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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 5 "     " }{TEXT 256 0 "" }{TEXT 
257 11 "MATH 2270-2" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "
" }{TEXT 259 39 "Symmetric matrices, conics and quadrics" }}{PARA 258 
"" 0 "" {TEXT -1 16 "December 3, 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 14 "Coni
c sections" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "with(linalg):
with(plots):#for computations and pictures\nwith(student):#to do algeb
ra computations like completing the square" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Let's do the problem we did in \+
class this past Wednesday, using Maple:  " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "A:=matrix(2,2,[2,5/2,5/2,2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"##\"\"&F*7$F+F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eigenvalues(A); #compute eigenvalue
s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"*\"\"##!\"\"F%" }}}{PARA 0 "
" 0 "" {TEXT -1 170 "Since the eigenvalues have the opposite sign,  th
e conic must be a hyperbola (or two lines crossing) .  We can verify t
his immediately by using the implicitplot command: " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "implicitpl
ot(2*x^2+2*y^2+5*x*y= 1,\n       x=-3..3,y=-3..3, grid=[100,100],\n   \+
    color=`black`);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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$FjinFg]n7$F_hoFago7$7$FhwFfw7$FcwFaw7$Fcho7$FbxF`x7$7$FhxFfxFeho7$Fgh
oFcgo7$7$FhwF^in7$F[inF`_n7$F[ioF^ho7$7$FhuFfu7$FbuF`u7$7$FbvF`v7$FgvF
ev7$7$F]wF[wFbio7$7$F\\xF[wFbho7$7$FhuFdgn7$FagnFban7$7$FbinFbanFjho7$
7$FctFat7$F^tF\\t7$F^jo7$FhuF\\v7$Fhio7$FjgnFhgn7$7$FctF^hnFbjo7$7$Fhr
Ffr7$FcrFar7$Fgjo7$FbsF`s7$7$FhsFfsFijo7$7$F\\uFjt7$FctFgt7$7$FhrFhfn7
$FefnFjcn7$Fa[p7$FctFbhn7$7$F`qF^q7$F[qFip7$Ff[p7$FgqFeq7$7$F]rF[rFh[p
7$Fj[p7$FhrF\\s7$7$F`qF_en7$F\\enFhdn7$F_\\pF`[p7$7$FhoFfo7$FcoFao7$Fc
\\p7$$\"3^JFFFxk67F*F]p7$7$FepFcpFe\\p7$Fi\\pFe[p7$F^\\p7$FeenFcen7$7$
FhoFienF\\]p7$7$FQFO7$FLFJ7$7$FXFV7$FgnFen7$7$F]oF[oFd]p7$Ff]pFb\\p7$7
$FQFbdn7$F_dnFccn7$7$F_dnF]fnF^]p7$7$F0F.7$F+F(7$7$F;F(7$$\"3Od'zOlzON
\"F*$!3%fBP%HsVHHF*7$7$FFFDFb^p7$Fh^pF`]p7$7$F0F_bn7$F\\bnFjan7$7$F\\b
nF[anFi]p7$7$F7F4F^^p7$F[_p7$FebnFcbn7$7$F[cnFibnFb_p7$7$Fj_nFh_n7$Fe_
nFi^n7$7$F_cnFi^nFd_p7$Ff_p7$F``nF^`n7$7$Ff`nFd`nF[`p7$7$Fd^nFb^n7$F_^
nF`]n7$F``pF]`p7$7$F\\\\nFj[n7$Fg[nFhjm7$Fd`pF_`p7$Fc`p7$$\"37!o%3$p%3
$p\"F*$!3o]l\"yi;yi'Fdu7$7$Fh\\nFf\\nFg`p7$7$FgimFeim7$FbimFehm7$F`ap7
$Fh\\nF\\]n7$F_ap7$F]jmF[jm7$7$FcjmFajmFdap7$7$F]hmF[hm7$FhgmFbgm7$Fia
pFfap7$7$FfemFdem7$FaemFadm7$7$FahmFadmFhap7$7$F\\fmFjem7$FafmF_fm7$7$
FgfmFefmFbbp7$7$F`cmF^cm7$F[cmF_bm7$FgbpFdbp7$Ffbp7$Ffcm$!30]?@@@@@\"*
Fdu7$7$F\\dmFjcmFjbp7$7$F^amF\\am7$Fi`mF]`m7$FacpF^cp7$F`cp7$FdamFbam7
$7$FjamFhamFdcp7$7$Fe_mFc_m7$F`_mFj^m7$FicpFfcp7$7$F`]mF^]m7$F[]mF_\\m
7$7$Fi_mF_\\mFhcp7$F\\dp7$Ff]mFd]m7$7$F\\^mFj]mFadp7$7$F^[mF\\[m7$Fijl
F]jl7$7$F`^mF]jlFcdp7$Fedp7$$\"3)f<S2FNI`#F*$!3](y+o(e4R6F*7$7$Fj[mFh[
mFjdp7$7$FihlFghl7$FdhlF^hl7$FcepF`ep7$7$FihlF]il7$FbilF`il7$7$FhilFfi
lFgep7$7$FgflFefl7$FbflFcel7$F\\fpFiep7$F[fp7$F]glF[gl7$7$FcglFaglF_fp
7$7$F_dlF]dl7$FjclF^cl7$FdfpFafp7$Fcfp7$FedlFcdl7$7$F[elFidlFgfp7$7$Fj
blFhbl7$FeblF_bl7$7$F_elF_blFifp-%'COLOURG6&%$RGBGF6F6F6-%+AXESLABELSG
6$%\"xG%\"yG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 
"Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "     Yes, it was a hyperbol
a." }}{PARA 0 "" 0 "" {TEXT -1 346 "     Now, what about all that work
 we did to explicitly pick new coordinates in which there was no cross
 term?  If we really need to do that, we can try to let Maple help, an
d although the commands seem a little cumbersome at first, when we're \+
done we'll have a template that will work for any quadratic equation w
ith only minor modifictations.  " }}{PARA 0 "" 0 "" {TEXT -1 88 "     \+
Let's do #30 page 501 of Kolman: We will write our quadratic as x(^T)A
x + Bx + C=0:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "A:=matrix(
2,2,[8,-8,-8,8]);\n    #matrix for quadratic form\nB:=matrix(1,2,[33*s
qrt(2),-31*sqrt(2)]);\n    #row matrix for linear term\nC:=70;\n    # \+
constant term" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7
$\"\")!\")7$F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#
7#7$,$*$-%%sqrtG6#\"\"#\"\"\"\"#L,$F+!#J" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"CG\"#q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "fmat:=
v->evalm(transpose(v)&*A&*v\n                 + B&*v +C) ;\n     #this
 function takes a vector v and computes \n     # transpose(v)Av + Bv +
 C ,\n     #which will be a one by one matrix\nf:=v->simplify(fmat(v)[
1]); #this extracts the entry of the one\n     #by one matrix fmat(v)
\n           " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fmatGf*6#%\"vG6\"6
$%)operatorG%&arrowGF(-%&evalmG6#,(-%#&*G6$-F16$-%*transposeG6#9$%\"AG
F8\"\"\"-F16$%\"BGF8F:%\"CGF:F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"fGf*6#%\"vG6\"6$%)operatorG%&arrowGF(-%)simplifyG6#&-%%fmatG6#9$6
#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "fmat(vect
or([x,y])); #should be a one by one matrix\nf(vector([x,y]));  #should
 get its entry, which is what we want.\n   #should agree with our exam
ple" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7#,,*&-%%sqrtG6#\"
\"#\"\"\"%\"xGF-\"#L*(\"#JF-F)F-%\"yGF-!\"\"*&,&F.\"\")*&F6F-F2F-F3F-F
.F-F-*&,&F.!\")*&F6F-F2F-F-F-F2F-F-\"#qF-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,.*&-%%sqrtG6#\"\"#\"\"\"%\"xGF)\"#L*(\"#JF)F%F)%\"yGF)
!\"\"*&\"\")F))F*F(F)F)*(\"#;F)F*F)F.F)F/*&F1F))F.F(F)F)\"#qF)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqtn:=f([x,y])=0;\n   #shoul
d be our equation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqtnG/,.*&-%%s
qrtG6#\"\"#\"\"\"%\"xGF,\"#L*(\"#JF,F(F,%\"yGF,!\"\"*&\"\")F,)F-F+F,F,
*(\"#;F,F-F,F1F,F2*&F4F,)F1F+F,F,\"#qF,\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 93 "implicitplot(eqtn,x=-4..4,y=-4..4, color=black,
\n    grid=[100,100]);\n    #what are we in for?" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 
2 "6%-%'CURVESG6W7$7$$!\"%\"\"!$!3.GAI/+Ut;!#<7$$!3kcw(34\"\\$*RF-$!3r
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$!3!4ND31KF'RF-$!3zAA$\\^DIh\"F-7$7$$!3gBa\"*RY<KRF-$!3Ytvvvvvv:F-F=7$
FC7$$!3x?.SM<vFRF-$!3.s\"\\0w(>n:F-7$7$$!3c>>>>>>>RF-$!3=<VQL<Lc:F-FI7
$7$F($!3-^L&=lE[X)!#=7$$!3&\\^4>>*eXRF-$!3*))\\swvv3A)FX7$7$$!36>>>>>>
>RF-$!3$fw+***H<D!)FXFY7$7$FP$!3S<VQL<Lc:F-7$$!3k'fl$))4l'*QF-$!3h9exD
/\\<:F-7$7$$!3)y!Ho4)o\"yQF-$!3e#\\\\\\\\\\\\\\\"F-Fbo7$7$$!3L3Ho4)o\"
yQF-F[p7$$!3QMM37]hkQF-$!3l'*)\\7K=(o9F-7$7$$!3nQQQQQQQQF-$!3x!H#Rj)>a
V\"F-Fap7$Fin7$$!3a,ij+8^#*QF-$!3KD[KiQdVzFX7$7$$!37r*f3I_E&QF-$!3f\\w
wwwwwwFXF]q7$Fcq7$$!3/$Rk_:uE%QF-$!3Z,@'z]kQj(FX7$7$$!3AQQQQQQQQF-$!3:
y2!\\<d?g(FXFiq7$7$F`rFjp7$$!3<We!f\"*p0$QF-$!3a0%>mLb>U\"F-7$7$$!33$R
]%zH;CQF-$!3q69999999F-Ffr7$F\\s7$$!3*)[lw*Gy9!QF-$!3:?1&>))Q-P\"F-7$7
$$!3!yPejf0Cx$F-$!3\"3LLLLLLL\"F-Fbs7$Fhs7$$!3AO'4S%Q%zw$F-$!3;_%**oCl
HK\"F-7$7$$!3!pvvvvvvv$F-$!3D<bT$4%)*38F-F^t7$F_r7$$!3'4)3\"**)GA#y$F-
$!3/4kT`jHIuFX7$7$Fet$!3_f&exg9`E(FXFjt7$Fdt7$$!3ih'\\ucT$QPF-$!3WX8lU
%f<F\"F-7$7$$!3#H%z2`'*=BPF-$!3$*\\______7F-Fdu7$7$$!3OVz2`'*=BPF-F]v7
$$!3Ecg***)HA4PF-$!38qoHR*p+A\"F-7$7$$!3XwwwwwwwOF-$!3KI%H,h/j<\"F-Fcv
7$7$$!3NdddddddPF-Fau7$$!3WV*R\"el#>s$F-$!3#4)\\/j)y^A(FX7$7$$!3,wwwww
wwOF-$!33O%e0)*)\\!)pFXFbw7$Fiv7$$!3AvF8X[?vOF-$!3\\q?N.+Gt6F-7$7$$!3D
5vz4P(Rn$F-$!3/prrrrrr6F-F^x7$Fdx7$$!3(*yC)f8-0l$F-$!35'G%pJY<<6F-7$7$
$!33&Q=SCC5j$F-$!3;)3444444\"F-Fjx7$F`y7$$!3W\\IT_2!Gi$F-$!3&QjbW$z1k5
F-7$7$$!37&fffffff$F-$!3!Q'4'\\w))R-\"F-Ffy7$7$FjvF[x7$$!3#p=k@z]Am$F-
$!3>O<s9d&Q,(FX7$7$$!3.>iB3F50OF-$!3&=%oooooooFXFcz7$Fiz7$$!39w%p,\"[)
of$F-$!3lS!)eEZVfoFX7$7$$!3e&fffffff$F-$!3]A?JI<UaoFXF_[l7$7$Ff[l$!3-k
4'\\w))R-\"F-7$$!3m,*o>G\"y\"f$F-$!3=,<4C$zU,\"F-7$7$$!3[AR-XFo)e$F-$!
3G25555555F-F^\\l7$Fd\\l7$$!3'=Pjw#zVrNF-$!3/,:*e(f9Q&*FX7$7$$!3LwI!Go
:]b$F-$!31l#HHHHHH*FXFj\\l7$F`]l7$$!3<:(H`O`4b$F-$!3]is9\"z5\\$*)FX7$7
$$!3Q$zZMrRl_$F-$!3Kd%[[[[[[)FXFf]l7$F\\^l7$$!33ku)fs/*HNF-$!3Si*)[wjJ
P$)FX7$7$$!3p9::::::NF-$!3TzH)GtC.w(FXFb^l7$7$Fi^l$!3u$*\\-de=>wFX7$$!
3TvKC$R^(*e$F-F\\[l7$7$Fb_l$!3'H%oooooooFX7$Ff[l$!3R@?JI<UaoFX7$Fh^l7$
$!3z$oma.+V^$F-$!3sffhtCG&o(FX7$7$$!3ZtR*)\\\"QQ^$F-FfqF\\`l7$Fb`l7$Fi
^l$!3%[*\\-de=>wFX-%+AXESLABELSG6$%\"xG%\"yG-%'COLOURG6&%$RGBGF*F*F*" 
1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 43 "Now let's go about the change of variable
s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "data:=eigenvects(A);#g
et eigenvectors" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dataG6$7%\"#;\"
\"\"<#-%'vectorG6#7$!\"\"F(7%\"\"!F(<#-F+6#7$F(F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 111 "You pick things out of the object above systematically, \+
using inidices to work through the nesting of brackets:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "data[1];#first piece of data" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"#;\"\"\"<#-%'vectorG6#7$!\"\"F%" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "data[1][1];#eigenvalue" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "data[1][2];#algebraic multiplicity" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "
data[1][3];#basis for eigenspace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#
-%'vectorG6#7$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "data[1][3][1];#actual eigenvector" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'vectorG6#7$!\"\"\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 42 "So that's how to extract the eigenvectors
:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "v1:=data[1][3][1];#fir
st eigenvector\nv2:=data[2][3][1];#second eigenvector\nw1:=v1/norm(v1,
2);#normalized\nw2:=v2/norm(v2,2);#normalized" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v1G-%'vectorG6#7$!\"\"\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v2G-%'vectorG6#7$\"\"\"F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#w1G,$*&%#v1G\"\"\"-%%sqrtG6#\"\"#F(#F(F," }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#w2G,$*&%#v2G\"\"\"-%%sqrtG6#\"\"#F(#F(F,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "P:=augment(w1,w2):#our \+
orthogonal matrix\n  if det(P)<0 then P:=augment(w2,w1):\n  fi:\nevalm
(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,$*$-%%sqrtG6#
\"\"#\"\"\"#F.F-,$F)#!\"\"F-7$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "f(evalm(P&*[u,v]));#do the change of variables" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"uG\"\"#*&\"#k\"\"\"%\"vGF(!\"\"*&
\"#;F()F)F%F(F(\"#qF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "si
mplify(%);#simplify it!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"uG\"\"
#*&\"#k\"\"\"%\"vGF(!\"\"*&\"#;F()F)F%F(F(\"#qF(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "completesquare(%,u); #complete the square in \+
u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"uG\"\"#*&\"#k\"\"\"%\"vGF(!
\"\"*&\"#;F()F)F%F(F(\"#qF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "completesquare(%,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&%\"
vG\"\"\"\"\"#!\"\"F)F(\"#;\"\"'F(*&F)F(%\"uGF(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "Eqtn:=%=0;\n  #this is the new equation\n  " 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EqtnG/,(*$),&%\"vG\"\"\"\"\"#!\"
\"F,F+\"#;\"\"'F+*&F,F+%\"uGF+F+\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 259 "" 1 "" {TEXT -1 274 "Let's collect everything into one \+
template .   You can see from where the colons and semicolons are that
 the output will be  the eigenvalues,  the transition matrix,  a plot,
 and an equation just short of standard form.  (You can improve it by \+
picking a denser grid size.)  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "restart;with(linalg):with(plots):with(student):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 816 "\nA:=matrix(2,2,[8,-8,-8,8]);\nB:=
matrix(1,2,[33*sqrt(2),-31*sqrt(2)]);  \nC:=70;\nfmat:=v->evalm(transp
ose(v)&*A&*v\n                 + B&*v + C):\nf:=v->simplify(fmat(v)[1]
):\neigenvals(A); #show the eigenvalues\ndata:=eigenvectors(A):\nv1:=d
ata[1][3][1]:#first eigenvector\nv2:=data[2][3][1]:#second eigenvector
\nw1:=v1/norm(v1,2):#normalized\nw2:=v2/norm(v2,2):#normalized\nP:=aug
ment(w1,w2):#our orthogonal matrix,\n  #unless we want to switch colum
ns\n  if det(P)<0 then\n     P:=augment(w2,w1):\n  fi:\nevalm(P);\nimp
licitplot(f([x,y])=0,x=-5..5,y=-5..5,\n     grid=[100,100],color=`blac
k`);\n     #increase grid size for better picture, but too big\n     #
takes too long \nf(evalm(P&*[u,v])):#do the change of variables\nsimpl
ify(%):#simplify it!\ncompletesquare(%,u): #complete the square in u\n
completesquare(%,v):#and v\nEqtn:=%=0;\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7$7$\"\")!\")7$F+F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7#7$,$*$-%%sqrtG6#\"\"#\"\"\"\"#L,$F+
!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG\"#q" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"!\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7$7$,$*$-%%sqrtG6#\"\"#\"\"\"#F.F-,$F)#!\"\"F-7$F(F(" }}{PARA 13 "" 
1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6cr7$7$$!\"&\"\"!$
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7$Ffhm7$$!3)ebb12A=a$F-$!3X%y(yFF7;!)Fddl7$7$$!39(Q!z@Ey@NF-FfamFjhm7$
7$$!3f(Q!z@Ey@NF-FfamF_gm-%'COLOURG6&%$RGBGF*F*F*-%+AXESLABELSG6$%\"xG
%\"yG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+
1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EqtnG/,(*$),&%\"vG\"\"\"\"\"#
!\"\"F,F+\"#;\"\"'F+*&F,F+%\"uGF+F+\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 
77 "You could check your book problems about conic sections using such
 a template" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
261 17 "Quadric Surfaces:" }}{PARA 0 "" 0 "" {TEXT -1 17 "Bonus Questi
on : " }{TEXT 262 23 "worth 10 midterm points" }{TEXT -1 366 ":  Creat
e a template which will diagonalize, graph, and put into standard form
, the solution set to any quadratic equation in 3 variables - i.e. ana
logous to the template above which works for conic sections.  You need
 only create a template which works when there are 3 distinct eigenval
ues.  (If you successfully tackle the  case of higher geometric multip
licity y" }{TEXT -1 229 "ou will need several conditionals to allow fo
r the structure of the eigenvects array.  Also, in that case you will \+
need to use the Maple Gram-Schmidt command or your own, to get an orth
onormal basis of the offending eigenspace.  " }}{PARA 0 "" 0 "" {TEXT 
-1 490 "     To make 3-d plots you can use implicitplot3d.  If you mak
e your grid too fine the plot will take a long time to make, so try th
e default grid at first and adjust if necessary.  You might also have \+
to adjust your limits in the plot to get a better picture.  You can ma
nipulate 3d plots with your mouse.  There are a lot of interesting plo
t options which you write into the command or access from the plotting
 toolbar.  One that helps me see things is to use the ``boxed'' axes o
ption.  " }}{PARA 0 "" 0 "" {TEXT -1 359 "     The 3d plotting routine
s have been known to crash Maple, so save your file often.  Sometimes \+
what really happens is that a dialog box gets opened behind one of you
r windows, where you can't see it, so that it will appear that Maple h
as frozen.  If this happens, minimize your maple window (the black dot
 a the upper left corner), and then re-display it. " }}{PARA 0 "" 0 "
" {TEXT -1 134 "     Here's a plot to play with, #1 on page 511.  You \+
should be able to move it around to make it look like a one-sheeted hy
perboloid." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "implicitplot3
d(x^2 + y^2 +2*z^2 - 2*x*y -4*x*z -4*y*z + 4*x = 8,\n              x=-
5..5,y=-5..5,z=-5..5, axes=`boxed`);" }}{PARA 13 "" 1 "" {GLPLOT3D 
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