{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 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 2 2 2 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 "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 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 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica " 1 18 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3 " -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 5" -1 260 1 {CSTYLE "" -1 -1 "Helvetica " 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 6" -1 261 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 7" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 263 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 9" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 10" -1 265 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 11" -1 266 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 73 "************************************************** ***********************" }}{PARA 0 "" 0 "" {TEXT -1 69 "A worksheet pr epared by: Prof. Maria Girardi and Douglas Meade" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Math 550 and 550A -- Spring 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Maple V Release 4 for Unix " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "URL: http://www.math.sc.edu/~meade/math5 50-S97/maple/sec4-3.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 18 "plot vector fields" }}{PARA 15 "" 0 "" {TEXT -1 15 "plot flow lines" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "*********************************************** **************************" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Initialization and Setup " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "First we clear out Maple's mem ory." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "The default viewing window f or all plots will be -3 <= x <= 3 and -3 <= y <= 3. If you want to \+ change this, simply change the appropriate numbers in the following as signments." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "xview := -3 . . 3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "yview := -3 .. 3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 20 "Define vector fields" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Definition of vector fields for all 11 problems in this section " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "vf[1] := [ 2, 2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vf[2] :=[4 , 0 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "vf[3] :=[x ,y ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vf[4] :=[ -x, y] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "vf[5] :=[ 2*y, x] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vf[6] :=[ y,-2*x ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "vf[7] :=[x/sqrt(x^2+y^2) ,y/sqrt(x^2+y^2) ] ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "vf[8] :=[y/sqrt(x^2+y^2) , x /sqrt(x^2+y^2)] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "vf[9] \+ :=[ y, -x ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vf[10] :=[ x, -y ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "vf[11] :=[ x, x^2 ] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Vector Fields" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "The VECTOR FIELD for a specified problem is obtained sim ply by specifying the problem number in the next assignment, then exec uting the " }{HYPERLNK 17 "fieldplot" 2 "plots,fieldplot" "" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "problem := 6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fieldplot( vf[problem] , x = xview \+ , y = yview," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " title = \+ `Vector Field for Section 4.3, #`.problem );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Flow L ines" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Definitions of Supplement al Procedures" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "It's also possib le to FLOW LINES for the direction field. However, this takes a little more effort. The biggest obstacle is to specify which flow lines shou ld be plotted. If you want to use a uniform grid, the makeIC procedure should be useful." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "makeI C := proc( nx:posint, ny:posint, xrange:range, yrange:range )" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " local dx, dy, i, x0, x1, y0, y1, \+ X, Y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " x0 := op(1,xrange); x1 : = op(2,xrange); dx := (x1-x0)/nx;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " y0 := op(1,yrange); y1 := op(2,yrange); dy := (y1-y0)/ny;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " X := [ x0+i*dx $ i=0..nx ];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " Y := [ y0+i*dy $ i=0..ny ];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " RETURN( \{ seq( seq( [0,x,y], y=Y ), x=X) \} )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "makeODE := (F,t,x,y) -> [ diff(x(t),t)=F[1], diff(y(t ),t)=F[2] ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Tests and Demonstrations of " }{TEXT 19 6 "makeIC" }{TEXT -1 5 " and " }{TEXT 19 7 "makeODE" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "makeIC( 2, 2, xview, yview );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "makeODE( vf[4], t, x, y );" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SYS := makeODE( vf[problem], t, x, y );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IC := makeIC( 2, 2, xview, y view );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "DEplot( SYS, [x,y], t=0..1, IC, arr ows=SMALL," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " title=`Flow L ines for Section 4.3, Problem `.problem );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 249 "Flow lines , superimposed on the vector field, are obtained with the following co mmand. Try to identify all of the flow lines in the plot. It may be he lpful to shorten or lengthen the \"time interval\" and to modify (or e ven delete) the viewing window." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "DEplot( SYS, [x, y], t=0..1, IC, arrows=SMALL, view=[xview,yview]," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 63 " title=`Flow Lines for Section 4.3, Problem \+ `.problem );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Caution for Problems 7 and 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The above commands will not work for the vector fields in Problems 7 and 8. But, if either of the first two ar guments to " }{TEXT 19 6 "makeIC" }{TEXT -1 28 " is odd there is no pr oblem." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 59 "What feature of these \+ problems causes the original problem?" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 26 "Why does this change work?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{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 }