 "." }}
}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "Phi:=(u,v)->[u, 0.5*sin(v), .6*v];\n\nu1:=-1;\nu2:=1;\n\nv1:=0;
\nv2:=2*Pi;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "A continuaci\363
n se calculan el gr\341fico de " }{TEXT 262 3 "Phi" }{TEXT -1 14 " y l
os bordes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sup:=plot3d(P
hi(u,v),u=u1..u2,v=v1..v2,scaling=constrained,style=patchnogrid):\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "borde1:=spacecurve(Phi(u1,v),v=v1
..v2,scaling=constrained,color=green,thickness=4):\n\nborde2:=spacecur
ve(Phi(u2,v),v=v1..v2,scaling=constrained,color=blue,thickness=3):\n\n
borde3:=spacecurve(Phi(u,v1),u=u1..u2,scaling=constrained,color=red,th
ickness=4):\n\nborde4:=spacecurve(Phi(u,v2),u=u1..u2,scaling=constrain
ed,color=yellow,thickness=3):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "display3d([sup,borde1,borde2,borde3,borde4]);" }}{PARA 13 "" 
0 "" {TEXT -1 0 "" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 29 "Vectores tangentes y normales" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "A partir
 de la parametriazaci\363n " }{TEXT 264 3 "Phi" }{TEXT -1 15 " definid
a antes" }{TEXT 263 2 ", " }{TEXT -1 74 "se graficar\341 la imagen del
 segmento que une dos puntos dados del dominio  " }{TEXT 266 1 "(" }
{TEXT 256 7 "punto0 " }{TEXT -1 1 "y" }{TEXT 257 10 " punto1). " }
{TEXT -1 111 "Adem\341s se animar\341 el desplazamiento de los vectore
s tangente y normal a la superficie a lo largo de esta curva." }{TEXT 
265 2 " \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "D\351 dos puntos de
ntro del dominio:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "punto0
:=[-1,0]:\npunto1:=[1,2*Pi]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "nframes:=23:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "i:='i':\n
\npunto:=(punto0[1]+(punto1[1]-punto0[1])*i/nframes,\n         punto0[
2]+(punto1[2]-punto0[2])*i/nframes):\n\ncurva:=spacecurve(Phi(punto),i
=0..nframes,color=yellow,thickness=2):\n\ndu:=unapply(diff(Phi(u,v),u)
,(u,v)):\ndv:=unapply(diff(Phi(u,v),v),(u,v)):\n\ni:=0:\n" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 136 "tangu:=line(Phi(punto),Phi(punto)+du(punto)
,color=blue,thickness=2):\ntangv:=line(Phi(punto),Phi(punto)+dv(punto)
,color=red,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "\np:=po
intplot3d(Phi(punto),color=black,symbol=circle):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 174 "\nn:=[du(punto)[2]*dv(punto)[3]-du(punto)[3]*dv(punt
o)[2],\n    du(punto)[3]*dv(punto)[1]-du(punto)[1]*dv(punto)[3],\n    \+
du(punto)[1]*dv(punto)[2]-du(punto)[2]*dv(punto)[1]]  :" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "nor:=line(Phi(punto),Phi(punto)+n,color=black,
thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "\n\ncuadro[0]:=dis
play3d([sup,p,tangu,tangv,nor,curva]):\n\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "for i from 1 to nframes do \n\np:=pointplot3d(Phi(pun
to),color=black,symbol=circle):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "du:=unapply(diff(Phi(u,v),u),(u
,v));\ndv:=unapply(diff(Phi(u,v),v),(u,v));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 136 "tangu:=line(Phi(punto),Phi(punto)+du(punto),color=bl
ue,thickness=2):\ntangv:=line(Phi(punto),Phi(punto)+dv(punto),color=re
d,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 173 "n:=[du(punto)[2]*dv(punto)[3]-du(punto)[3]*dv(p
unto)[2],\n    du(punto)[3]*dv(punto)[1]-du(punto)[1]*dv(punto)[3],\n \+
   du(punto)[1]*dv(punto)[2]-du(punto)[2]*dv(punto)[1]]  :" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 59 "nor:=line(Phi(punto),Phi(punto)+n,color=bla
ck,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "cuadro[i]:=display3d([sup,p,tangu,tangv,nor,cu
rva,cuadro[0]]);\n\nod:\n\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "di
splay3d(seq(cuadro[i],i=0..nframes),insequence=true,\nscaling=constrai
ned);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Ejercitaci\363n" }}
{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 267 3 "1.-" }{TEXT -1 1 " " }
{TEXT 268 0 "" }{TEXT 269 7 "Esfera:" }{TEXT 270 0 "" }{TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "Phi(u,v) = [cos(u)*sin(v), sin(u)*sin(v), cos(v)];" 
"6#/-%$PhiG6$%\"uG%\"vG7%*&-%$cosG6#F'\"\"\"-%$sinG6#F(F.*&-F06#F'F.-F
06#F(F.-F,6#F(" }{TEXT -1 124 ",  u = 0..2*Pi,  v = 0..Pi.\n\n        \+
a) Suponga que orientamos positivamente la esfera con respecto a la no
rmal exterior.  \277" }{XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT -1 78 " \+
conserva la orientaci\363n?\n     \n        b) Grafique s\363lo el hem
isferio norte.\n" }}{PARA 0 "" 0 "" {TEXT -1 46 "        c) Grafique e
l hemisferio occidental.\n" }}{PARA 0 "" 0 "" {TEXT -1 69 "        d) \+
Grafique una banda en torno del ecuador.\n    \n        e) \277" }
{XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT -1 94 " es suave en el polo nor
te?  Observe el comportamiento de la normal cuando se desplaza desde \+
\n" }{TEXT 256 19 "             punto0" }{TEXT -1 16 "=[0,Pi/2] hasta \+
" }{TEXT 257 6 "punto1" }{TEXT -1 9 "=[0, 0].\n" }}{PARA 0 "" 0 "" 
{TEXT -1 86 "        f) D\351 una parametrizaci\363n de la esfera en l
a que se preserve la orientaci\363n.\n\n" }{TEXT 285 4 "2.- " }{TEXT 
286 5 "Cono:" }{TEXT 287 0 "" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Phi(u,v
) = [u*cos(v), u*sin(v), u];" "6#/-%$PhiG6$%\"uG%\"vG7%*&F'\"\"\"-%$co
sG6#F(F+*&F'F+-%$sinG6#F(F+F'" }{TEXT -1 107 " ,  u = -1..1,  v = 0..2
*Pi. \n\n         a) D\351 una parametrizaci\363n que invierta la orie
ntaci\363n inducida por " }{XPPEDIT 18 0 "Phi;" "6#%$PhiG" }{TEXT -1 
100 ".\n\n         b) Elija un camino que pase por el v\351rtice del c
ono. \277Qu\351 pasa con la normal?\n         \n" }{TEXT 271 4 "3.- " 
}{TEXT 272 5 "Toro:" }{TEXT 273 0 "" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "
Phi(u,v) = [2*cos(v)+.5*sin(u)*cos(v), 2*sin(v)+.5*sin(v)*sin(u), .5*c
os(u)];" "6#/-%$PhiG6$%\"uG%\"vG7%,&*&\"\"#\"\"\"-%$cosG6#F(F-F-*(-%&F
loatG6$\"\"&!\"\"F--%$sinG6#F'F--F/6#F(F-F-,&*&F,F--F86#F(F-F-*(-F36$F
5F6F--F86#F(F--F86#F'F-F-*&-F36$F5F6F--F/6#F'F-" }{TEXT -1 31 ",  u = \+
0..2*Pi,  v = 0..2*Pi.\n\n" }{TEXT 274 4 "4.- " }{TEXT 275 17 "Cinta d
e Moebius:" }{TEXT 276 0 "" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Phi(u,v) \+
= [2*cos(u)+v*sin(u/2)*cos(u), 2*sin(u)+v*sin(u/2)*sin(u), v*cos(u/2)]
;" "6#/-%$PhiG6$%\"uG%\"vG7%,&*&\"\"#\"\"\"-%$cosG6#F'F-F-*(F(F--%$sin
G6#*&F'F-F,!\"\"F--F/6#F'F-F-,&*&F,F--F36#F'F-F-*(F(F--F36#*&F'F-F,F6F
--F36#F'F-F-*&F(F--F/6#*&F'F-F,F6F-" }{TEXT -1 132 ",                 \+
                                                                  u = \+
0..2*Pi,   v = -2..2.\n\n        a) Considere " }{TEXT 277 6 "punto0" 
}{TEXT -1 9 "=[0,0] y " }{TEXT 278 6 "punto1" }{TEXT -1 32 "=[Pi,0]. \+
\n\n        b) Considere " }{TEXT 256 6 "punto0" }{TEXT -1 9 "=[0,0] y
 " }{TEXT 257 6 "punto1" }{TEXT -1 87 "=[2*Pi,0]. \n\n        c) \277C
u\341ntas caras tiene la cinta? \277Es una superficie orientable?\n\n
" }{TEXT 279 4 "5.- " }{TEXT 280 10 "Helicoide:" }{TEXT 281 0 "" }
{TEXT -1 3 "   " }{XPPEDIT 18 0 "Phi(u,v) = [u*cos(v), u*sin(v), v];" 
"6#/-%$PhiG6$%\"uG%\"vG7%*&F'\"\"\"-%$cosG6#F(F+*&F'F+-%$sinG6#F(F+F(
" }{TEXT -1 29 ",  u = 0..4,  v = 0..3*Pi.\n\n\n" }{TEXT 282 4 "6.- " 
}{TEXT 283 23 "Gr\341fico de una funci\363n:" }{TEXT 284 0 "" }{TEXT 
-1 106 "  Supondremos que el gr\341fico de una funci\363n esta orienta
do positivamente si la normal tiene su coordenada " }{TEXT 294 1 "z" }
{TEXT -1 80 " positiva.\n\n     a) D\351 una parametrizaci\363n positi
va de la gr\341fica de la funci\363n " }{XPPEDIT 18 0 "f(x,y) = (x^2-y
^2)/2;" "6#/-%\"fG6$%\"xG%\"yG*&,&*$F'\"\"#\"\"\"*$F(F,!\"\"F-F,F/" }
{TEXT -1 132 "  donde x = -1..1 e  y = -1..1.\n\n     b) D\351 una par
ametrizaci\363n que induzca una orientaci\363n negativa en la superfic
ie del item a).\n\n" }{TEXT 288 4 "7.- " }{TEXT 289 25 "Superficie de \+
revoluci\363n:" }{TEXT 290 0 "" }{TEXT -1 97 " \n\n      a) Parametric
e la superficie engendrada al rotar 360\272 alrededor del eje x la gr
\341fica de " }{XPPEDIT 18 0 "f(x) = x^2+1;" "6#/-%\"fG6#%\"xG,&*$F'\"
\"#\"\"\"F+F+" }{TEXT -1 53 " con x = -1..1.\n\n      b) Pruebe con ot
ras funciones." }}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }