{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
272 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1
" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 
1 0 0 8 4 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 44 "This works
heet is called \"vectorvalued.mws\"." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 69 "Type in the commands below.  You do not need to type in the com
ments." }}{PARA 0 "" 0 "" {TEXT -1 22 "Or, visit my web page " }{TEXT 
270 38 "http://www-math.bgsu.edu/~zirbel/calc3" }{TEXT -1 35 " and loo
k for the worksheet called " }{TEXT 271 16 "vectorvalued.mws" }{TEXT 
-1 64 ".  You can download it following the directions on the web page
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 31 "If you have \+
trouble printing:  " }{TEXT 272 398 "Make your worksheets shorter; spl
it one assignment into several worksheets so none of them has very man
y graphs.  If you still have trouble, at the bottom left of a Macintos
h screen is a tab.  Click and pull it out.  Click on the colored bars \+
and set the number of colors to 256 (rather than millions).  This avoi
ds the memory trouble that apparently causes Maple to mess up when pri
nting graphics." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Vector-valued
 functions in two dimensions" }}{PARA 0 "" 0 "" {TEXT -1 53 "The first
 step is to define a vector-valued function " }{TEXT 256 1 "r" }{TEXT 
-1 104 "(t).  Based on the way Maple does real-valued functions of one
 variable, one might think you would type " }{XPPEDIT 18 0 "r := proc \+
(t) options operator, arrow; vector([x(t), y(t)]) end;" "6#>%\"rGR6#%
\"tG7\"6$%)operatorG%&arrowG6\"-%'vectorG6#7$-%\"xG6#F'-%\"yG6#F'F,F,F
," }{TEXT -1 87 ", but you don't.  Instead, you put a function in each
 component of a vector as follows:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "r:=vector([t->2*cos(2*t)*cos(t),t->2*cos(2*t)*sin(t)]
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "r(t); r[1](t); r[2](t
); " }{TEXT -1 23 "This is how you get at " }{TEXT 260 1 "r" }{TEXT 
-1 27 " and the two components of " }{TEXT 257 1 "r" }{TEXT -1 1 "." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(r(t),t=0..20);" }
{TEXT -1 60 "This is what we might expect to do to plot r, but it is n
ot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot([r[1](t),r[2](t
),t=0..2*Pi]); " }{TEXT -1 102 "One needs to specify the two component
s here.  You can't just use r(t).  See p. 151 of Gresser's book." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v:=map(D,r); " }{TEXT -1 36 
"This is how you differentiate.  The " }{TEXT 261 3 "map" }{TEXT -1 
46 " command applies the differentiation operator " }{TEXT 278 1 "D" }
{TEXT -1 0 "" }{TEXT -1 22 " to each component of " }{TEXT 279 1 "r" }
{TEXT -1 0 "" }{TEXT -1 46 " separately.  It means \"multiple applicat
ion\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r(1); v(1); " }
{TEXT -1 54 "Compute the position and derivative vectors at time 1." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(r(1)); evalf(v(1));
" }{TEXT -1 41 "Evaluate (approximate) these numerically." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }{TEXT -1 0 "" }
{TEXT 258 85 "This is really important!  This loads functions that  do
 things like adding vectors. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "l:=map(unapply,matadd(r(1), scalarmul(v(1),s)),s);" }{TEXT -1 
65 "This is the parametric form of the tangent line at time 1.  Here \+
" }{TEXT 263 2 "l " }{TEXT -1 17 "is a function of " }{TEXT 262 1 "s" 
}{TEXT -1 15 ".  The command " }{TEXT 273 7 "unapply" }{TEXT -1 37 " t
urns an expression into a function." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "p1:=plot([r[1](t),r[2](t),t=0..2*Pi]):" }{TEXT -1 20 
" Store the graph of " }{TEXT 274 1 "r" }{TEXT -1 4 " as " }{TEXT 275 
2 "p1" }{TEXT -1 18 ".  Note the colon!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "p2:=plot([l[1](s),l[2](s),s=-0.75..0.75]): " }{TEXT 
-1 19 "Store the graph of " }{TEXT 276 1 "l" }{TEXT -1 4 " as " }
{TEXT 277 2 "p2" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "with(plots):" }{TEXT -1 86 " Load some additional plotting fea
tures.  This is necessary for the next line to work!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p1,p2\});" }{TEXT -1 35 " Displ
ay these two graphs together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "speed:=t->norm(v(t),2);" }
{TEXT -1 40 "  The speed function, called v in class." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(speed,0..2*Pi);" }{TEXT -1 26 
" Graph of speed over time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "arclength:=evalf(Int(speed(t),t=0..2*Pi)); " }{TEXT -1 39 "Evaluat
e arc length numerically.  Slow." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "a:=map(D,v); " }{TEXT -1 24 "The acceleration vector.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "curvature:=t->abs(v[1](
t)*a[2](t)-v[2](t)*a[1](t))/speed(t)^3; " }{TEXT -1 156 "Curvature as \+
a function of time.  Since these are two-dimensional vectors, imagine \+
adding a third component that is zero before taking the cross product \+
of " }{TEXT 266 1 "v" }{TEXT -1 5 " and " }{TEXT 267 1 "a" }{TEXT -1 
36 ".  The cross product is then in the " }{TEXT 268 1 "k" }{TEXT -1 
80 " direction, and its length is the absolute value in the definition
 of curvature." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(curv
ature,0..2*Pi); " }{TEXT -1 77 "The curvature is largest at the far ri
ght, left, top, bottom of the graph of " }{TEXT 269 1 "r" }{TEXT -1 1 
"." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Building more interesting
 functions" }}{PARA 0 "" 0 "" {TEXT -1 108 "Here is how to build up so
mething more interesting based on what we did above.  Enter the lines \+
above first." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q:=vector([t
->cos(40*t),t->sin(40*t)]);" }{TEXT -1 58 " A new vector-valued functi
on doing quick circular motion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "p:=scalarmul(q,0.2);" }{TEXT -1 35 " Make the circle have radi
us 0.2.  " }{TEXT 265 10 "p:=0.2*q; " }{TEXT -1 13 "doesn't work." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s:=matadd(p,r);" }{TEXT -1 
103 "  Add this motion on top of the cloverleaf above.  This is how to
 add vector-valued functions in Maple." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "plot([s[1](t),s[2](t),t=0..20]);" }{TEXT -1 38 "  Plo
t the new vector-valued function." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 43 "Vector-valued functions in three dimensions" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "This example includes all the necessary commands, so it c
an be run separately from what is above." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "with(plots): with(linalg):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 46 "r:=vector([t->2*cos(t), t->t/4, t->2*sin(t)]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "p1:=spacecurve(r(t),t=0..1
2*Pi,axes=BOXED,thickness=2,numpoints=800,labels=[\"x axis\",\"y axis
\",\"z axis\"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display
(p1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "v:=map(D,r);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "l:=map(unapply,matadd(r(Pi/3
), scalarmul(v(Pi/3),s)),s); " }{TEXT -1 42 "The equation for the tang
ent line at time " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 61 "/3.  H
int: same formula as for two-dimensional tangent lines." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p2:=spacecurve(l(s),s=-1..1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{p1,p2\});" }}}}}
{MARK "0 0 1" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }