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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 256 "REVIEW FOR MATH 4530-1   \+
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EG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 4 1 
1.000000 56.000000 38.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 213 "curv:= proc(alpha)\nlocal alphap, alphapp;\nalphap
:=map(diff,alpha,t);\nalphapp:=map(diff,alphap,t);\nsimplify(Norm(Cros
sProduct(alphap,alphapp),Euclidean) / \nNorm(alphap,Euclidean, conjuga
te=false)^3, symbolic);\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "he
l2:=< c*cos(t) | c*sin(t) | d1*t>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%hel2G-%'RTABLEG6%\"([l$>-%'VECTORG6#7%*&%\"cG\"\"\"-%$cosG6#%\"tGF/
*&F.F/-%$sinGF2F/*&%#d1GF/F3F/&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 11 "curv(hel2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&%\"cG\"\"\",&*$)%#d1G\"\"#F%F%*$)F$F*F%F%!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 305 "tor:=proc(alpha)\nlocal alphap, alphapp, a
lphappp;\nalphap:=map(diff,alpha,t);\nalphapp:=map(diff,alphap,t);\nal
phappp:=map(diff,alphapp,t);\nsimplify(DotProduct(CrossProduct(alphap,
alphapp),alphappp,conjugate=false) / \nNorm(CrossProduct(alphap,alphap
p),Euclidean, conjugate=false)^2, sqrt,trig,symbolic);\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tor(hel2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#*&%#d1G\"\"\",&*$)F$\"\"#F%F%*$)%\"cGF)F%F%!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 70 "cand1:= < (1- r*cos(m*t))*cos(t)| (1-r*cos(m*t))*si
n(t) | r*sin(m*t)>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&cand1G-%'RTA
BLEG6%\"(7GM#-%'VECTORG6#7%*&,&\"\"\"F/*&%\"rGF/-%$cosG6#*&%\"mGF/%\"t
GF/F/!\"\"F/-F36#F7F/*&F.F/-%$sinGF:F/*&F1F/-F=F4F/&%'VectorG6#%$rowG
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" {XPPMATH 20 "6#-%'RTABLEG6%\"(CNT#-%'VECTORG6#7%,$*&,&\"\"\"!\"\"*&%
\"rGF.-%$cosG6#*&%\"mGF.%\"tGF.F.F.F.-F36#F7F.F/,$*&F-F.-%$sinGF9F.F/*
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F'F%)F,F'F%F%*(F'F%F4F%)F0F'F%F%**F'F%F(F%F,F%F7F%F2**\"\")F%F(F%F,F%)
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4F%)F0\"\"'F%F%*(F'F%FCF%F;F%F%*(FHF%FCF%F?F%F%#F%F<,*F%F%*(F<F%F)F%F,
F%F2*&FCF%F?F%F%*&FCF%F;F%F%#!\"$F<" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#**,6-%$cosG6#*&%\"mG\"\"\"%\"tGF*!\"\"*&)%\"rG\"\"#F*)F%\"\"$F*F,*&F/
F*)F)\"\"%F*F,**F5F*F.F*)F)F0F*F%F*F,*(F.F*F1F*F7F*F**(F0F*F/F*)F%F0F*
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55555555A401D55555555555A0000000000000000-%+LIGHTMODELG6#%(LIGHT_3G-%*
AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 5 
4 1 1.000000 87.000000 96.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 212 "EFG:=proc(X)\nlocal Xu, Xv, E, F, G;\nXu:=map
(diff,X,u);\nXv:=map(diff,X,v);\nE:=DotProduct(Xu,Xu,conjugate=false);
\nF:=DotProduct(Xu,Xv,conjugate=false);\nG:=DotProduct(Xv,Xv,conjugate
=false);\nsimplify([E,F,G]);\nend:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 11 "EFG(Ennep);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,
.\"\"\"F%*&\"\"#F%)%\"uGF'F%F%*&F'F%)%\"vGF'F%F%*$)F)\"\"%F%F%*(F'F%F(
F%F+F%F%*$)F,F/F%F%\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
237 "UN:=proc(X)\nlocal Xu, Xv, Z, s;\nXu:=map(diff,X,u);\nXv:=map(dif
f,X,v);\nZ:=CrossProduct(Xu,Xv);\ns:=simplify(sqrt(DotProduct(Z,Z,conj
ugate=false)),symbolic);\nsimplify(simplify(simplify(<Z[1]/s | Z[2]/s \+
| Z[3]/s>,sqrt),trig),symbolic);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "UN(Ennep);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTAB
LEG6%\"(/***e-%'VECTORG6#7%,$*(\"\"#\"\"\"%\"uGF.,(F.F.*$)F/F-F.F.*$)%
\"vGF-F.F.!\"\"F6,$*(F-F.F5F.F0F6F.,$*&,(F1F.F3F.F.F6F.F0F6F6&%'Vector
G6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Ennep;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(!GQM-%'VECTORG6#7%,(%\"
uG\"\"\"*&#F-\"\"$F-*$)F,F0F-F-!\"\"*&F,F-)%\"vG\"\"#F-F-,(F6F-*&#F-F0
F-*$)F6F0F-F-F3*&F6F-)F,F7F-F-,&*$F>F-F-*$F5F-F3&%'VectorG6#%$rowG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify(DotProduct(UN(Enne
p),UN(Ennep),conjugate=false));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "lmn:=proc(X)\nloca
l Xu, Xv, Xuu, Xuv, Xvv, U, l, m, n, Z, s;\nXu:=map(diff,X,u);\nXv:=ma
p(diff,X,v);\nXuu:=map(diff,Xu,u);\nXuv:=map(diff,Xu,v);\nXvv:=map(dif
f,Xv,v);\nU:=UN(X);\nl:=DotProduct(Xuu,U,conjugate=false);\nm:=DotProd
uct(Xuv,U,conjugate=false);\nn:=DotProduct(Xvv,U,conjugate=false);\nsi
mplify(simplify(simplify([l,m,n],sqrt),trig),symbolic);\nend:\n\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "lmn(Ennep);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%\"\"#\"\"!!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 309 "shape:=proc(X)\nlocal Y, Z, a,b,c,d, s;\nY:=EFG(X);
\nZ:=lmn(X);\ns:=simplify( Y[1]*Y[3]-Y[2]^2,sqrt,trig,symbolic);\na:=s
implify((Z[1]*Y[3]-Z[2]*Y[2])/s);\nb:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/s
);\nc:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/s);\nd:=simplify((Z[3]*Y[1]-Z[2]
*Y[2])/s);\n[S(x_u)=a*x_u+b*x_v,S(x_v)=c*x_u+d*x_v];\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "shape(Ennep);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$/-%\"SG6#%$x_uG,$*(\"\"#\"\"\",.F,F,*&F+F,)%\"uGF+F,F
,*&F+F,)%\"vGF+F,F,*$)F0\"\"%F,F,*(F+F,F/F,F2F,F,*$)F3F6F,F,!\"\"F(F,F
,/-F&6#%$x_vG,$*(F+F,F-F:F>F,F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 298 "shape_matrix:=proc(X)\nlocal Y, Z, a,b,c,d, s;\nY:=EFG(X);\nZ
:=lmn(X);\ns:=simplify( Y[1]*Y[3]-Y[2]^2,sqrt,trig,symbolic);\na:=simp
lify((Z[1]*Y[3]-Z[2]*Y[2])/s);\nb:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/s);
\nc:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/s);\nd:=simplify((Z[3]*Y[1]-Z[2]*Y
[2])/s);\nMatrix([[a,c],[b,d]]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "shape_matrix(Ennep);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'RTABLEG6%\"(G)op-%'MATRIXG6#7$7$,$*&\"\"#\"\"\",.F/F/*&F.F/)%\"u
GF.F/F/*&F.F/)%\"vGF.F/F/*$)F3\"\"%F/F/*(F.F/F2F/F5F/F/*$)F6F9F/F/!\"
\"F/\"\"!7$F>,$*&F.F/F0F=F=%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "Torus:=< (2+cos(u))*cos(v)|(2+cos(u))*sin(v)|sin(u)>;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot3d(convert(Torus,list),u=-P
i..Pi,v=-Pi..Pi,scaling=constrained, axes=normal,lightmodel=light1);" 
}}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
TorusG-%'RTABLEG6%\"(/A*p-%'VECTORG6#7%*&,&\"\"#\"\"\"-%$cosG6#%\"uGF0
F0-F26#%\"vGF0*&F.F0-%$sinGF6F0-F:F3&%'VectorG6#%$rowG" }}{PARA 13 "" 
1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%%MESHG6#X.%)anythingG6\"6
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3BFEEE8DD4748BF19BCED1A62633145C03CD91A62633145C0BFF0000000000000BCED1
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00006BFEBB67AE8584CA7BCED1A62633145C03FE6A09E667F3BD0BFE6A09E667F3BCAB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