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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }{TEXT 
270 34 "Vector valued functions in Maple 7" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 260 45 "This worksheet is called \"vectorvalued7.
mws\"." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Type in the commands be
low.  You do not need to type in the comments." }}{PARA 0 "" 0 "" 
{TEXT -1 22 "Or, visit my web page " }{TEXT 261 38 "http://www-math.bg
su.edu/~zirbel/calc3" }{TEXT -1 35 " and look for the worksheet called
 " }{TEXT 262 17 "vectorvalued7.mws" }{TEXT -1 64 ".  You can download
 it following the directions on the web page." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }{TEXT 257 31 "If you have trouble printing:  " }
{TEXT 263 398 "Make your worksheets shorter; split one assignment into
 several worksheets so none of them has very many graphs.  If you stil
l have trouble, at the bottom left of a Macintosh screen is a tab.  Cl
ick and pull it out.  Click on the colored bars and set the number of \+
colors to 256 (rather than millions).  This avoids the memory trouble \+
that apparently causes Maple to mess up when printing graphics." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Vector-valued functions in two di
mensions" }}{PARA 0 "" 0 "" {TEXT -1 53 "The first step is to define a
 vector-valued function " }{TEXT 256 1 "r" }{TEXT -1 6 "(t).  " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 85 " This
 clears Maple's memory and previous definitions.  It helps Maple avoid
 crashing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r:=t->2*cos(2
*t)*[cos(t),sin(t)]; " }{TEXT -1 47 "This says that r is defined to be
 (that's what " }{TEXT 271 2 ":=" }{TEXT -1 334 " means) a function ta
king t to the expression 2cos(2t)[cos(t),sin(t)].  We all know that th
e vector [cos(t),sin(t)] gives you circular motion.  Multiplying by th
e scalar 2cos(2t) changes the radius of the circle as a function of ti
me, and even makes the \"radius\" negative.  The resulting graph, as y
ou will see below, is a cloverleaf." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&-%$cosG6#,$9$\"\"#\"\"\"
7$-F/6#F2-%$sinGF7F4F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "r(t); r(t)[1]; r(t)[2];" }{TEXT -1 113 " Display what r(t) is.  \+
If you want the first and second components of the vector, you say thi
s with [1] and [2] " }{TEXT 268 5 "after" }{TEXT -1 15 " you type r(t)
." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$cosG6#,$%\"tG\"\"#\"\"\"7$
-F&6#F)-%$sinGF.F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&,$*&-%$cosG6#
,$%\"tG\"\"#\"\"\"7$-F'6#F*-%$sinGF/F,F+6#F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#&,$*&-%$cosG6#,$%\"tG\"\"#\"\"\"7$-F'6#F*-%$sinGF/F,F+6
#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([evalm(r(t))[1]
,evalm(r(t))[2],t=0..2*Pi]); " }{TEXT -1 223 "This is the best way to \+
plot r(t), since the same command will work for more complicated vecto
r valued functions, as you will see below.  In this plot command, the \+
two components of r(t) are listed separately.  The command " }{TEXT 
272 5 "evalm" }{TEXT -1 73 " does scalar multiplication and vector add
ition, if any needs to be done." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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/$!30MiH'4&\\>aF87$$\"3M,Dgf%[\\\\\"F/$!3))Q'G>s%H>^F87$$\"3#\\_yJ%H$)
o;F/$!3j1#4*eT6.XF87$$\"3=9m@dpK0=F/$!3'R!HA4gKgOF87$$\"33M]f'=\"*z!>F
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ioqU:X^8F87$$\"3kWFF:.=%*>F/$!3c[cL(o,)3oF27$F($\"3A\\E7Gu#3k\"!#E-%'C
OLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q!6\"Fi^m-%%VIEWG6$%(DEFAU
LTGF^_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "v:=t->-4*sin(2*t)*[
cos(t),sin(t)] + 2*cos(2*t)*[-sin(t),cos(t)]; " }{TEXT -1 265 "I have \+
tried and tried, but I have not found a satisfactory way to have Maple
 compute the derivative of r(t).  (There are lots of ways of trying, b
ut none of them makes v a function of the same sort that r is.)  So I \+
computed the derivative by hand and typed it in." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&-%$sinG6#,
$9$\"\"#\"\"\"7$-%$cosG6#F2-F/F8F4!\"%*(F3F4-F7F0F47$,$F9!\"\"F6F4F4F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "l:=s->r(1) + s*v(1)
;" }{TEXT -1 66 " This is the parametric form of the tangent line at t
ime 1.  Here " }{TEXT 259 2 "l " }{TEXT -1 17 "is a function of " }
{TEXT 258 1 "s" }{TEXT -1 122 ".  Please note how easy it is to make u
p the tangent line.  It is a function taking s to r(1) + s v(1), just \+
what we want." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lGf*6#%\"sG6\"6$%
)operatorG%&arrowGF(,&-%\"rG6#\"\"\"F0*&9$F0-%\"vGF/F0F0F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "l(s); l(2); eval(l);" }
{TEXT -1 44 " This is how I check that the definition of " }{TEXT 273 
1 "l" }{TEXT -1 105 " is doing what I want it to do.  You don't need t
o type this in if you just want to see the tangent line." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*&-%$cosG6#\"\"#\"\"\"7$-F&6#F)-%$sinGF,F)F(*&
%\"sGF),&*&-F.F'F)F*F)!\"%*(F(F)F%F)7$,$F-!\"\"F+F)F)F)F)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,(*&-%$cosG6#\"\"#\"\"\"7$-F&6#F)-%$sinGF,F)F(*
(\"\")F)-F.F'F)F*F)!\"\"*(\"\"%F)F%F)7$,$F-F2F+F)F)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#f*6#%\"sG6\"6$%)operatorG%&arrowGF&,&-%\"rG6#\"\"\"F.
*&9$F.-%\"vGF-F.F.F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 
"p1:=plot([evalm(r(t))[1],evalm(r(t))[2],t=0..2*Pi]):" }{TEXT -1 20 " \+
Store the graph of " }{TEXT 264 1 "r" }{TEXT -1 4 " as " }{TEXT 265 2 
"p1" }{TEXT -1 44 ".  Note the colon at the end of the command!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "p2:=plot([evalm(l(s))[1],eva
lm(l(s))[2],s=-0.5..0.5],color=black): " }{TEXT -1 19 "Store the graph
 of " }{TEXT 266 1 "l" }{TEXT -1 4 " as " }{TEXT 267 2 "p2" }{TEXT -1 
1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }
{TEXT -1 86 " Load some additional plotting features.  This is necessa
ry for the next line to work!" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "display(\{p1,p2\});" }{TEXT -1 119 " Display these
 two graphs together.  This is the standard way to display two vector \+
valued functions in the same graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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LSG6$Q!6\"Facn-%%VIEWG6$;F($\"+3`=$G'F,%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "arclength:=evalf(Int(speed(t),t=0..2*Pi)); " }
{TEXT -1 57 "Evaluate arc length numerically by integrating the speed.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*arclengthG$\"+W'*oP>!\")" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Building more interesting functio
ns" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here is how to build up something m
ore interesting based on what we did above." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "r:=t->2*cos(2*t)*[cos(t),sin(t)];" }{TEXT -1 31 " The same cloverl
eaf  as above." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$
%)operatorG%&arrowGF(,$*&-%$cosG6#,$9$\"\"#\"\"\"7$-F/6#F2-%$sinGF7F4F
3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "q:=t->[cos(t),si
n(t)]; " }{TEXT -1 16 "Circular motion." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"qGf*6#%\"tG6\"6$%)operatorG%&arrowGF(7$-%$cosG6#9$-%$sinGF/F
(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s:=t->r(t) + 0.1*q
(80*t); " }{TEXT -1 68 "Start with r(t), then add a scaled down and sp
eeded up version of q." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%
\"tG6\"6$%)operatorG%&arrowGF(,&-%\"rG6#9$\"\"\"*&$F1!\"\"F1-%\"qG6#,$
F0\"#!)F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([
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"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7air7$$\"33++
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45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 0 {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 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"with(plots): " }{TEXT -1 55 "Load the package of functions that inclu
des spacecurve." }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chan
gecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "r:=t->[2*cos(t),t/4,2*sin(t)];" }{TEXT -1 81 " Define a helix th
at circles in the xz plane and moves linearly along the y axis." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrow
GF(7%,$-%$cosG6#9$\"\"#,$F1#\"\"\"\"\"%,$-%$sinGF0F2F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "p1:=spacecurve(evalm(r(t)),t=0..12
*Pi,axes=BOXED,thickness=2,numpoints=800,labels=[\"x axis\",\"y axis\"
,\"z axis\"]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display(p
1);" }{TEXT -1 87 " Typing evalm here will be helpful in situations wh
ere r has a more complex definition." }}{PARA 13 "" 1 "" {GLPLOT3D 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "v:=t->[-2*sin(t),1/4,2*cos(t)];" }
{TEXT -1 51 " I computed the derivative by hand and typed it in." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)operatorG%&arrow
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "l:=s->r(Pi/3)+s*v(Pi/3); " }{TEXT 
-1 42 "The equation for the tangent line at time " }{XPPEDIT 18 0 "pi;
" "6#%#piG" }{TEXT -1 61 "/3.  Hint: same formula as for two-dimension
al tangent lines." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lGf*6#%\"sG6
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F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p2:=spacecurv
e(evalm(l(s)),s=-1..1,color=black):" }{TEXT -1 45 " Prepare and save a
 plot of the tangent line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "display(\{p1,p2\});" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
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