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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "MATH 243-015" }}{PARA 0 "
" 0 "" {TEXT -1 32 "Analytic Geometry and Calculus C" }}{PARA 0 "" 0 "
" {TEXT -1 19 "Prof. D. A. Edwards" }}{PARA 0 "" 0 "" {TEXT -1 14 "Cop
yright 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 
"" {TEXT -1 47 "Sections 14.1 and 14.2: Vector-Valued Functions" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Today we'
ll be using different plot devices as well as vectors." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 52 "restart;\nwith(plots):\nwith(plottools):\nwith
(linalg):" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 13 "Visualization" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "W
e define vector-valued functions just as we defined regular vectors.  \+
We begin by defining a helix and the simple (" }{XPPEDIT 18 0 "t,t^2,t
^3" "6%%\"tG*$F#\"\"#*$F#\"\"$" }{TEXT -1 8 ") curve:" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "avec:=vector([co
s(t),sin(t),t]);\nbvec:=vector([t,t^2,t^3]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 28 "To display them, we use the " }{TEXT 0 10 "spacecurve" 
}{TEXT -1 66 " command as before.  All we have to do is specify the ra
nge for t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "spacecurve(avec,t=0..
2*Pi,axes=normal,orientation=[-15,55],color=blue);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 68 "spacecurve(bvec,t=0..1,axes=normal,orientat
ion=[31,101],color=blue);" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 257 "" 0 "
" {TEXT -1 12 "Manipulation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "To \+
make substitutions into vectors, we must use a peculiar combination of
 " }{TEXT 0 4 "eval" }{TEXT -1 5 " and " }{TEXT 0 5 "evalm" }{TEXT -1 
1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eval(evalm(avec),t=1);\n" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Basically, we are telling " }
{TEXT 0 4 "eval" }{TEXT -1 33 " to expect a vector by using the " }
{TEXT 0 5 "evalm" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Recall that we had to use the " }
{TEXT 0 5 "evalm" }{TEXT -1 152 " command when doing arithmetic with v
ectors so that Maple would know what to do.  We must make similar adju
stments when doing calculus; we must use the " }{TEXT 0 3 "map" }
{TEXT -1 60 " command.  For our purposes, the three arguments of map a
re " }{TEXT 0 4 "diff" }{TEXT -1 4 " or " }{TEXT 0 3 "int" }{TEXT -1 
151 ", depending on whether we want to differentiate or integrate, the
 vector with which we are working, and the variable of differentiation
 or integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "davec:=map(diff,avec,t);\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "dbvec:=map(diff,bvec,t);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "iavec:=map(int,avec,t);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 61 "Note that if we are doing an indefinite integral, Mapl
e does " }{TEXT 256 3 "NOT" }{TEXT -1 67 " insert the constants of int
egration.  One has to do that by hand.\n" }}{PARA 0 "" 0 "" {TEXT -1 
89 "If we are doing a definite integral, we specify the range of the v
ariable of integration:" }{MPLTEXT 1 0 28 "ibvec:=map(int,bvec,t=0..3)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 47 "1.  If the velocity of some object is given by " }
{TEXT 257 1 "v" }{TEXT -1 86 "(t)=(sin(t),cos(t),t) and the object sta
rts at the origin, calculate the acceleration " }{TEXT 258 1 "a" }
{TEXT -1 25 "(t) and the displacement " }{TEXT 259 1 "r" }{TEXT -1 4 "
(t)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "
We just have to make sure to get the constants right when doing the in
tegration to find " }{TEXT 260 1 "r" }{TEXT -1 47 ".  We can do that b
y doing a definite integral." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 97 "vvec:=vector([sin(t),cos(t),t]);\navec:=map(diff,vvec,t);\nmap(i
nt,vvec,t=0..x);\nrvec:=eval(%,x=t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%vvecG-%'vectorG6#7%-%$sinG6#%#t|irG-%$cosGF+F," }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%avecG-%'vectorG6#7%-%$cosG6#%#t|irG,$-%$sinGF+!\"
\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&-%$cosG6#%
\"xG!\"\"\"\"\"F--%$sinGF*,$*$)F+\"\"#F-#F-F3" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%rvecG-%'vectorG6#7%,&-%$cosG6#%#t|irG!\"\"\"\"\"F/-%
$sinGF,,$*$)F-\"\"#F/#F/F5" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 28 "2.  On the same graph, plot " }{TEXT 
261 1 "v" }{TEXT -1 12 "(t) in red, " }{TEXT 262 1 "a" }{TEXT -1 17 "(
t) in blue, and " }{TEXT 263 1 "r" }{TEXT -1 21 "(t) in green for 0<t<
" }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 31 ".  Be s
ure to include the axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 161 "vp:=spacecurve(vvec,t=0..2*Pi,color=red):\na
p:=spacecurve(avec,t=0..2*Pi,color=blue):\nrp:=spacecurve(rvec,t=0..2*
Pi,color=green):\ndisplay(\{vp,ap,rp\},axes=normal);" }}{PARA 13 "" 1 
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STYLEG6#%'NORMALG" 1 2 0 1 10 0 2 1 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 38 "Section 14.3: \+
Arc Length and Curvature" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "\nIn \+
everything we're going to do in this class, s and t are assumed to be \+
real variables.  Maple needs to know this for some arcane reasons, so \+
we use the " }{MPLTEXT 1 0 6 "assume" }{TEXT -1 94 " command.  The fir
st argument is the variable; the second is the property you wish to gi
ve it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(t,real):\n
assume(s,real):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We next define
 a helix curve:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "h:=vector([cos(t),sin(t),t]);" }}{PARA 0 "" 0 "" 
{TEXT -1 107 " Note that the t has a tilde after it.  This lets you kn
ow that Maple has associated some property with it." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 177 "There are no set commands to compute the arc l
ength of a curve.  We have to generate the length ourselves using the \+
formula.  First we calculate the derivative vector using the " }{TEXT 
0 3 "map" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "v:=map(diff,h,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 34 "Then we take its length using the " }{TEXT 0 7 "dotprod" }
{TEXT -1 44 " command (it works out better in this case):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "len:=simplify(sqrt(dotprod(v,v)));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "We note that in the case the
 length does not depend upon t.  However, in general it will, so next \+
we substitute in t=" }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 1 ":" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "len:=simplify(eval(len,t=tau));" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Now we integrate the result from
 the base point " }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 
100 " to t in order to calculate s.  Here we take the base point to be
 (1,0,1), which corresponds to t=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "s=int(len,tau=0..t);" }}}{PARA 0 "" 0 "" {TEXT -1 71 "For later pu
rposes, it will be convenient to solve this equation for t:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eq1:=t=solve(%,t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "In this example we take the base point to
 be (-1,0," }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 26 "), which corre
sponds to t=" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "s=int(len,tau=Pi..t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Once we h
ave the vector " }{TEXT 270 1 "v" }{TEXT -1 5 ", we " }{TEXT 0 9 "norm
alize" }{TEXT -1 4 " it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "T:=simp
lify(normalize(v));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "There are
 two ways to calculate the normal vector.  The first is to use the def
inition, which is that " }{TEXT 266 1 "N" }{TEXT -1 50 " is the unit v
ector pointing in the direction of d" }{TEXT 269 1 "T" }{TEXT -1 60 "/
ds.  Therefore, we must substitute the expression eq1 into " }{TEXT 
267 1 "T" }{TEXT -1 8 " to get " }{TEXT 268 1 "T" }{TEXT -1 20 " as a \+
function of s:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eval(evalm(T),eq1
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Once that's been done, we \+
differentiate this vector and calculate its length, which is the defin
tion of the curvature " }{XPPEDIT 18 0 "kappa" "6#%&kappaG" }{TEXT -1 
1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dTds:=map(diff,%,s);\nkappa
:=simplify(norm(dTds,2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Then
 using the definition, we have that" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "N:=evalm(dTds/kappa);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Note
 that we had to use the " }{TEXT 0 5 "evalm" }{TEXT -1 72 " command.  \+
We next want to show a curve along with the two vectors at t=" }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 25 ", which corresponds to s=" 
}{XPPEDIT 18 0 "Pi*sqrt(2)" "6#*&%#PiG\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 
-1 86 ".  Therefore, we must first construct the vectors, and we also \+
construct the vector h(" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 54 ")
, which is the tail from which the arrows will point:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "tvec:=evalf(eval(evalm(T),t=Pi));
\nnvec:=eval(evalm(N),s=Pi*sqrt(2));\nhvec:=evalf(eval(evalm(h),t=Pi))
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "We must use " }{TEXT 0 5 "ev
alf" }{TEXT -1 92 " since the plot commands need numeric values.  Then
 we plot the arrows along with the helix." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 196 "p1:=spacecurve(h,t=0..2*Pi,color=black,thickness=3):
\np2:=arrow(hvec,tvec,0.05,0.1,0.1,color=blue):\np3:=arrow(hvec,nvec,0
.05,0.1,0.1,color=red):\ndisplay(\{p1,p2,p3\},axes=normal,orientation=
[59,43]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 " We may also use the
 more convenient expression that " }{TEXT 272 1 "N" }{TEXT -1 50 " is \+
the unit vector pointing in the direction of d" }{TEXT 271 1 "T" }
{TEXT -1 4 "/dt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "map(diff,T,t);
\nN1:=simplify(normalize(%));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 208 
"Of course, the two expressions don't match exactly since one is a fun
ction of t and the other is a function of s.  However, by substituting
 our relationship between t and s we can show that they are the same:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eval(evalm(N1),eq1);\nevalm(N);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Another way to calculate the \+
curvature is to use the expression " }{XPPEDIT 18 0 "kappa" "6#%&kappa
G" }{TEXT -1 2 "=|" }{TEXT 264 1 "v" }{TEXT -1 7 " cross " }{TEXT 265 
1 "a" }{TEXT -1 2 "|/" }{XPPEDIT 18 0 "abs(v)^3;" "6#*$-%$absG6#%\"vG
\"\"$" }{TEXT -1 57 ".  Of course, to do this, we must calculate the v
ector a:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=map(diff,v,t
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Then we use the formula:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "norm(crossprod(v,a),2)/len^3;\nkap
pa1:=simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "kappa" "6#%&kappaG" }{TEXT -1 86 " is constant, we can \+
immediately see that the two expressions for the curvature match:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "kappa;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "1.  Calculate t
he length of the curve described by " }{TEXT 279 1 "r" }{TEXT -1 4 "(t
)=" }{XPPEDIT 18 0 "vector([cos t,t^2,sin(Pi-t^3)])" "6#-%'vectorG6#7%
*&%$cosG\"\"\"%\"tGF)*$F*\"\"#-%$sinG6#,&%#PiGF)*$F*\"\"$!\"\"" }
{TEXT -1 13 " from t=0 to " }{XPPEDIT 18 0 "t = Pi;" "6#/%\"tG%#PiG" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "First we define the curve and calculate the length of the
 tangent vector." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "rvec:=vector([c
os(t),t^2,sin(Pi-t^3)]);\nvvec:=map(diff,rvec,t);\nspeed:=sqrt(dotprod
(vvec,vvec));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rvecG-%'vectorG6#7
%-%$cosG6#%#t|irG*$)F,\"\"#\"\"\"-%$sinG6#*$)F,\"\"$F0" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%vvecG-%'vectorG6#7%,$-%$sinG6#%#t|irG!\"\",$F-
\"\"#,$*&-%$cosG6#*$)F-\"\"$\"\"\"F9)F-F0F9F8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&speedG*$-%%sqrtG6#,(*$)-%$sinG6#%#t|irG\"\"#\"\"\"F1
*&\"\"%F1)F/F0F1F1*(\"\"*F1)-%$cosG6#*$)F/\"\"$F1F0F1)F/F3F1F1F1" }}}
{PARA 0 "" 0 "" {TEXT -1 55 "Then we calculate the length of the curve
, numerically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(int(
speed,t=0..Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%R@^I#!\")" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "2.  Calculate " }{TEXT 
274 1 "T" }{TEXT -1 2 ", " }{TEXT 275 1 "N" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "kappa" "6#%&kappaG" }{TEXT -1 15 " for the curve " }
{TEXT 273 1 "r" }{TEXT -1 4 "(t)=" }{XPPEDIT 18 0 "vector([cos t,t^2,s
in(Pi-t^3)])" "6#-%'vectorG6#7%*&%$cosG\"\"\"%\"tGF)*$F*\"\"#-%$sinG6#
,&%#PiGF)*$F*\"\"$!\"\"" }{TEXT -1 8 " at t=1." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Soluti
on" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Since we have the velocity v
ector, calculating " }{TEXT 276 1 "T" }{TEXT -1 5 " and " }{TEXT 277 
1 "N" }{TEXT -1 67 " is easy.  The trick to saving time is to do the s
ubstitution into " }{TEXT 278 1 "N" }{TEXT -1 23 " first, then normali
ze." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "tvec:=simplify(normalize(vv
ec)):\nt0:=evalf(eval(evalm(tvec),t=1));\nnvec:=eval(map(diff,tvec,t),
t=1):\nn0:=evalf(normalize(nvec));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#t0G-%'vectorG6#7%$!+0r*o5$!#5$\"+GEW%Q(F+$\"+qqu%)fF+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#n0G-%'vectorG6#7%$!+t\")Gt<!#5$\"+4P\\NdF
+$!+h@Z(*zF+" }}}{PARA 0 "" 0 "" {TEXT -1 30 "To calculate kappa, we n
eed a:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "avec:=map(diff,vv
ec,t);\na0:=evalf(eval(evalm(avec),t=1));\nv0:=evalf(eval(evalm(vvec),
t=1));\nkap:=evalf(norm(crossprod(v0,a0),2)/(eval(speed,t=1)^3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%avecG-%'vectorG6#7%,$-%$cosG6#%#t|i
rG!\"\"\"\"#,&*&-%$sinG6#*$)F-\"\"$\"\"\"F8)F-\"\"%F8!\"**(\"\"'F8-F+F
4F8F-F8F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G-%'vectorG6#7%$!+fI
-.a!#5$\"\"#\"\"!$!+G]UJV!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0
G-%'vectorG6#7%$!+[)4ZT)!#5$\"\"#\"\"!$\"+=p!4i\"!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$kapG$\"+'fbnT'!#5" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 "Sections" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 433 "In order to sepa
rate your Maple worksheets into parts, you may also wish to use sectio
ns, which are indicated by the boxes with minus signs above.  Note tha
t extending from the box is a line enclosing a series of commands. If \+
you click on the minus box, the section will collapse.  Only the title
 will appear and now the minus sign is a plus sign, indicating there i
s more to see.  Clicking on the plus sign will reverse the process." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "There a
re two ways to make sections.  The first is to select Insert...Section
 from the menu bar.  Then begin typing the section title, then the com
mands you wish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 117 "The second is to type the lines as normal.  Then select \+
them and hit Format...Indent.  Try this with the lines below." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sin(x),x=0..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "map(diff,avec+bvec,t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Now insert a section below and giv
e it a title." }{MPLTEXT 1 0 0 "" }}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }