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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 49 "Math 251 Assignment 3 Sol
utions - Maple Component" }}{PARA 19 "" 0 "" {TEXT -1 32 "by J. Hebron
, SFU, February 2000" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "This is \+
the Maple component of the solutions to Math 251 Homework Assignment #
03, which was due Wednesday, February 2nd, 2000." }}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 30 "First, load the plots package:" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Section 11.
7 #16" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x = \+
sin(t);" "6#/%\"xG-%$sinG6#%\"tG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y =
 cos(t);" "6#/%\"yG-%$cosG6#%\"tG" }{TEXT -1 6 ", and " }{XPPEDIT 18 
0 "z = sin(t)^2;" "6#/%\"zG*$-%$sinG6#%\"tG\"\"#" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x:=sin(t);y:=cos(t);z:=sin(t
)^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Now let's see if these co
rrespond to the intersection of the circular cylinder " }{XPPEDIT 18 
0 "x^2+y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F(\"\"\"" }{TEXT 
-1 28 " and the parabolic cylinder " }{XPPEDIT 18 0 "z-x^2 = 0;" "6#/,
&%\"zG\"\"\"*$%\"xG\"\"#!\"\"\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "x^2+y^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 6 "z-x^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Indeed, we see tha
t the space curve is the intersection of the two cylinders." }}{PARA 
0 "" 0 "" {TEXT -1 66 "Let's plot these cylinders so we can visualize \+
their intersection:" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Plotting \+
the circular cylinder" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "im
plicitplot3d('x'^2+'y'^2=1,'x'=-1..1,'y'=-1..1,'z'=-0.5..2,scaling=con
strained,axes=frame,labels=['x','y','z']);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 122 "Notice the quotes around the x, y, and z.  This is to \"
delay the evaluation\" of these names, because they are assigned to " 
}{XPPEDIT 18 0 "sin(t);" "6#-%$sinG6#%\"tG" }{TEXT -1 2 ". " }
{XPPEDIT 18 0 "cos(t);" "6#-%$cosG6#%\"tG" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "sin(t)^2;" "6#*$-%$sinG6#%\"tG\"\"#" }{TEXT -1 60 ".  W
ithout the quotes, Maple would produce an error message." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 31 "Plotting the parabolic cylinder" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "plot3d('x'^2,'x'=-1.5..1.5,'y'=-1.5..1.5,scaling=cons
trained,axes=frame,labels=['x','y','z']);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 141 "Again, notice the quotes around the x, y, and z.  This i
s again to \"delay the evaluation\" of these names, because they are s
till assigned to " }{XPPEDIT 18 0 "sin(t);" "6#-%$sinG6#%\"tG" }{TEXT 
-1 2 ". " }{XPPEDIT 18 0 "cos(t);" "6#-%$cosG6#%\"tG" }{TEXT -1 6 ", a
nd " }{XPPEDIT 18 0 "sin(t)^2;" "6#*$-%$sinG6#%\"tG\"\"#" }{TEXT -1 
60 ".  Without the quotes, Maple would produce an error message." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 42 "Putting the two cylinders on the same plot" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 170 "In order to put both cylinders on the sa
me plot, we assign each plot to a name, say \"p1\" and \"p2\", and the
n use the \"display\" command to put both graphs on the same plot:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p1:=implicitplot3d('x'^2+'y
'^2=1,'x'=-1..1,'y'=-1..1,'z'=-0.5..2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "p2:=plot3d('x'^2,'x'=-1.5..1.5,'y'=-1.5..1.5):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 220 "Notice the colon used at the end \+
of these statements, rather than a semi-colon.  This \"suppresses the \+
output\" of the command.  If you don't do this, you will see screenful
ls of the internal details of the plot structure." }}{PARA 0 "" 0 "" 
{TEXT -1 35 "Now we can \"display\" the two plots:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "display(\{p1,p2\},scaling=constrained,axes=
frame,labels=['x','y','z']);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "O
ne can now see clearly what the spacecurve of intersection looks like.
  Let's plot it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Plotting the space curve of inter
section" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "spacecurve([x,y,
z],t=0..2*Pi,color=red,thickness=3,scaling=constrained,axes=frame,labe
ls=['x','y','z']);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "This does \+
indeed look like the intersection of the parabolic cylinder and the ci
rcular cylinder.  To illustrate this, let's put all three on the same \+
graph." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 62 "Putting the two cylinders and the space c
urve on the same plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p1:
=implicitplot3d('x'^2+'y'^2=1,'x'=-1..1,'y'=-1..1,'z'=-0.5..2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p2:=plot3d('x'^2,'x'=-1.5..1
.5,'y'=-1.5..1.5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p3:=s
pacecurve([x,y,z],t=0..2*Pi,color=red,thickness=3):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 72 "display(\{p1,p2,p3\},scaling=constrained,
axes=frame,labels=['x','y','z']);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
10 "Beautiful!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Section 11.7 #20" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 28 "The plot, using \"spacecurve\"" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "spacecurve([sin(t),sin(2*t),sin(3*
t)],t=0..2*Pi,thickness=3,scaling=constrained,axes=frame,labels=['x','
y','z'],orientation=[65,80]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 
"It may be a little hard to visualize this in 3 dimensions, so let's a
lso plot a \"tubeplot\" of the same function." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "The pl
ot, using \"tubeplot\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "t
ubeplot([sin(t),sin(2*t),sin(3*t)],t=0..2*Pi,radius=1/4,scaling=constr
ained,axes=frame,labels=['x','y','z'],orientation=[65,80]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The tube plot perhaps gives us a b
etter idea of what the 3-dimensional structure is." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "
Section 11.7 #56" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Let's start by
 clearing the names \"x\", \"y\", and \"z\", which were assigned in 11
.7 #16" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:='x'; y:='y'; z
:='z';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 31 "Plotting the parabolic cylinder" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot3d([x,x^2,z],x=-1.5..1.5
,z=-0.5..2.5,axes=frame,labels=['x','y','z'],orientation=[60,55]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 38 "Plotting the top half of the ellipsoid" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 110 "implicitplot3d(x^2+4*y^2+4*z^2=16,x=-4..
4,y=-2..2,z=0..2,axes=frame,labels=['x','y','z'],orientation=[60,55]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 34 "Putting them both on the same plot" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p1:=plot3d([x,x^2,z],x=-1.5..1.5,z=
-0.5..2.5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p2:=implicit
plot3d(x^2+4*y^2+4*z^2=16,x=-4..4,y=-2..2,z=0..2):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "display(\{p1,p2\},axes=frame,labels=['x','y
','z'],orientation=[60,55]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "T
his gives us a pretty good idea of what the curve of intersection look
s like." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 40 "Plotting the space curve of intersection
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "To get the curve of intersecti
on, let " }{XPPEDIT 18 0 "x = t;" "6#/%\"xG%\"tG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "y = t^2;" "6#/%\"yG*$%\"tG\"\"#" }{TEXT -1 17 ", and th
en solve " }{XPPEDIT 18 0 "x^2+4*y^2+4*z^2 = 16;" "6#/,(*$%\"xG\"\"#\"
\"\"*&\"\"%F(*$%\"yG\"\"#F(F(*&\"\"%F(*$%\"zG\"\"#F(F(\"#;" }{TEXT -1 
5 " for " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 18 " as a function o
f " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 30 ".  The curve is thus g
iven by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "intcurve:=[t,t^
2,1/2*sqrt(16-t^2-4*t^4)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We \+
also need to find the " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 11 " a
nd which " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(16-t^2-4*t^4=0,t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 73 "The last two solutions are imaginary so we can \+
ignore them.  We thus let " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 
23 " go from -1.37 to 1.37:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
99 "spacecurve(intcurve,t=-1.37..1.37,thickness=3,axes=frame,labels=['
x','y','z'],orientation=[60,55]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Looking good!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Putting the cylinder, ellipsoid, \+
and space curve on the same plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "p1:=plot3d([x,x^2,z],x=-1.5..1.5,z=-0.5..2.5):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p2:=implicitplot3d(x^2+4*y^2+4*z^2=
16,x=-4..4,y=-2..2,z=0..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "p3:=spacecurve(intcurve,t=-1.37..1.37,thickness=3,color=red):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(\{p1,p2,p3\},axes=fr
ame,labels=['x','y','z'],orientation=[60,55]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 10 "Beautiful!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Section 11.7 #jh1" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "circ1:=[cos(t),sin(t),0];" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "circ2:=[cos(t),0,sin(t)];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "circ3:=[0,cos(t),sin(t)
];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "tubeplot(\{circ1,cir
c2,circ3\},t=0..2*Pi,radius=1/5,scaling=constrained,\naxes=frame,label
s=['x','y','z'],orientation=[22,75]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Section 11.
7 #jh2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "tsp:=[sin(8*t),cos
(t)*(3+cos(8*t)),sin(t)*(3+cos(8*t))];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "tubeplot(tsp,t=0..2*Pi,radius=1/3,scaling=constraine
d,\nnumpoints=100,axes=frame,labels=['x','y','z'],orientation=[40,75])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 16 "Section 11.9 #16" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "r:=[t+t^3/6,1+t^4/12,1/4+t-1/4*cos(2*t)];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "spacecurve(r,t=0..Pi,thickness=3,a
xes=normal,labels=['x','y','z'],shading=zhue,orientation=[-20,70]);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "It is a little difficult to vis
ualize the 3-dimensional structure of the path from a 3-dimensional gr
aph.   Sometimes it helps to plot the path with its projection in the \+
xy-plane.  This is done below:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "The path with its pro
jection in the xy-plane." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
r_xy:=[t+t^3/6,1+t^4/12,0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
108 "spacecurve(\{r,r_xy\},t=0..Pi,thickness=3,axes=normal,labels=['x'
,'y','z'],shading=zhue,orientation=[-20,70]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }