{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 26 "Paths (and a few Surfaces)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Maple Worksheet for Secti ons 2.4 and 2.7 of Marsden and Tromba" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "Prepared by Douglas B. Meade" }} {PARA 0 "" 0 "" {TEXT -1 15 "30 January 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "URL: http://www.math.sc.edu/~me ade/math550-S97/maple/sec2-4.mws" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Objectives" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 96 "provide plots o f functions that appear in selected Examples and Problems in sections \+ 2.4 and 2.7" }}{PARA 15 "" 0 "" {TEXT -1 71 "provide an animated view \+ of the motion of a point on the rim of a wheel" }}{PARA 15 "" 0 "" {TEXT -1 77 "demonstrate how Maple can be used to examine functions al ong specified curves" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "A Helpful Procedure (for parametric paths)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 287 "It makes most sense to enter a parametric plot as a list of two o r three elements. However, for plotting purposes (in 2D) the range of \+ parametrs needs to be the third element of the list. The following pro cedure can be used to do this in a general way (see the figures for sa mple usage)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "makeparamet ric := proc( V:list, prange:name=range )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " RETURN( [ op(V), prange ] )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Figure 2.4.4 (p. 126)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c := [ t, t^2 ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot( makeparametric( c, t=-2..2 ), title=`Figure 2.4.4, p. 126` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Figure 2.4. 6 (p. 127)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The path is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "c := [ v*t - r*sin(v*t/r), R - r*cos(v*t/r) ];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "The plot w ill be more meaningful if the moving circle, and the line segment betw een the center of the circle and the point, are animated. Here are the circle and line segment," }}{PARA 0 "" 0 "" {TEXT -1 19 "in parametri c form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "C := [ v*T + R*c os(theta), R + R*sin(theta) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "L := simplify( [ (1-t)*v*T + t*subs( t=T, c[1] ), (1-t)*R + t* subs( t=T, c[2] ) ] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Enter \+ the specific choice of parameters ( v : velocity of wheel, r : distanc e from point to center, R : radius of circle )" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "PARAM := [ v=1, r=1, R=1 ]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "And, now, the complete graph " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs( t=tau*T, c );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cc := makeparametric( subs( PARAM, \" ), tau=0..1 ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs( theta=2*Pi*tau, C );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CC := makeparametric( subs( PARAM, \+ \" ), tau=0..1 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs( t=tau, L );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "LL := makeparametric( subs( \+ PARAM, \" ), tau=0..1 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "animate ( \{ cc, CC, LL \}, T=0..10*Pi, color=RED );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Figure 2.4.9 (p. 130)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "This helix is d efined by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "c := [ cos(t), sin(t), t ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This can be plotted with plot3d, b ut it requires two ranges (here the second range is meaningless)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d( c, t=0..4*Pi, a=0..1 , axes=BOXED, title=`Figure 2.4.9` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Figure 2.7. 1 (p. 161)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x*y/sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot3d( f, x=-1..1, y=-1..1, axes=B OXED, title=`Figure 2.7.1` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Figure 2.7.2 (p. 163) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := 2*x*y^2/(x^2+y^4);; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot3d( f, x=-1..1, y=- 1..1, axes=BOXED, title=`Figure 2.7.2` );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }