%% Created by Maple 7.00 (IBM INTEL NT)
%% Source Worksheet: puntos4y5.mws
%% Generated: Sun Oct 23 04:06:20 2005
\documentclass{article}
\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Bullet Item}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Maple Output}
\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  
\end{Title}

\begin{Title}
Geometría Diferencial MATE2410
\end{Title}

\begin{Title}
{\small Prof. José Ricardo ARTEAGA BEJARANO}\QTR{2D Comment}{
}
\end{Title}

\end{maplegroup}

\section{Puntos 4 y 5 del parcial 2}

\subsection{Coeficientes de la Primera Forma Fundamental}

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

\begin{mapleinput}
\mapleinline{active}{1d}{EFG:=proc(X)
local Xu,Xv,E,F,G;
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
E:=DotProduct(Xu,Xu,conjugate=false);
F:=DotProduct(Xu,Xv,conjugate=false);
G:=DotProduct(Xv,Xv,conjugate=false);
simplify([E,F,G]);
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{UN:=proc(X)
local Xu,Xv,Z,s;
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
Z:=CrossProduct(Xu,Xv);
s:=VectorNorm(Z,Euclidean,conjugate=false);
simplify(<Z[1]/s|Z[2]/s|Z[3]/s>,sqrt,trig,trigh,simbolic);
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{lmn:=proc(X)
local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
Xuu:=<diff(Xu[1],u),diff(Xu[2],u),diff(Xu[3],u)>;
Xuv:=<diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)>;
Xvv:=<diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)>;
U:=UN(X);
l:=DotProduct(U,Xuu,conjugate=false);
m:=DotProduct(U,Xuv,conjugate=false);
n:=DotProduct(U,Xvv,conjugate=false);
simplify([l,m,n],sqrt,trig,trigh,symbolic);
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{shape:=proc(X)
local Y,Z,a,b,c,d;
Y:=EFG(X);
Z:=lmn(X);
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
[S(x_u)=a*x_u+b*x_v,S(x_v)=c*x_u+d*x_v];
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{shape_matrix:=proc(X)
local Y,Z,a,b,c,d;
Y:=EFG(X);
Z:=lmn(X);
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));
Matrix([[a,c],[c,d]]);
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{MK:=proc(X)
local E,F,G,l,m,n,S,T;
S:=EFG(X); T:=lmn(X);
E:=S[1]; F:=S[2]; G:=S[3];
l:=T[1]; m:=T[2]; n:=T[3];
simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2),sqrt,trig,trigh,symbolic);
end:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{GK:=proc(X)
local E,F,G,l,m,n,S,T;
S:=EFG(X); T:=lmn(X);
E:=S[1]; F:=S[2]; G:=S[3];
l:=T[1]; m:=T[2]; n:=T[3];
simplify((l*n-m^2)/(E*G-F^2),sqrt,trig,trigh,symbolic);
end:}{%
}
\end{mapleinput}

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

\end{maplegroup}

\subsection{Punto 4}

\subsubsection{Pseudoesfera}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{sup:=[(1/cosh(v))*cos(u),(1/cosh(v))*sin(u),v-tanh(v)];}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{PFF:=EFG(sup):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{I_:=simplify(Matrix([[PFF[1],PFF[2]],[PFF[2],PFF[3]]]));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{lmn:=lmn(sup):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{II_=simplify(Matrix([[lmn[1],lmn[2]],[lmn[2],lmn[3]]]));}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{shape(sup):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{S:=shape_matrix(sup);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{H:=MK(sup);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{K:=GK(sup);
plot3d(subs(R=4,r=1,sup),u=0..2*Pi,v=0..2*Pi,scaling=constrained);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{sup := [1/cosh(v)*cos(u), 1/cosh(v)*sin(u), v-tanh(v)];}{%
\[
\mathit{sup} := [{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{
cosh}(v)}} , \,v - \mathrm{tanh}(v)]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{I_ := _rtable[12705076];}{%
\[
\mathit{I\_} :=  \left[ 
{\begin{array}{cc}
{\displaystyle \frac {1}{\mathrm{cosh}(v)^{2}}}  & 0 \\ [2ex]
0 & {\displaystyle \frac {\mathrm{cosh}(v)^{2} - 1}{\mathrm{cosh}
(v)^{2}}} 
\end{array}}
 \right] 
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{II_ = _rtable[13110316];}{%
\[
\mathit{II\_}= \left[ 
{\begin{array}{c}
 - {\displaystyle \frac {\sqrt{\mathrm{cosh}(v)([{\displaystyle 
\frac {\mathrm{cos}(u)}{\mathrm{cosh}(v)}} , \,{\displaystyle 
\frac {\mathrm{sin}(u)}{\mathrm{cosh}(v)}} , \, - {\displaystyle 
\frac { - v\,\mathrm{cosh}(v) + \mathrm{sinh}(v)}{\mathrm{cosh}(v
)}} ])^{2} - 1}}{\mathrm{cosh}(v)([{\displaystyle \frac {\mathrm{
cos}(u)}{\mathrm{cosh}(v)}} , \,{\displaystyle \frac {\mathrm{sin
}(u)}{\mathrm{cosh}(v)}} , \,{\displaystyle \frac {v\,\mathrm{
cosh}(v) - \mathrm{sinh}(v)}{\mathrm{cosh}(v)}} ])^{2}}} \,, \,0
 \\ [2ex]
0\,, \,{\displaystyle \frac {\sqrt{\mathrm{cosh}(v)([
{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{cosh}(v)}} , \,
{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{cosh}(v)}} , \, - 
{\displaystyle \frac { - v\,\mathrm{cosh}(v) + \mathrm{sinh}(v)}{
\mathrm{cosh}(v)}} ])^{2} - 1}}{\mathrm{cosh}(v)([{\displaystyle 
\frac {\mathrm{cos}(u)}{\mathrm{cosh}(v)}} , \,{\displaystyle 
\frac {\mathrm{sin}(u)}{\mathrm{cosh}(v)}} , \,{\displaystyle 
\frac {v\,\mathrm{cosh}(v) - \mathrm{sinh}(v)}{\mathrm{cosh}(v)}
} ])^{2}}} 
\end{array}}
 \right] 
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{S := _rtable[12629144];}{%
\maplemultiline{
S :=  \left[ 
{\begin{array}{c}
 - {\displaystyle \frac {\sqrt{\mathrm{cosh}(v)(\mathrm{\%1})([
{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{cosh}(v)}} , \,
{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{cosh}(v)}} , \, - 
{\displaystyle \frac { - v\,\mathrm{cosh}(v) + \mathrm{sinh}(v)}{
\mathrm{cosh}(v)}} ])^{2} - 1}\,\mathrm{cosh}(v)^{2}}{\mathrm{
cosh}(v)(\mathrm{\%1})(\mathrm{\%1})^{2}}} \,, \,0 \\ [2ex]
0\,, \,{\displaystyle \frac {\sqrt{\mathrm{cosh}(v)(\mathrm{\%1})
([{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{cosh}(v)}} , \,
{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{cosh}(v)}} , \, - 
{\displaystyle \frac { - v\,\mathrm{cosh}(v) + \mathrm{sinh}(v)}{
\mathrm{cosh}(v)}} ])^{2} - 1}\,\mathrm{cosh}(v)^{2}}{\mathrm{
cosh}(v)(\mathrm{\%1})(\mathrm{\%1})^{2}\,(\mathrm{cosh}(v)^{2}
 - 1)}} 
\end{array}}
 \right]  \\
\mathrm{\%1} := [{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {v\,\mathrm{cosh}(v) - 
\mathrm{sinh}(v)}{\mathrm{cosh}(v)}} ] }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{H := -1/2*(cosh(v)([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])^2-1)^(1/2)*(cosh(v)^2-2)*cosh(v)^2/cosh(
v)([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])^2/(cosh(v)^2-1);}{%
\maplemultiline{
H :=  - {\displaystyle \frac {1}{2}} \,{\displaystyle \frac {
\sqrt{\mathrm{cosh}(v)(\mathrm{\%1})(\mathrm{\%1})^{2} - 1}\,(
\mathrm{cosh}(v)^{2} - 2)\,\mathrm{cosh}(v)^{2}}{\mathrm{cosh}(v)
(\mathrm{\%1})(\mathrm{\%1})^{2}\,(\mathrm{cosh}(v)^{2} - 1)}} 
 \\
\mathrm{\%1} := [{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {v\,\mathrm{cosh}(v) - 
\mathrm{sinh}(v)}{\mathrm{cosh}(v)}} ] }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{K := -1/cosh(v)([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])([1/cosh(v)*cos(u), 1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])^4*(cosh(v)([1/cosh(v)*cos(u),
1/cosh(v)*sin(u), (v*cosh(v)-sinh(v))/cosh(v)])([1/cosh(v)*cos(u),
1/cosh(v)*sin(u),
(v*cosh(v)-sinh(v))/cosh(v)])^2-1)/(cosh(v)^2-1)*cosh(v)^4;}{%
\maplemultiline{
K :=  - {\displaystyle \frac {(\mathrm{cosh}(v)(\mathrm{\%1})(
\mathrm{\%1})^{2} - 1)\,\mathrm{cosh}(v)^{4}}{\mathrm{cosh}(v)(
\mathrm{\%1})(\mathrm{\%1})^{4}\,(\mathrm{cosh}(v)^{2} - 1)}} 
 \\
\mathrm{\%1} := [{\displaystyle \frac {\mathrm{cos}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {\mathrm{sin}(u)}{\mathrm{
cosh}(v)}} , \,{\displaystyle \frac {v\,\mathrm{cosh}(v) - 
\mathrm{sinh}(v)}{\mathrm{cosh}(v)}} ] }
%
}
\end{maplelatex}

\begin{center}
\mapleplot{puntos4y501.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{restart:}{%
}
\end{mapleinput}

\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{?simplify}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{?trig}{%
}
\end{mapleinput}

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

\end{maplegroup}

\subsection{Ejercicios 2:}

\begin{flushleft}
\textbf{Calcular las curvaturas: }Curvatura Normal\textbf{
k(}u\textbf{) }donde\textbf{ u }es el vector en el plano tangente
en\textbf{ p }talque es\textbf{ \textit{"bisectriz " }}de\textbf{ X\_u
}y\textbf{ X\_v, Curvaturas Principales} (k1,k2), \textbf{Curvatura
Media} (H), y \textbf{Curvatura de Gauss} (K) para:
\end{flushleft}

\begin{Bullet Item}
Un cilindro vertical (eje de simetría eje z).
\end{Bullet Item}

\begin{Bullet Item}
La superficie de Enneper.
\end{Bullet Item}

\begin{Bullet Item}
La catenoide.
\end{Bullet Item}

\begin{Bullet Item}
La superficie "rara".
\end{Bullet Item}

\begin{Bullet Item}
Una silla de montar.
\end{Bullet Item}

\begin{Bullet Item}
Un helicoide.
\end{Bullet Item}

\begin{Bullet Item}
Una esfera.
\end{Bullet Item}

\begin{Bullet Item}
Un cono.
\end{Bullet Item}

\begin{Bullet Item}
Un toro.
\end{Bullet Item}

\begin{flushleft}
De todas estas superfices qué puede decir de cada una de ellas
conociendo sus curvaturas. Clasifíquelas según sean superfices
mínimas, superficies "flat" o superficies de revolución.
\end{flushleft}

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