{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 0 1 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 59 "PHYS2030 Computational M ethods for Physicists and Engineers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 100 "A Maple Primer: this introduction to Ma ple 7/8/9 provides a tutorial and reference to the question:" }} {PARA 0 "" 0 "" {TEXT -1 53 "how does one do Calculus and Linear Algeb ra in Maple?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "While at some point you should know most of the commands expla ined by heart, you can use this worksheet during homework and tests." }}{PARA 0 "" 0 "" {TEXT -1 119 "Note that there are tutorials in Maple 's help system, and that each command can be queried by ?commandname, \+ e.g., ?diff" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "1.1 Basics:" }}{PARA 0 "" 0 "" {TEXT -1 147 "Commands are ente red into Maple at the prompt (> symbol), and are terminated by a semic olon or colon. A colon suppresses the display of the result:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3+9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "To assign \+ the result to a variable:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:=3+9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Want to know what " }{TEXT 19 1 "a" } {TEXT -1 4 " is?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=4+10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "b-a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "b*a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "b^a ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"/'HvB\"Rpc" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5 "b**a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"/' HvB\"Rpc" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Both versions work, b ut we will avoid the Fortran-style " }{TEXT 19 2 "**" }{TEXT -1 10 " o perator!" }}{PARA 0 "" 0 "" {TEXT -1 111 "At this point we know how to do arithmetic. We can save our session for future reference. Maple st ores it as a " }{TEXT 19 5 "*.mws" }{TEXT -1 195 " file. Note that whe n we re-open the file, it will display its content, but it will not be active unless we execute (enter) the commands. This can also be done \+ for an existing worksheet from the " }{TEXT 19 4 "Edit" }{TEXT -1 16 " menu in one go." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Let us continue to the level of " }{TEXT 257 19 "graphing calculator" }{TEXT -1 68 " proficiency: use the trig and exponential \+ functions and graph them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f1:=sin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G-%$sinG6#,$ *&\"\"#\"\"\"%\"xGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f 2:=exp(-3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G-%$expG6#,$*&\" \"$\"\"\"%\"xGF+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 19 2 "f1" } {TEXT -1 5 " and " }{TEXT 19 2 "f2" }{TEXT -1 83 " are not functions, \+ but EXPRESSIONS in Maple. 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dl$\"3a>TJ?6L:SFgu7$Fjdl$\"3K)4q=&Q)4*oFgu7$F_el$\"3;9N7xQYi()Fgu7$$\" 3'3]<@A'yXHF*$\"3>YW'fBwWK*Fgu7$Fdel$\"3!\\@&pAl50(*Fgu7$$\"3!oO7f/K1/ $F*$\"3uP\\;K`IL)*Fgu7$$\"3ub1^?tCsIF*$\"3*)GTRq5SB**Fgu7$$\"3pW*3^fiQ 5$F*$\"3;M-MF$Gz(**Fgu7$Fiel$\"3k!\\#4\"4O%****Fgu7$$\"3EY(*=T\\_oJF*$ \"3!fu$3E&z$*)**Fgu7$$\"3LfAn7?d,KF*$\"3GR;c:X*)[**Fgu7$$\"3SsZ:%3>YB$ F*$\"3#ps1?(4!3))*Fgu7$F^fl$\"3K,%>x-[yy*Fgu7$$\"3g6Bg)HgPL$F*$\"3xW3c K\\\"z`*Fgu7$Fcfl$\"31T\"z'p(R\">#*Fgu7$Fhfl$\"3%eh!**42Ck%)Fgu7$F]gl$ \"3h)3fZ<'oEwFgu7$Fbgl$\"3QDIkM\\!4%fFgu7$Fggl$\"3W\"=*=*pYTZSPFgu7$Feil$\"3XM4_+C*3\"RFgu7$F jil$\"33Swyu?GTUFgu7$F_jl$\"39)QmnC#H;ZFgu7$Fdjl$\"3h51<:uV6fFgu7$Fijl $\"3G10$3uF&evFgu7$Fc[m$\"3EJ>URMKS\"*Fgu7$F]\\m$\"2y***************F* -Fb\\m6&Fd\\mFh\\mFe\\mFh\\m-%+AXESLABELSG6$Q\"s6\"Q!Fgcp-%%VIEWG6$;$! +3`=$G'!\"*$\"+3`=$G'F_dp%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 15 "Exercise 1.2.1:" }}{PARA 0 "" 0 "" {TEXT -1 185 "Observe the behavior of the function and its derivatives at the critical points (vanishing derivative=stationary point, vanish ing 2nd derivative=turning point), by inspecting the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "How do we find \+ the critical points? This is a question of computer algebra. The " } {TEXT 19 5 "solve" }{TEXT -1 5 " and " }{TEXT 19 6 "fsolve" }{TEXT -1 46 " commands will be explored to find the zeroes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sol:=solve(f1,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 153 "Note that solve found only one root! The function is periodic, and this root should be repeated infinitely many times. \+ In addition there are other roots." }}{PARA 0 "" 0 "" {TEXT -1 87 "The message: be careful with answers from Maple (or any other computer al gebra system)." }}{PARA 0 "" 0 "" {TEXT -1 65 "Let us start with the o ne root Maple found. We want to verify it:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "subs(s=sol,f1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&-%$cosG6#,$*&\"\"#!\"\"%#PiG\"\"\"F,F,-%$expG6#-%$sinGF&F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We \+ learned the substitute command, and that it does not automatically sim plify the result. Alternatively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(f1,s=sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Do we have a minimum or a \+ maximum here?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(f2,s= sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#\"\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!+G=G=F!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "We have a negative second derivative, and thus a maximum. (use the pa rabola x^2/-x^2 to remember the condition!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 237 "Now let us learn a numerical m ethod to arrive at the roots of the derivative function. We use a blac k-box method. fsolve requires that you bracket the desired root, we us e 1 and 2 as brackets to find the above solution in numerical form." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=fsolve(f1,s=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G$\"+Fjzq:!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(f1,s=sol1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!+HzGvb!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 " Note that the function evaluation does not yield zero. This is caused \+ by the limited precision, which is set by default to 10. Digits is an \+ internal variable which allows us to increase the precision." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=14;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=fsolve(f1,s=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G$\"/\\zEjzq:!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(f1,s=sol1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ !/\"pW1#))*=*!#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "We find that we can push the accuracy closer towards the desired result. In contra st to floating point number system based computing languages we are no t limited to precision restrictions such as 8 or 16 or 32. We can set \+ " }{TEXT 19 6 "Digits" }{TEXT -1 38 " to almost anything we like (or n eed)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Nevertheless, the problem remains: when do we recognize a number to b e close enough to zero that it may actually be zero?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(10^(-Digits));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/++++++5!#F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 15 "Exercise 1.2.2:" }}{PARA 0 "" 0 "" {TEXT -1 213 "Use the graph as a g uide to find the other critical points numerically, and verify the min imum/maximum property. Observe what happens when no root can be found \+ (e.g., by trying to solve for a root of the function " }{TEXT 19 1 "f " }{TEXT -1 9 " itself)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Now we calculate some anti-derivatives:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(f,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$expG6#-%$sinG6#%\"sGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "This is the display when Maple can't do \+ it. It just shows the integral sign. In contrast to derivatives, which can always be obtained mechanically, integrals of elementary function s often cannot be found in terms of elementary functions." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Exercise 1.2.3:" }}{PARA 0 "" 0 "" {TEXT -1 60 "Use known integrals of elementary funct ions to test Maple's " }{TEXT 19 3 "int" }{TEXT -1 130 " engine. It is an engine, because it does not use look-up, but algorithmic approache s, something you learned in 1st year Calculus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "An important fac t, however, is that definite integrals can be approximated numerically in these cases where the anti-derivative is not known as a simple fun ction. We use a black-box approach, and an inert version of the integr ation command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(Int (f,s=0..Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/6rN!e(3i!#8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "This number represents the area un der the curve between the vertical lines at " }{TEXT 19 3 "s=0" } {TEXT -1 5 " and " }{TEXT 19 4 "s=Pi" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 15 "Exercise 1.2.4:" }} {PARA 0 "" 0 "" {TEXT -1 173 "Come up with examples where the function crosses the x-axis in the range of integration, and verify that the s ign of the function matters, i.e., calculate the areas between " } {TEXT 19 1 "a" }{TEXT -1 5 " and " }{TEXT 19 2 "x0" }{TEXT -1 5 " and \+ " }{TEXT 19 2 "x0" }{TEXT -1 5 " and " }{TEXT 19 1 "b" }{TEXT -1 26 " \+ as well as the area from " }{TEXT 19 1 "a" }{TEXT -1 4 " to " }{TEXT 19 1 "b" }{TEXT -1 19 " and explain! Here " }{TEXT 19 3 "a,b" }{TEXT -1 40 " are the boundaries of integration, and " }{TEXT 19 2 "x0" } {TEXT -1 46 " is the location of the zero. You can use the " }{TEXT 19 6 "sin(x)" }{TEXT -1 37 " function, and bracket only one root." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "What else can we do?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sum(n^2,n=1. .N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"$!\"\",&%\"NG\"\"\"F)F )F%F)*&\"\"#F&F'F+F&*&\"\"'F&F(F)F)#F)F-F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Comparable to integration: Maple can calculate closed-for m sums." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(sin(a*x)/x,x=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Maple can calculate limits. The statement " }{TEXT 19 7 " a:='a':" }{TEXT -1 33 " serves to unassign the variable " }{TEXT 19 1 "a" }{TEXT -1 36 ", i.e., to turn it back into symbol " }{TEXT 19 1 "a " }{TEXT -1 34 ". Maple also does one-sided limits" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=abs(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$absG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " f1:=diff(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G-%$absG6$\"\" \"%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "limit(f1,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "limit(f1,x=0,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "lim it(f1,x=0,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot([f,f1],x=-2..2,color=[r ed,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 913 367 367 {PLOTDATA 2 "6&-% 'CURVESG6$7U7$$!\"#\"\"!$\"\"#F*7$$!3MLLL$Q6G\">!#<$\"3MLLL$Q6G\">F07$ $!3bmm;M!\\p$=F0$\"3bmm;M!\\p$=F07$$!3MLLL))Qj^'***!#=$\"3w++++()>'***Fbo7$$!3E++++0\"*H\"*Fbo$\"3E++++0\"*H\" *Fbo7$$!35++++83&H)Fbo$\"35++++83&H)Fbo7$$!3\\LLL3k(p`(Fbo$\"3\\LLL3k( p`(Fbo7$$!3Anmmmj^NmFbo$\"3Anmmmj^NmFbo7$$!3)zmmmYh=(eFbo$\"3)zmmmYh=( eFbo7$$!3+,++v#\\N)\\Fbo$\"3+,++v#\\N)\\Fbo7$$!3commmCC(>%Fbo$\"3commm CC(>%Fbo7$$!39*****\\FRXL$Fbo$\"39*****\\FRXL$Fbo7$$!3t*****\\#=/8DFbo $\"3t*****\\#=/8DFbo7$$!3=mmm;a*el\"Fbo$\"3=mmm;a*el\"Fbo7$$!3komm;Wn( o)!#>$\"3komm;Wn(o)Fjr7$$!3$G++]7bDW%Fjr$\"3$G++]7bDW%Fjr7$$!3IqLLL$eV (>!#?$\"3IqLLL$eV(>Fes7$$\"3V[mmT+07UFjrFis7$$\"3)Qjmm\"f`@')FjrF\\t7$ $\"3%z****\\nZ)H;FboF_t7$$\"3ckmm;$y*eCFboFbt7$$\"3f)******R^bJ$FboFet 7$$\"3'e*****\\5a`TFboFht7$$\"3'o****\\7RV'\\FboF[u7$$\"3Y'*****\\@fke FboF^u7$$\"3_ILLL&4Nn'FboFau7$$\"3A*******\\,s`(FboFdu7$$\"3%[mm;zM)>$ )FboFgu7$$\"3M*******pfa<*FboFju7$$\"39HLLeg`!)**FboF]v7$$\"3w****\\#G 2A3\"F0F`v7$$\"3;LLL$)G[k6F0Fcv7$$\"3#)****\\7yh]7F0Ffv7$$\"3xmmm')fdL 8F0Fiv7$$\"3bmmm,FT=9F0F\\w7$$\"3FLL$e#pa-:F0F_w7$$\"3!*******Rv&)z:F0 Fbw7$$\"3ILLLGUYo;F0Few7$$\"3_mmm1^rZF0F^x7$F+F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fhx-F$6$7hn7$ F($!\"\"F*7$F.F]y7$F4F]y7$F9F]y7$F>F]y7$FCF]y7$FHF]y7$FMF]y7$FRF]y7$FW F]y7$FfnF]y7$F[oF]y7$F`oF]y7$FfoF]y7$F[pF]y7$F`pF]y7$FepF]y7$FjpF]y7$F _qF]y7$FdqF]y7$FiqF]y7$F^rF]y7$FcrF]y7$FhrF]y7$F^sF]y7$FcsF]y7$$\"381i T&Q.d\"y!#@$\"\"\"F*7$$\"3`6mT5!*\\PNFesF\\[l7$$\"3W-;H#oFMH'FesF\\[l7 $$\"3O$fmTNc$\\!*FesF\\[l7$$\"3_d;zp87c9FjrF\\[l7$$\"3qbm;/rI2?FjrF\\[ l7$$\"31_m\"Hdy'4JFjrF\\[l7$FisF\\[l7$$\"3:Tm;zHz;kFjrF\\[l7$F\\tF\\[l 7$$\"3mILLL1+Y7FboF\\[l7$F_tF\\[l7$FbtF\\[l7$FetF\\[l7$FhtF\\[l7$F[uF \\[l7$F^uF\\[l7$FauF\\[l7$FduF\\[l7$FguF\\[l7$FjuF\\[l7$F]vF\\[l7$F`vF \\[l7$FcvF\\[l7$FfvF\\[l7$FivF\\[l7$F\\wF\\[l7$F_wF\\[l7$FbwF\\[l7$Few F\\[l7$FhwF\\[l7$F[xF\\[l7$F^xF\\[l7$F+F\\[l-Fbx6&FdxFhxFhxFex-%+AXESL ABELSG6$Q\"x6\"Q!Fe^l-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 46.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 15 "Exercise 1.2.5:" }}{PARA 0 "" 0 "" {TEXT -1 116 "Explore other functions with cusps, e.g., |cos(x)|, and \+ explore the behavior of the derivative function at the cusp." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 18 "Improp er integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 511 "Many integrals in physics involve an infinite range, e.g., we \+ could have an expression for the radiated power of a light source at a given wavelength (frequency). Suppose we want to know the entire radi ated intensity. This requires an integral over all wavelengths (or fre quencies). Problems with the integrand at the endpoints lead to potent ially infinite results. Modern physics (quantum mechanics) was discove red as a result of the demand to avoid problems with a classical calcu lation (Planck's hypothesis)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(x^2*exp(-x),x=0..infinit y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(x^2*exp(-x),x=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 905 307 307 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"!F)F(7$$\"3 GLLL3x&)*3\"!#=$\"31bx7@C9l5!#>7$$\"3emmm;arz@F-$\"3ZdG1'zO1#QF07$$\"3 v***\\7y%*z7$F-$\"3[$\\H$44CcrF07$$\"3[LL$e9ui2%F-$\"3#p+X!)QT`5\"F-7$ $\"3z***\\(oMrU^F-$\"3G!yQFn\"R\"e\"F-7$$\"3nmmm\"z_\"4iF-$\"3;7P\"H\" 42s?F-7$$\"3Unmmm6m#G(F-$\"3qlz%ogV.c#F-7$$\"39ommT&phN)F-$\"3N&3`Fv%o FIF-7$$\"3A,+v=ddC%*F-$\"3GW6$HgU6Y$F-7$$\"3KLLe*=)H\\5!#<$\"3?\"*RZyH hbQF-7$$\"3-++v=JN[6Ffn$\"3)\\Ob\")[=C=%F-7$$\"3smm\"z/3uC\"Ffn$\"3KR= 6z;mpWF-7$$\"3ULLe*ot*\\8Ffn$\"3C!GWq*QfCZF-7$$\"3!****\\7LRDX\"Ffn$\" 3?\\?>&zxl$\\F-7$$\"3%om;zR'ok;Ffn$\"33r*R9hfWC&F-7$$\"3OLL3_(>/x\"Ffn $\"32tog\")HlO`F-7$$\"33++D1J:w=Ffn$\"33 Ffn$\"3gK$[#\\$HnS&F-7$$\"3+n;HdG\"\\)>Ffn$\"3>'))yvs,JT&F-7$$\"37+D\" Gt#HR?Ffn$\"3no@JD#\\8T&F-7$$\"3oLLL3En$4#Ffn$\"3&zThQt1>S&F-7$$\"3_++ Dc#o%*=#Ffn$\"3W9\"Q]\\')yO&F-7$$\"3#pmmT!RE&G#Ffn$\"3R4EI'zxOJ&F-7$$ \"3D+++D.&4]#Ffn$\"3I\"**GGnO$H^F-7$$\"3;+++vB_ro([F-7$ $\"33+++v'Hi#HFfn$\"37d6>))Qc*e%F-7$$\"3&om;z*ev:JFfn$\"3uq4T*=\")\\I% F-7$$\"3_LLL347TLFfn$\"3W8xBJuU^RF-7$$\"3nLLLLY.KNFfn$\"3GlVe&>O%[OF-7 $$\"33++D\"o7Tv$Ffn$\"3\"GfGd\"y$3I$F-7$$\"3?LLL$Q*o]RFfn$\"3*fdBr]#>. IF-7$$\"3m++D\"=lj;%Ffn$\"3=?a\"RCj?p#F-7$$\"3S++vV&RF-7$$\"3emm;/T1&*\\Ffn$\"3c`:Hn6[*o\"F-7$$\"3=nm\"zRQb@&Ffn$\"3iYw [g-Zx9F-7$$\"3:++v=>Y2aFfn$\"3W)3q=Ef3J\"F-7$$\"3Znm;zXu9cFfn$\"3ofA$e 6-([6F-7$$\"34+++]y))GeFfn$\"3#)\\q[@*)\\$***F07$$\"3H++]i_QQgFfn$\"3k =8*[F'p(p)F07$$\"3b++D\"y%3TiFfn$\"3%zyo)\\]n'e(F07$$\"3+++]P![hY'Ffn$ \"3ha#>ouxC]'F07$$\"3iKLL$Qx$omFfn$\"3=\\')yg1Q\\cF07$$\"3Y+++v.I%)oFf n$\"3'RTj&pU$=&[F07$$\"3?mm\"zpe*zqFfn$\"3uMa_+3j>UF07$$\"3;,++D\\'QH( Ffn$\"3ed\"G'RV,;OF07$$\"3%HL$e9S8&\\(Ffn$\"3DD%[Fr>A7$F07$$\"3s++D1#= bq(Ffn$\"3M^@Jeh'Qn#F07$$\"3\"HLL$3s?6zFfn$\"3-L2`*=?XH#F07$$\"3a***\\ 7`Wl7)Ffn$\"3;%>'32N3_>F07$$\"3enmmm*RRL)Ffn$\"3Etj]eIYo;F07$$\"3%zmmT vJga)Ffn$\"3QMa3)[q\">9F07$$\"3]MLe9tOc()Ffn$\"3y'*fyXUF27F07$$\"31,++ ]Qk\\*)Ffn$\"3!ROUl`9&R5F07$$\"3![LL3dg6<*Ffn$\"3!\\eOw\\,ru)!#?7$$\"3 %ymmmw(Gp$*Ffn$\"3g%Q$GW!*H)[(F_]l7$$\"3C++D\"oK0e*Ffn$\"3V4Ed!*ozQjF_ ]l7$$\"35,+v=5s#y*Ffn$\"3kkJ\"p^,$*R&F_]l7$$\"#5F)$\"3Z%[[i(H**RXF_]l- %'COLOURG6&%$RGBG$Fa^l!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F^_l-%%VIEWG6$ ;F(F`^l%(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 "" {TEXT -1 40 "What a bout the following generalization:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(x^2*exp(-a*x),x=0..infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&limitG6$,$*&,**()%\"aG\"\"#\"\"\")%\"xGF,F--%$expG 6#,$*&F+F-F/F-!\"\"F-F-**F,F-F+F-F/F-F0F-F-*&F,F-F0F-F-F,F5F-F+!\"$F5/ F/%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Why can't Maple \+ do the limit?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "int(x^2*ex p(-a*x),x=0..infinity) assuming a>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&\"\"#\"\"\"%\"aG!\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 15 " Exercise 1.2.6:" }}{PARA 0 "" 0 "" {TEXT -1 35 "Choose bell-shaped cur ves [such as " }{TEXT 19 11 "1/(a^2+x^2)" }{TEXT -1 94 "] and find imp roper integrals over the entire x-axis. Graph the integrand for some c hoices of " }{TEXT 19 1 "a" }{TEXT -1 63 " and convince yourself why t he area under the curve depends on " }{TEXT 19 1 "a" }{TEXT -1 33 " as given by the symbolic answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 261 19 "1.3 Linear Algebra:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 216 "Step 1: call the \+ LinearAlgebra package. Before calling it we reset the Maple session to clear all variables. Call the LinearAlgebra package with a semicolon \+ terminator so that the names of defined procedures come up." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart; with(LinearAlgebra) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "The two most important obje cts defined in the package are Matrix and Vector. We define variables \+ by entering content using lists and lists of lists:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "V:=Vector([1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'RTABLEG6%\")gIZO-%'MATRIXG6#7%7#\"\"\"7#\"\"#7 #\"\"$&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A:=Matrix([[a11,a12,a13],[a21,a22,a23],[a31,a32,a33]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")O,kO-%'MATRIXG6#7%7%%$a 11G%$a12G%$a13G7%%$a21G%$a22G%$a23G7%%$a31G%$a32G%$a33G%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "a11:=1: a12:=5: a13:=7: a21: =0: a22:=1: a23:=2: a31:=9: a32:=6: a33:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "X:=LinearSolve(A,V );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'RTABLEG6%\")'\\Dn$-%'MA TRIXG6#7%7##\"\"*\"\"#7#!\"(F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A.X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\" )!=%*o$-%'MATRIXG6#7%7#,(*(\"\"*\"\"\"\"\"#!\"\"%$a11GF/F/*&\"\"(F/%$a 12GF/F1*(F.F/F0F1%$a13GF/F/7#,(*(F.F/F0F1%$a21GF/F/*&F4F/%$a22GF/F1*(F .F/F0F1%$a23GF/F/7#,(*(F.F/F0F1%$a31GF/F/*&F4F/%$a32GF/F1*(F.F/F0F1%$a 33GF/F/&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")O,kO-%'MATRI XG6#7%7%\"\"\"\"\"&\"\"(7%\"\"!F,\"\"#7%\"\"*\"\"'F,%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(A.X - V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG6#7%,**(\"\"*\"\"\"\"\"#!\"\"%$a11GF* F**&\"\"(F*%$a12GF*F,*(F)F*F+F,%$a13GF*F*F*F,,**(F)F*F+F,%$a21GF*F**&F /F*%$a22GF*F,*(F)F*F+F,%$a23GF*F*F+F,,**(F)F*F+F,%$a31GF*F**&F/F*%$a32 GF*F,*(F)F*F+F,%$a33GF*F*\"\"$F,Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalm(A &* X - V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'vectorG6#7%\"\"!F'F'Q(pprint16\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "We see that Maple9 is not free of problems (at the level of version 9.01). The above two lines should work in the same fashion ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 15 "Exe rcise 1.3.1:" }}{PARA 0 "" 0 "" {TEXT -1 204 "Set up your own inhomoge neous systems of equations of dimensionalities 2,3,4, etc.. Check that the coefficient matrix is non-singular by calculating the determinant , then find the solution, and verify it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "When one needs to solve many syst ems with the same coefficient matrix, but different right-hand sides, \+ it may be opportune to calculate the inverse of the matrix:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ai:=MatrixInverse(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AiG-%'RTABLEG6%\")3&4t$-%'MATRIXG6# 7%7%#!#6\"#;#\"#PF0#\"\"$F07%#\"\"*\"\")#!#JF8#!\"\"F87%#!\"*F0#\"#RF0 #\"\"\"F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Ai . A ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")[_EO-%'MATRIXG6#7% 7%,(*(\"#6\"\"\"\"#;!\"\"%$a11GF/F1*(\"#PF/F0F1%$a21GF/F/*(\"\"$F/F0F1 %$a31GF/F/,(*(F.F/F0F1%$a12GF/F1*(F4F/F0F1%$a22GF/F/*(F7F/F0F1%$a32GF/ F/,(*(F.F/F0F1%$a13GF/F1*(F4F/F0F1%$a23GF/F/*(F7F/F0F1%$a33GF/F/7%,(*( \"\"*F/\"\")F1F2F/F/*(\"#JF/FKF1F5F/F1*&FKF1F8F/F1,(*(FJF/FKF1F;F/F/*( FMF/FKF1F=F/F1*&FKF1F?F/F1,(*(FJF/FKF1FBF/F/*(FMF/FKF1FDF/F1*&FKF1FFF/ F17%,(*(FJF/F0F1F2F/F1*(\"#RF/F0F1F5F/F/*&F0F1F8F/F/,(*(FJF/F0F1F;F/F1 *(FenF/F0F1F=F/F/*&F0F1F?F/F/,(*(FJF/F0F1FBF/F1*(FenF/F0F1FDF/F/*&F0F1 FFF/F/%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "yikes, not aga in!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(Ai &* A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7 %F)F)F(Q(pprint26\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Let us ex plore a little the depths of symbolic computation scoping rules. The m atrix A was defined from symbolic entries, and then the entries were a ssigned to be numbers. We can unassign them and watch what happens." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "a11:='a11': a12:='a12': a 13:='a13': a21:='a21': a22:='a22': a23:='a23': a31:='a31': a32:='a32': a33:='a33':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")O,kO-%'MATRIXG6#7%7%%$a11G%$a 12G%$a13G7%%$a21G%$a22G%$a23G7%%$a31G%$a32G%$a33G%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "We are able to turn back to the original general result. Not so for the matrix inverse, because it was compute d based on the knowledge of the matrix entries:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "Ai;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABL EG6%\")3&4t$-%'MATRIXG6#7%7%#!#6\"#;#\"#PF.#\"\"$F.7%#\"\"*\"\")#!#JF6 #!\"\"F67%#!\"*F.#\"#RF.#\"\"\"F.%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The matrix inverse can be calculated symbolically:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MatrixInverse(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?![u$-%'MATRIXG6#7%7%*&,&*&%$ a22G\"\"\"%$a33GF0F0*&%$a23GF0%$a32GF0!\"\"F0,.*(%$a31GF0%$a12GF0F3F0F 0*(F8F0%$a13GF0F/F0F5*(%$a21GF0F9F0F1F0F5*(F=F0F;F0F4F0F0*(%$a11GF0F/F 0F1F0F0*(F@F0F3F0F4F0F5F5,$*&,&*&F9F0F1F0F0*&F;F0F4F0F5F0F6F5F5*&,&*&F 9F0F3F0F0*&F;F0F/F0F5F0F6F57%,$*&,&*&F8F0F3F0F5*&F=F0F1F0F0F0F6F5F5*&, &*&F8F0F;F0F5*&F@F0F1F0F0F0F6F5,$*&,&*&F=F0F;F0F5*&F@F0F3F0F0F0F6F5F57 %*&,&*&F8F0F/F0F5*&F=F0F4F0F0F0F6F5,$*&,&*&F8F0F9F0F5*&F@F0F4F0F0F0F6F 5F5*&,&*&F=F0F9F0F5*&F@F0F/F0F0F0F6F5%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "An important message:" }}{PARA 0 "" 0 "" {TEXT -1 162 "The symbolic matrix inverse requires a substantial amount of stor age, which does not arise when the entries are numbers. The number of \+ terms grows to the tune of " }{TEXT 19 2 "n!" }{TEXT -1 22 " for an n- by-n matrix." }}{PARA 0 "" 0 "" {TEXT -1 171 "Suppose that we want to \+ try out how far Maple can go - it would be clumsy to define the matrix by hand. We will learn a sequence command that allows us to automate \+ things:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq(i^2,i=1..10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"%\"\"*\"#;\"#D\"#O\"#\\ \"#k\"#\")\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "To assemble a matrix with symbolic entries we also need to know how to concatenate \+ symbols:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a||m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#a8G" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Let us pick the matrix size:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "[seq(a1||j,j=1..N)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'%$a11G%$ a12G%$a13G%$a14G%$a15G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "That is neat. Now let us do the same with the other index and put it all toge ther:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=Matrix([seq([se q(a||i||j,j=1..N)],i=1..N)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A G-%'RTABLEG6%\")kJ]P-%'MATRIXG6#7'7'%$a11G%$a12G%$a13G%$a14G%$a15G7'%$ a21G%$a22G%$a23G%$a24G%$a25G7'%$a31G%$a32G%$a33G%$a34G%$a35G7'%$a41G%$ a42G%$a43G%$a44G%$a45G7'%$a51G%$a52G%$a53G%$a54G%$a55G%'MatrixG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Wow! The line below writes screens full when terminated with a semicolon:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ai:=MatrixInverse(A):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "You can exhaust your computational resources easily by in creasing " }{TEXT 19 1 "N" }{TEXT -1 218 ". While the numerical invers ion of a non-singular 20-by-20 matrix is trivial (within Maple or othe rwise), the symbolic inversion of a general 9-by-9 matrix will make ev en the most recent computers chew up their memory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 246 "Inhomogeneous systems of linear equations are relatively straightforward to deal with. Let us \+ proceed with homogeneous systems of equations that arise most often in the eigenvalue-eigenvector problem context. Let us go back to a numer ical matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=Matrix([ [1,2,3],[1,4,9],[0,1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-% 'RTABLEG6%\")?UJ8-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%F.\"\"%\"\"*7%\"\"! F.F5%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinan t(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "V:=Vector([1,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'RTABLEG6%\")+l08-%'MATRIXG6#7%7#\"\"\"F-F-&%'V ectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "We chose a \+ vector which points along the diagonal in Euclidean 3-space R3. When w e multiply this vector with A, we obtain a new vector with different l ength (Euclidean vector norm), and which points in a different directi on:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "V1:=A . V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G-%'RTABLEG6%\")+:YO-%'MATRIXG6#7%7#\" \"'7#\"#97#\"\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "VectorNorm(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VectorNorm(V1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "This is clearly the maximum norm, and not the Euclidean norm!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "VectorNorm(V,Euclidean),V ectorNorm(V1,Euclidean);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$\"\"$#\" \"\"\"\"#*$\"$L#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 302 "This is cl umsy. Let us define our own procedure to compute the norm. First we ne ed a way to measure the number of components in our vectors, then we u se a simple add function that works in analogy to the sequence constru ct. Finally we need to know that procedures always return the last com puted entry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For lists one can use the " }{TEXT 19 4 "nops" }{TEXT -1 85 " c ommand to find out how many entries there are. However, this does not \+ work for the " }{TEXT 19 6 "Vector" }{TEXT -1 28 " constructs. We need to use " }{TEXT 19 9 "Dimension" }{TEXT -1 10 " from the " }{TEXT 19 13 "LinearAlgebra" }{TEXT -1 9 " package." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "nops([1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nops(Vector([1,2, 3,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(V1),seq(op(i,V1),i=1..nops(V));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"$F#<%/6#\"\"\"\"\"'/6#\"\"#\"#9/6# F#F'/%)datatypeG%)anythingG/%(storageG%,rectangularG/%&orderG%.Fortran _orderG/%&shapeG7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Dime nsion(V1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Dimension(Vector([1,2,3,4,5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "VNorm:=proc(W::Vector): sqrt(add(W[i]^2,i=1..Dimension(W))); e nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "VNorm(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "VNorm([1,2,3]);" }}{PARA 8 "" 1 "" {TEXT -1 102 "Error, invalid input: VNorm expects its 1st argument, W, to be of typ e Vector, but received [1, 2, 3]\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "This is the result of our restriction for the dummy argument " }{TEXT 19 1 "W" }{TEXT -1 15 " to be of type " }{TEXT 19 6 "Vector" } {TEXT -1 26 ", it won't work on a list." }}{PARA 0 "" 0 "" {TEXT -1 111 "Interestingly: we have been able to use loop constructs without d efining loops. This is accomplished using the " }{TEXT 19 13 "seq, add , mul" }{TEXT -1 48 " functions. We will get to explicit loops later. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now t he problem of eigenvectors: for each N-by-N matrix " }{TEXT 19 1 "A" } {TEXT -1 21 " we can find at most " }{TEXT 19 1 "N" }{TEXT -1 52 " dis tinct vector directions such that the result of " }{TEXT 19 3 "A.V" } {TEXT -1 37 " will point in the same direction as " }{TEXT 19 1 "V" } {TEXT -1 42 ". Such vectors are called eigenvectors of " }{TEXT 19 1 " A" }{TEXT -1 39 ", and the problem is how to find them. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The problem is usu ally formulated as" }}{PARA 0 "" 0 "" {TEXT 19 14 "A.V = lambda V" } {TEXT -1 12 " , where " }{TEXT 19 6 "lambda" }{TEXT -1 75 " is an e lement of the number field; it is a placeholder for a scale factor." } }{PARA 0 "" 0 "" {TEXT -1 211 "Read this equation carefully: it says t hat for a given matrix A we are asking to find vectors V and scale fac tors lambda, such that the action of A onto these vectors corresponds \+ to a stretch with factor lambda." }}{PARA 0 "" 0 "" {TEXT -1 122 "We c an manipulate the right-hand side of the equation by squeezing in an i ndentity matrix, and moving it over to the left:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C:=IdentityMatrix(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\")Si%o$-%'MATRIXG6#7%7%\"\"\"\"\"!F/ 7%F/F.F/7%F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "B:=A-lambda*C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG 6%\")k5nN-%'MATRIXG6#7%7%,&\"\"\"F/%'lambdaG!\"\"\"\"#\"\"$7%F/,&\"\"% F/F0F1\"\"*7%\"\"!F/,$F0F1%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Now we have the problem" }}{PARA 0 "" 0 "" {TEXT 19 7 "B.V = 0 " }{TEXT -1 36 " (where 0 stands for a zero-vector)" }}{PARA 0 "" 0 " " {TEXT -1 76 "and we are left with the problem to solve a homogeneous system of equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We use the placeholder " }{TEXT 19 6 "lambda" }{TEXT -1 84 " to our advantage. We recall the result of an important theorem from linear algebra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "A system of equations has a unique solution if the de terminant of the coefficient matrix does not vanish, i.e., if the coef ficient matrix is non-singular." }}{PARA 0 "" 0 "" {TEXT -1 157 "In th e context of the homogeneous system of equations this means that the o nly solution acceptable in this case is the trivial solution where all components " }{TEXT 19 5 "V_i=0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Interesting solutions can be found only if the determinant of the coefficient matrix " }{TEXT 19 1 "B" }{TEXT -1 22 " vanishes, i.e., when " }{TEXT 19 1 "B" }{TEXT -1 53 " is singular. We use the freedom of the undetermined " }{TEXT 19 6 "lambda" }{TEXT -1 28 " to gurantee such solutions," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cp:=Determinant(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG,**&\"\"(\"\"\"%'lambdaGF(F(*&\"\"&F()F)\"\" #F(F(\"\"'!\"\"*$)F)\"\"$F(F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " The values of " }{TEXT 19 6 "lambda" }{TEXT -1 41 " that allow interes ting solution vectors " }{TEXT 19 1 "V" }{TEXT -1 79 " are obtained fr om the roots of this polynomial (called the characteristic p.)." }} {PARA 0 "" 0 "" {TEXT -1 30 "An analysis of the cubic (for " }{TEXT 19 3 "N=3" }{TEXT -1 163 ") shows that there can be three real roots, \+ or one real root and a complex conjugate pair. A limiting case is when the minimum in the parabola touches down on the " }{TEXT 19 6 "lambda " }{TEXT -1 44 "-axis, and the pair becomes a repeated root." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(cp,lambda=-4..7);" }} {PARA 13 "" 1 "" {GLPLOT2D 1045 417 417 {PLOTDATA 2 "6%-%'CURVESG6$7U7 $$!\"%\"\"!$\"$5\"F*7$$!33LL3_c6!)Q!#<$\"3f#))p`+\"!#:7$$!3imm;/8Bg PF0$\"3'*pye.lAa\"*!#;7$$!3#)*\\iSd?fl$F0$\"3/c$*pJ$F0$\"3.XVx=)H)GiF97$$!3gmmT]8# 33$F0$\"3`\"yyAZ/L\"\\F97$$!3EL$e9*>xXGF0$\"3w(o>j*QzhPF97$$!3em;HZ6&y i#F0$\"37\"HObM%*z#GF97$$!34+]iNn?-CF0$\"3I&H4=jq**)>F97$$!3ym;Hi\\%)o @F0$\"3#)f#eGC`RD\"F97$$!35+]7$eJi$>F0$\"3AU>V!zg-X'F07$$!3GLL$38gpp\" F0$\"3;aMpY*QjS\"F07$$!3ammT0(4i[\"F0$!3ok0]JGgw?F07$$!3y****\\UY&*[7F 0$!3CTf\"*=],&*\\F07$$!3S++](QD2,\"F0$!3DY-sphsMpF07$$!3%3++]dt9\"y!#= $!3n;+*HHE/%zF07$$!3Uqm\"H7&oEdFhp$!3sO*=kEF6=)F07$$!3IMLL3+nZKFhp$!3u JTPNmu6xF07$$!3#eLLL.>w9\"Fhp$!3\"yd!*HSqft'F07$$\"3!y**\\P\\R_H\"Fhp$ !3py`dRIi6]F07$$\"3edmm;KedMFhp$!38_$[;m#GBIF07$$\"3:++v$*p,IeFhp$!3i* of!o@!p<%Fhp7$$\"3_.+D\")\\8*3)Fhp$\"3A@0Y=Wz/CF07$$\"3'om;/wGY/\"F0$ \"3,%3s+@(pGcF07$$\"3%pmTN&*)3h7F0$\"3![H(*oY#yt()F07$$\"3yKLe90d%\\\" F0$\"3;E4rC:AH7F97$$\"3mK$3xB#4PF0$\"3-$R7'=y0A>F97$$\"3ULL3P!>i<#F0$\"3)\\))>^-w1E#F97$$\"3&*)****\\ jwg&*GF97$$\"3O* *\\PfK>lGF0$\"3k7T\"))4p\"eJF97$$\"3Z***\\7%Gw7JF0$\"3`m>yuK`2MF97$$\" 3*emm;7:_L$F0$\"3%p[)yl1]'e$F97$$\"3Y****\\7/tsNF0$\"3/Hg;_ctAPF97$$\" 3%GL3xcazy$F0$\"3_&4%)Ho$o!z$F97$$\"3$4++vT^K-%F0$\"3io7Kh^H(z$F97$$\" 3il;/;ukWUF0$\"38cb>NVR\\F0$\"3Q4:KTvw0IF97$$\" 3;MLLjRLn^F0$\"3u$o_YmH.d#F97$$\"38LLeH\\j+aF0$\"3u$*e#\\D=>,#F97$$\"3 rm;/YS+KcF0$\"3]9IG@&>xL\"F97$$\"3\"3++]B3Y%eF0$\"3E&*HqR(341'F07$$\"3 omm\"ziw#)3'F0$!3+Y()*GeN8s$F07$$\"3/LLLVl@1jF0$!3Gm?G>$e-Q\"F97$$\"3P **\\P\\feQlF0$!3YU[D(oA4g#F97$$\"3o+]i?J*4w'F0$!3V*H1gR&)p\"RF97$$\"\" (F*$!#bF*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F\\\\l-%+AXESLABELSG6$Q'lamb da6\"Q!Fa\\l-%%VIEWG6$;F(Fa[l%(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 "" {TEXT -1 90 "In our example we have three real roots, but note that th e entries in the original matrix " }{TEXT 19 1 "A" }{TEXT -1 63 " coul d have been shifted a little to push the minimum onto the " }{TEXT 19 6 "lambda" }{TEXT -1 17 "-axis, and above." }}{PARA 0 "" 0 "" {TEXT -1 76 "The roots are called the eigenvalues. For each eigenvalue a sol ution vector " }{TEXT 19 1 "V" }{TEXT -1 74 " can be found, and it is \+ undetermined in its length. First find the roots:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "Evals:=solve(cp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&EvalsG6%\"\"',&*&\"\"#!\"\"\"\"&#\"\"\"F)F-#F-F)F*,& *&F)F*F+F,F*#F-F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval f(Evals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"'\"\"!$\"+!))R.='!#5 $!+))R.=;!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalue s(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")s " 0 "" {MPLTEXT 1 0 28 "B1:=eval(B,lambda=Evals[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1G-%'RTABLEG6%\")WMs?-%'MATRIXG6#7%7%!\"&\"\"#\"\"$ 7%\"\"\"!\"#\"\"*7%\"\"!F2!\"'%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "V1:=LinearSolve(B1,Vector([0,0,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G-%'RTABLEG6%\")k.z?-%'MATRIXG6#7%7#,$*&\"\"$ \"\"\"&%$_t0G6#F0F1F17#,$*&\"\"'F1F2F1F17#F2&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "The eigenvector associated with t he first eigenvalue is determined up to an arbitrary scale parameter _ t0[3]. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V1:=simplify(1/V Norm(V1) * V1) assuming _t0[3]>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#V1G-%'RTABLEG6%\"')[Q&-%'MATRIXG6#7%7#,$*(\"\"$\"\"\"\"#Y!\"\"F2#F1 \"\"#F17#,$*(F0F1\"#BF3F2F4F17#,$*&F2F3F2F4F1&%'VectorG6#%'columnG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "VNorm(V1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "We have eliminated the freedom by normalizing the eigenvector to unit Eu clidean norm, i.e., unit length in real 3-space." }}{PARA 0 "" 0 "" {TEXT -1 37 "Now calculate the second eigenvector:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "B2:=eval(B,lambda=Evals[2]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#B2G-%'RTABLEG6%\"(?\"[7-%'MATRIXG6#7%7%,&#\" \"$\"\"#\"\"\"*&F1!\"\"\"\"&#F2F1F4F1F07%F2,&#\"\"*F1F2*&F1F4F5F6F4F:7 %\"\"!F2,&*&F1F4F5F6F4F6F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "V2:=LinearSolve(B2,Vector([0,0,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%'RTABLEG6%\"'Gom-%'MATRIXG6#7%7#,&*&#\"\"& \"\"#\"\"\"*&&%$_t0G6#\"\"$F3F1#F3F2F3!\"\"*&#\"#6F2F3F5F3F:7#,&*&F9F3 F4F3F3*&#F3F2F3F5F3F:7#F5&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "V2:=simplify(1/VNorm(V2) * V2) assuming _t0[3] >0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%'RTABLEG6%\"'/\"f'-%'MA TRIXG6#7%7#,$*(\"\"#!\"\",&*&\"\"&\"\"\"F4#F5F0F5\"#6F5F5,&\"#kF5*&\"# FF5F4F6F5#F1F0F17#,$*(F0F1,&*$F4F6F5F5F1F5F8F " 0 "" {MPLTEXT 1 0 15 "evalf([V1,V2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'RTABLEG6%\"'k@l-%'MATRIXG6# 7%7#$\"+%oeKU%!#57#$\"+pt^Y))F/7#$\"+h&>WZ\"F/&%'VectorG6#%'columnG-F% 6%\"'?Nu-F)6#7%7#$!+gyGW**F/7#$\"+!ph " 0 "" {MPLTEXT 1 0 28 "simplify(DotProduct(V1,V2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"##*!\"\"\"#Y#\"\"\"\"\"#,&*&\"\"*F)\"\"&F(F)\"#PF )F),&\"#kF)*&\"#FF)F.F(F)#F&F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+5_:wP! #5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(add(V1[i]*V2[i] ,i=1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+3_:wP!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The two vectors are not collinear, i.e., \+ they are linearly independent. They are not orthogonal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 15 "Exercise 1.3.2:" }} {PARA 0 "" 0 "" {TEXT -1 41 "Find the third eigenvector of the matrix \+ " }{TEXT 19 1 "A" }{TEXT -1 61 " and show that it is linearly independ ent from the other two." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 364 "If the matrix is real, symmetric, and has distinct \+ eigenvalues, then the eigenvectors will turn out to be perpendicular. \+ They are always guaranteed to be linearly independent, and that is the most important feature. One can always apply an orthonormalization pr ocedure (such as Gram-Schmidt) to turn them into an orthonormal basis \+ for the vector space in question." }}{PARA 0 "" 0 "" {TEXT -1 135 "Thi s latter property makes the eigenvector-eigenvalue problem an importan t cornerstone for function space as used in quantum mechanics." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "The prob lem is so ubiquitous that we need a shortcut to obtain the complete so lution. In symbolic mode the eigenvectors are returned normalized with the maximum norm:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Eigen vectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"'ozt-%'MA TRIXG6#7%7#\"\"'7#,&*&\"\"#!\"\"\"\"&#\"\"\"F0F4#F4F0F17#,&*&F0F1F2F3F 1#F4F0F1&%'VectorG6#%'columnG-F$6%\"'!=&**-F(6#7%7%\"\"$*(,&#FEF0F4*&F 0F1F2F3F4F4F.F1,&#FEF0F1*&F0F1F2F3F4F1*(,&FHF4*&F0F1F2F3F1F4F7F1,&#FEF 0F1*&F0F1F2F3F1F17%F,,$*&FJF4F.F1F1,$*&FPF4F7F1F17%F4F4F4%'MatrixG" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "If we evaluate the matrix to floa ting-point entries first, the " }{TEXT 19 11 "Eigenvector" }{TEXT -1 118 " function applies a numerical diagonalization procedure which ret urns eigenvectors normalized with the Euclidean norm." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Evecs:=Eigenvectors(evalf(A));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&EvecsG6$-%'RTABLEG6%\"';Hq-%'MATRIX G6#7%7#^$$\"\"'\"\"!$F2F27#^$$\"3!o%*)\\())R.='!#=F37#^$$!3o%*)\\())R. =;!#@^'yGW**F8F3^$$\"3G+0C')o>NZ!#>F37%^$$\"39GQHpt^Y))F8F3^$$\" 3i#\\<9phWZ\"F8F3^$$\"3yqbEu (en'*)FRF3^$$\"3-_?b=QT^_F8F3%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The appearance of the placeholder for imaginary parts ap pears annoying. Let us experiment a little with complex numbers, befor e we proceed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eq:=z^2+1= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,&*$)%\"zG\"\"#\"\"\"F+F+ F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(eq,z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The imaginary unit is displayed and entered with the upper-case i = " }{TEXT 19 1 "I" }{TEXT -1 68 ", unlike in textbook m ath notation. This is done so that we can use " }{TEXT 19 1 "i" } {TEXT -1 39 " also for an index, or other occasions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "z1:=2*a+3*b*I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G,&*&\"\"#\"\"\"%\"aGF(F(*&^#\"\"$F(%\"bGF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Re(z1),Im(z1),conjugate(z1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%#ReG6#,&*&\"\"#\"\"\"%\"aGF)F)*& ^#\"\"$F)%\"bGF)F)-%#ImGF%-%*conjugateGF%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "evalc([%]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$ *&\"\"#\"\"\"%\"aGF'F',$*&\"\"$F'%\"bGF'F',&*&F&F'F(F'F'*&^#!\"$F'F,F' F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 19 5 "ev alc" }{TEXT -1 371 " cannot operate on the sequence of expressions, bu t it can operate on the list formed from these. This is so because acc ording to the syntax, appending objects by commas (a sequence) is tant amount to passing several arguments. Putting the brackets around the e xpression sequence turns it into a single object, a list which is an o rdered set. Also note that the functions " }{TEXT 19 17 "Re, Im, conju gate" }{TEXT -1 35 " do not evaluate without a call to " }{TEXT 19 5 " evalc" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Now let us demonstrate the following. We extract the e igenvectors from " }{TEXT 19 5 "Evecs" }{TEXT -1 64 " in the form of a matrix made up of the eigenvectors as columns." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "Evecs[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\"(/56\"-%'MATRIXG6#7%7%^$$\"3&=\"pk%oeKU%!#=$\"\"!F1^$$!3c) >@^'yGW**F/F0^$$\"3G+0C')o>NZ!