\documentclass{article}
\usepackage[latin1]{inputenc}
\usepackage[spanish]{babel}
\usepackage{amssymb}
\usepackage{graphicx}
\setlength{\topmargin}{-2.5cm}  \setlength{\textheight}{25cm}
\setlength{\oddsidemargin}{0.0cm} \setlength{\evensidemargin}{0.0cm}
\setlength{\textwidth}{16cm} \thispagestyle{empty}
\NeedsTeXFormat{LaTeX2e}[1995/12/01]


\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Author}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Plot}
\DefineParaStyle{Title}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
\begin{document}
\pagestyle{empty}
\begin{maplegroup}
\begin{Title}
Laboratorios de Maple \\%
Geometría Diferencial MATE2410 \\%
Lab3 Representación de Weierstrass - Enneper II
\end{Title}

\begin{Author}
Prof. José Ricardo ARTEAGA BEJARANO
\end{Author}

\end{maplegroup}

\section{Funciones holomorfas}

\subsection{Función holomorfa}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{restart: with(plots):with(liesymm):with(linalg):}{%
}
\end{mapleinput}

\mapleresult
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\begin{maplettyout}
Warning, the protected name close has been redefined and unprotected
\end{maplettyout}

\begin{maplettyout}
Warning, the protected names norm and trace have been redefined and
unprotected
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{assume(u,real);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{assume(v,real);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Phi:=z^(3);
F(z):=subs(z=u+I*v,Phi);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Phi := z^3;}{%
\[
\Phi  := z^{3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{F(z) := (u+I*v)^3;}{%
\[
\mathrm{F}(z) := (\mathit{u\symbol{126}} + I\,\mathit{v
\symbol{126}})^{3}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{phi(u,v):=evalc(Re(F(z)));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{psi(u,v):=evalc(Im(F(z)));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{phi(u,v) := u^3-3*u*v^2;}{%
\[
\phi (\mathit{u\symbol{126}}, \,\mathit{v\symbol{126}}) :=
\mathit{u\symbol{126}}^{3} - 3\,\mathit{u\symbol{126}}\,\mathit{v
\symbol{126}}^{2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{psi(u,v) := 3*u^2*v-v^3;}{%
\[
\psi (\mathit{u\symbol{126}}, \,\mathit{v\symbol{126}}) := 3\,
\mathit{u\symbol{126}}^{2}\,\mathit{v\symbol{126}} - \mathit{v
\symbol{126}}^{3}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{laplacian(phi(u,v), [u,v]);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{laplacian(psi(u,v), [u,v]);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{diff(phi(u,v),u)-diff(psi(u,v),v);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{diff(phi(u,v),v)+diff(psi(u,v),u);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{0;}{%
\[
0
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{0;}{%
\[
0
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{0;}{%
\[
0
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{0;}{%
\[
0
\]
%
}
\end{maplelatex}

\end{maplegroup}

\subsection{Superfcie asociada}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X:=[u,v,phi(u,v),psi(u,v)];}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Xu:=diff(X,u);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Xv:=diff(X,v);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{E:=simplify(dotprod(Xu,Xu));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{F:=simplify(dotprod(Xu,Xv));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{G:=simplify(dotprod(Xv,Xv));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Xuu:=diff(Xu,u);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Xuv:=diff(Xu,v);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Xvv:=diff(Xv,v);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{X := [u, v, u^3-3*u*v^2, 3*u^2*v-v^3];}{%
\[
X := [\mathit{u\symbol{126}}, \,\mathit{v\symbol{126}}, \,
\mathit{u\symbol{126}}^{3} - 3\,\mathit{u\symbol{126}}\,\mathit{v
\symbol{126}}^{2}, \,3\,\mathit{u\symbol{126}}^{2}\,\mathit{v
\symbol{126}} - \mathit{v\symbol{126}}^{3}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Xu := [1, 0, 3*u^2-3*v^2, 6*u*v];}{%
\[
\mathit{Xu} := [1, \,0, \,3\,\mathit{u\symbol{126}}^{2} - 3\,
\mathit{v\symbol{126}}^{2}, \,6\,\mathit{u\symbol{126}}\,\mathit{
v\symbol{126}}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Xv := [0, 1, -6*u*v, 3*u^2-3*v^2];}{%
\[
\mathit{Xv} := [0, \,1, \, - 6\,\mathit{u\symbol{126}}\,\mathit{v
\symbol{126}}, \,3\,\mathit{u\symbol{126}}^{2} - 3\,\mathit{v
\symbol{126}}^{2}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{E := 1+9*u^4+18*u^2*v^2+9*v^4;}{%
\[
E := 1 + 9\,\mathit{u\symbol{126}}^{4} + 18\,\mathit{u
\symbol{126}}^{2}\,\mathit{v\symbol{126}}^{2} + 9\,\mathit{v
\symbol{126}}^{4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{F := 0;}{%
\[
F := 0
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{G := 1+9*u^4+18*u^2*v^2+9*v^4;}{%
\[
G := 1 + 9\,\mathit{u\symbol{126}}^{4} + 18\,\mathit{u
\symbol{126}}^{2}\,\mathit{v\symbol{126}}^{2} + 9\,\mathit{v
\symbol{126}}^{4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Xuu := [0, 0, 6*u, 6*v];}{%
\[
\mathit{Xuu} := [0, \,0, \,6\,\mathit{u\symbol{126}}, \,6\,
\mathit{v\symbol{126}}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Xuv := [0, 0, -6*v, 6*u];}{%
\[
\mathit{Xuv} := [0, \,0, \, - 6\,\mathit{v\symbol{126}}, \,6\,
\mathit{u\symbol{126}}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Xvv := [0, 0, -6*u, -6*v];}{%
\[
\mathit{Xvv} := [0, \,0, \, - 6\,\mathit{u\symbol{126}}, \, - 6\,
\mathit{v\symbol{126}}]
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{H:=simplify((1+G)*(Xuu[3]+Xuu[4])+(1+E)*(Xvv[3]+Xvv[4])-2*F*(Xuv[3]+X
uv[4]));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{H := 0;}{%
\[
H := 0
\]
%
}
\end{maplelatex}

\end{maplegroup}

\section{Representación de Weierstrass - Enneper}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{restart: with(plots): with(linalg):}{%
}
\end{mapleinput}

\mapleresult
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\begin{maplettyout}
Warning, the protected names norm and trace have been redefined and
unprotected
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{F(tau):=1/(2*tau^2);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X:=((int((1-tau^2)*F(tau),tau)));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Y:=((int(I*(1+tau^2)*F(tau),tau)));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Z:=((int(2*tau*F(tau),tau)));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{F(tau) := 1/2*1/(tau^2);}{%
\[
\mathrm{F}(\tau ) := {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {1}{\tau ^{2}}}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X := -1/2*tau-1/2*1/tau;}{%
\[
X :=  - {\displaystyle \frac {1}{2}} \,\tau  - {\displaystyle
\frac {1}{2}} \,{\displaystyle \frac {1}{\tau }}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Y := 1/2*I*(tau-1/tau);}{%
\[
Y := {\displaystyle \frac {1}{2}} \,I\,(\tau  - {\displaystyle
\frac {1}{\tau }} )
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Z := ln(tau);}{%
\[
Z := \mathrm{ln}(\tau )
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X:=subs(tau=exp(z),X);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Y:=subs(tau=exp(z),Y);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Z:=subs(tau=exp(z),Z);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{X := -1/2*exp(z)-1/2*1/exp(z);}{%
\[
X :=  - {\displaystyle \frac {1}{2}} \,e^{z} - {\displaystyle
\frac {1}{2}} \,{\displaystyle \frac {1}{e^{z}}}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Y := 1/2*I*(exp(z)-1/exp(z));}{%
\[
Y := {\displaystyle \frac {1}{2}} \,I\,(e^{z} - {\displaystyle
\frac {1}{e^{z}}} )
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Z := ln(exp(z));}{%
\[
Z := \mathrm{ln}(e^{z})
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X:=subs(z=u+I*v,X);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Y:=subs(z=u+I*v,Y);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{Z:=subs(z=u+I*v,Z);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{X := -1/2*exp(u+I*v)-1/2*1/exp(u+I*v);}{%
\[
X :=  - {\displaystyle \frac {1}{2}} \,e^{(u + I\,v)} -
{\displaystyle \frac {1}{2}} \,{\displaystyle \frac {1}{e^{(u + I
\,v)}}}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Y := 1/2*I*(exp(u+I*v)-1/exp(u+I*v));}{%
\[
Y := {\displaystyle \frac {1}{2}} \,I\,(e^{(u + I\,v)} -
{\displaystyle \frac {1}{e^{(u + I\,v)}}} )
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{Z := ln(exp(u+I*v));}{%
\[
Z := \mathrm{ln}(e^{(u + I\,v)})
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X1Re:=simplify(factor(evalc(Re(X))),trig);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X1Im:=simplify(factor(evalc(Im(X))),trig);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X2Re:=simplify(factor(evalc(Re(Y))),trig);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X2Im:=simplify(factor(evalc(Im(Y))),trig);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X3Re:=simplify(factor(evalc(Re(Z))),trig);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{X3Im:=simplify(factor(evalc(Im(Z))),trig);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{X1Re := -1/2*cos(v)*(exp(u)^2+1)/exp(u);}{%
\[
\mathit{X1Re} :=  - {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {\mathrm{cos}(v)\,((e^{u})^{2} + 1)}{e^{u}}
}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X1Im := -1/2*sin(v)*(exp(u)^2-1)/exp(u);}{%
\[
\mathit{X1Im} :=  - {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {\mathrm{sin}(v)\,((e^{u})^{2} - 1)}{e^{u}}
}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X2Re := -1/2*sin(v)*(exp(u)^2+1)/exp(u);}{%
\[
\mathit{X2Re} :=  - {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {\mathrm{sin}(v)\,((e^{u})^{2} + 1)}{e^{u}}
}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X2Im := 1/2*cos(v)*(exp(u)^2-1)/exp(u);}{%
\[
\mathit{X2Im} := {\displaystyle \frac {1}{2}} \,{\displaystyle
\frac {\mathrm{cos}(v)\,((e^{u})^{2} - 1)}{e^{u}}}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X3Re := u;}{%
\[
\mathit{X3Re} := u
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{X3Im := v;}{%
\[
\mathit{X3Im} := v
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{RRR:=evalm([X1Re,X2Re,X3Re]);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{RRI:=[X1Im,X2Im,X3Im];}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{RRR := vector([-1/2*cos(v)*(exp(u)^2+1)/exp(u),
-1/2*sin(v)*(exp(u)^2+1)/exp(u), u]);}{%
\[
\mathit{RRR} :=  \left[  \!  - {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {\mathrm{cos}(v)\,((e^{u})^{2} + 1)}{e^{u}}
} , \, - {\displaystyle \frac {1}{2}} \,{\displaystyle \frac {
\mathrm{sin}(v)\,((e^{u})^{2} + 1)}{e^{u}}} , \,u \!  \right]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{RRI := [-1/2*sin(v)*(exp(u)^2-1)/exp(u),
1/2*cos(v)*(exp(u)^2-1)/exp(u), v];}{%
\[
\mathit{RRI} := [ - {\displaystyle \frac {1}{2}} \,
{\displaystyle \frac {\mathrm{sin}(v)\,((e^{u})^{2} - 1)}{e^{u}}
} , \,{\displaystyle \frac {1}{2}} \,{\displaystyle \frac {
\mathrm{cos}(v)\,((e^{u})^{2} - 1)}{e^{u}}} , \,v]
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plot3d(RRR,u=-1.3..1.3,v=0..2*Pi,orientation=[134,86],numpoints=5000,
style=PATCH,title=`Superfcie mínima asociada a F(tau)`);}{%
}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{lab301.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
\end{document}
%% End of Maple 7.00 Output
