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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 1 "\n" }{TEXT 260 43 "An\341lisis I
 - Matem\341tica 1 - An\341lisis II (C)" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT 261 15 "TALLER DE MAPLE" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 14 " LABORATORIO 5" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 100 "En este laboratorio  trabajaremos con ca
mpos gradientes y su relaci\363n con las superficies de nivel.\n" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Los comandos " }{TEXT 262 8 "grad
plot" }{TEXT -1 3 " y " }{TEXT 263 10 "gradplot3d" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Los comandos " }
{HYPERLNK 17 "gradplot" 2 "gradplot" "" }{TEXT -1 3 " y " }{HYPERLNK 
17 "gradplot3d" 2 "gradplot3d" "" }{TEXT -1 98 " permiten graficar el \+
campo gradiente de funciones reales de una y dos variables respectivam
ente.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots):\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "gradplot(x^2+y^2,x=-2..2,y=-2..2,scaling=con
strained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "gradplot(x^2-
y^2,x=-2..2,y=-2..2,scaling=constrained,arrows=slim,\ngrid=[10,10]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "gradplot3d( (x^2+y^2-z^2)
,x=-2..2,y=-2..2,z=-2..2,grid=[5,5,5]);\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comand
o " }{TEXT 264 4 "grad" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "grad" 2 "grad
" "" }{TEXT -1 112 " permite hallar el gradiente de una funci\363n de \+
varias variables. Se encuentra en el  paquete de Algebra Lineal.\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=grad(3*x^2 + 2*y*z, [x,y,z]);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 2 "g;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "eval(g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eval(g,\{x
=1,y=-2,z=0\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval(eva
l(g),\{x=1,y=-2,z=0\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
h:=(u,v,w)->eval(eval(g),\{x=u,y=v,z=w\});" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "h(1,-2,0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "
Para evaluar expresiones que involucran matrices o vectores hay que us
ar el comando " }{HYPERLNK 17 "evalm" 2 "evalm" "" }{TEXT -1 2 ".\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "h(1,-2,0)+[1,2,3];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(h(1,-2,0)+[1,2,3]);\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "El comando " }{HYPERLNK 17 "d
otprod" 2 "dotprod" "" }{TEXT -1 69 "  permite hallar el producto esca
lar (o interno) entre dos vectores.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "dotprod([1,2,3],[3,0,1],'orthogonal');" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dotprod([1,2,3],[x,y,z],'orthogonal
');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dotprod(h(1,-2,0),h(
1,-2,0),'orthogonal');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "El comando " }{TEXT 265 5 "ar
row" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 11 "El comando " }{HYPERLNK 17 "arrow" 2 "arrow" "" }{TEXT -1 12 " \+
 genera un " }{TEXT 256 14 "objeto gr\341fico" }{TEXT 259 1 " " }
{TEXT -1 82 "que representa una flecha. Para visualizar este objeto de
be utilizarse el comando " }{TEXT 257 7 "display" }{TEXT -1 2 ".\n" }
{TEXT 266 5 "Arrow" }{TEXT -1 28 " se encuentra en el paquete " }
{TEXT 258 9 "plottools" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "restart:\nwith(plots):\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 24 "El  primer argumento de " }{TEXT 267 5 "arrow" }{TEXT -1 
87 " es el punto base de la flecha. El segundo es el punto extremo o e
l vector direcci\363n :\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
146 "l1 := plottools[arrow]([0,0,0],[10,10,10], .2, .5, .1, color=gree
n):\nl2 := plottools[arrow]([10,10,10], vector([0,0,5]), .2, .4, .21, \+
color=red):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display3d \+
(\{l1,l2\},axes=normal,orientation=[-7,83]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Relaci
\363n entre gradiente y curvas y superficies de nivel" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 154 "Graficaremos juntos el campo gradiente d
e una funci\363n de dos variables y algunas curvas de nivel. Recordar \+
que el gradiente es perpendicular a las curvas." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "\nf:=x^2+y^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "p:=contourplot(f,x=-1.3..1.3,y=-1.3..1.3,contours=[1/
2,1,1.5],\nscaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "g:=gradplot(f,x=-1.3..1.3,y=-1.3..1.3,scaling=constra
ined,arrows=slim,grid=[10,10]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "display(\{g,p\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Graficaremos juntos el camp
o gradiente de una funci\363n de tres variables y la superficie de niv
el 0. Recordar que el gradiente es perpendicular a la superficie.\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x^2+y-z+1;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 65 "p:=implicitplot3d(f=0,x=-3..3,y=-3..3,z=-3..
3,style=patchnogrid):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "g:=gradplo
t3d(f,x=-3..3,y=-3..3,z=-3..3,grid=[7,7,7],color=blue):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display3
d(\{p,g\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 308 "Sabiendo que el gradiente se\361ala la d
irecci\363n de mayor crecimiento de la funci\363n y es ortogonal a la \+
superficie de nivel, c\363mo cree que el gr\341fico anterior cambia si
 consideramos superficies de nivel mayores que 0. Trate de imaginarlo \+
y luego vea la siguiente animaci\363n de las superficies de nivel de 0
 a 4.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "for i from 1 to
 25 do\np:=implicitplot3d(f=i/5,x=-3..3,y=-3..3,z=-3..3,numpoints=500)
:\ncuadro[i]:=display3d([g,p]):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "display3d(seq([cuadro[i]],i=1..25),insequence=true,sc
aling=constrained,\nstyle=patchnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Plano tange
nte a una supeficie de nivel" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Graficaremos el vector ortogonal e
n un punto P de una superfcie de nivel de la funci\363n " }{TEXT 268 
1 "f" }{TEXT -1 30 " junto con el plano tangente.\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plott
ools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots):\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=(x,y,z)->x^2+y^2-z^2;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "F:=implicitplot3d(f(x,y,z)
=1/4,x=-2..2,y=-2..2,z=-1.2..1.2,numpoints=3000,scaling=constrained,or
ientation=[22,74]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "F;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Consideremos \+
un punto en la superficie de nivel.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "p:=(1/2,1/2,1/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 59 "Hallemos y grafiquemos el vector gradiente trasladado a P.\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "g:=(u,v,w)->eval(evalm(gra
d(f(x,y,z),[x,y,z])),\{x=u,y=v,z=w\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "v:=plottools[arrow]([p],evalm(g(p)),.06,.2,.1,color=r
ed):" }}}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "display3d(\{v,F\},scaling=constrained,orientation=[10
8,87]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Ahora dibujaremos el plano tang
ente a la superficie en el punto P.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "h:=dotprod(g(p),[x,y,z]-vector([p]),'orthogonal');" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plano:=implicitplot3d(h=0,
x=-2..2,y=-2..2,z=-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
108 "display3d([plano,v,F],scaling=constrained,view=[-2..2,-2..2,-2..2
],\norientation=[104,88],style=patchnogrid);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Ejercicio" }}{PARA 0 "" 0 "" 
{TEXT -1 60 "Realice una animaci\363n de las curvas de nivel de la fun
ci\363n  " }{XPPEDIT 18 0 "z = x^2-y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"
*$%\"yGF(!\"\"" }{TEXT -1 31 "  junto con el campo gradiente." }}}}
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