#>F07%^$$\"39GQHpt^Y))F/F0^$$\"3i#\\<9ph WZ\"F/F0^$$\"3yqbEu(en'*)F8F 0^$$\"3-_?b=QT^_F/F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "S:=Re(Evecs[2]);" }}{PARA 8 "" 1 "" {TEXT -1 213 "Error, invalid input: simpl/Re expects its 1st a rgument, x, to be of type algebraic, but received Matrix(3, 3, [[...], [...],[...]], datatype = complex[8], storage = rectangular, order = Fo rtran_order, shape = [])\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "yik es, this could have worked; evalf worked in this way after all!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S:=map(Re,Evecs[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'RTABLEG6%\"(K8=\"-%'MATRIXG6# 7%7%$\"3&=\"pk%oeKU%!#=$!3c)>@^'yGW**F0$\"3G+0C')o>NZ!#>7%$\"39GQHpt^Y ))F0$\"3i#\\<9phWZ\"F0$\"3yqbEu(en'* )F5$\"3-_?b=QT^_F0%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "T he matrix is not orthogonal. The columns (or rows) are normalized, but not mutually orthogonal. Therefore, the inverse is NOT obtained by tr ansposing it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Si:=Transp ose(S);" }}{PARA 8 "" 1 "" {TEXT -1 55 "Error, attempting to assign to `Si` which is protected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "OK, there are predefined names that we shouldn't use." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Str:=Transpose(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$StrG-%'RTABLEG6%\"(7rB\"-%'MATRIXG6#7%7%$\"3&=\"pk%o eKU%!#=$\"39GQHpt^Y))F0$\"3kr*[:c>WZ\"F07%$!3c)>@^'yGW**F0$\"3i#\\<9ph $\"3yqbEu(en'*)F:7%$\"3G+0C')o>NZF:$!3C$43u/mp\\)F0$\"3-_?b=QT^_ F0%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "S . Str;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")'R-/#-%'MATRIXG6#7%7%$ \"3gc@J%*Hy'=\"!#<$\"3o@b;rmgfH!#=$\"3gMnRe\"*!#@7%F/$\"3UzY-GTm2:F .$!3k$Ga`-p!3JF17%F2F8$\"39R;jw(Gb0$F1%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "Str . S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' RTABLEG6%\")7_Z?-%'MATRIXG6#7%7%$\"2))***************!#<$!35B=?5_:wP!# =$!3PEw(*Qq7LlF17%F/$\"3A+++++++5F.$!3*)G,[$4;)3Z!#>7%F2F7$\"2y******* ********F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Sinv :=MatrixInverse(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%SinvG-%'RTAB LEG6%\")7`i?-%'MATRIXG6#7%7%$\"3'\\]_^#oAa;!#=$\"3]@DwDT8r#)F0$\"3`.?7 g9QB8!#<7%$!3-Ja&p9()pE*F0$\"3fb&Go6u'RNF0$\"3uT\\*)Rn\"Hc'F07%$\"3`s! okq\")y6\"F0$!3W]^$HI_m#HF0$\"3Ko1#)poi?9F5%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sinv . S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Wjk?-%'MATRIXG6#7%7%$\"\"\"\"\"!$\"3qX9y!yyxQ\"!#M$F .F.7%$!3/PVMUjLjTF1F,$\"3s#yDJ7:6b&F17%F2$!3O\"*GchvbvFF1F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This is as close as we can get \+ to a unit matrix when working to 15 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "S . Sinv;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"'7jy-%'MATRIXG6#7%7%$\" \"\"\"\"!$F.F.F/7%$!3O\"*GchvbvF!#MF,F/7%F/F1F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The point of the exercise is to show that the original matrix " }{TEXT 19 1 "A" }{TEXT -1 119 " is transformed \+ to diagonal form by the transformation matrix made up from the eigenve ctors. The order matters, namely:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "S . A . Sinv;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'R TABLEG6%\"'_.%)-%'MATRIXG6#7%7%$\"3KSLMc&y*y=!#<$\"3!Q*fyz.=ip!#=$!3'Q cur!)yeN\"!#;7%$!3sp]'pU`n\\&F1$\"3mQzBnDH)e#F1$\"3'G6PD06>Z'F.7%$!3MY \"p\\i;T8*F1$\"3k3!*p:m*pZ#F1$\"3#f'G$p=#>iGF.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Sinv . A . S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(C#p7-%'MATRIXG6#7%7%$\"\"'\"\"!$\"338.D \\gW?A!#L$!3K_7+(>%y\")))F17%$!38g#*eoll,9!#K$\"3-\\*)\\())R.='!#=$\"3 ac^iCIA56F17%$!3#[tPp`M`m\"F1$\"3we<.I[A3OF1$!3!\\*)\\())R.=;!#<%'Matr ixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Therefore, the problem of finding the eigenvectors and eigenvalues is called also the diagonali zation problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 20 "The physics message:" }}{PARA 0 "" 0 "" {TEXT -1 19 "Sup pose the matrix " }{TEXT 19 1 "A" }{TEXT -1 140 " represents some phys ics variable in an arbitrary reference frame. This could be the moment of inertia tensor of some 3d object (the matrix " }{TEXT 19 1 "A" } {TEXT -1 94 " would be real and symmetric based on the definitions of \+ the principal and deviation moments)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 461 "Finding the eigenvectors is tantamou nt to finding coordinate axes such that when these axes are chosen the tensor is represented by a very simple matrix, namely the matrix of e igenvalues on the diagonal. In classical mechanics this proves that fo r any shape (a potato!) three axes can be found such that the object c an be spun without wobbling. When one chooses an eigenvector of the mo ments of inertia matrix as rotation axis, one can spin the object smoo thly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 360 "Your car mechanic knows about this in practical terms when balancing \+ the wheels of your car: a small weight is added (of a mass and to a lo cation as specified by a sophisticated machine) to ensure that the axl e direction becomes prinicipal axis for the wheel. With this technique imperfections in the rubber, as well as dents to the rim can be compe nsated for." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 15 "Exercise 1.3.3:" }}{PARA 0 "" 0 "" {TEXT -1 20 "Use simple mat rices " }{TEXT 19 1 "A" }{TEXT -1 277 " that are real and symmetric an d find their eigenvectors and eigenvalues. Work with the symbolic form only if the eigenvalues and eigenvectors are very simple. Verify that the matrix inverse is given by matrix transpose in this case, and ver ify the diagonalization calculation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Quadrat ic forms: (in 2-space, can be extended to 3-space)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=Matrix([[1,2],[2,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"'!ob%-%'MATRIXG6#7$7$\"\"\"\"\"# 7$F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "QF:=Tran spose(Vector([x,y])) . A . Vector([x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#QFG,&*&%\"xG\"\"\",&F'F(*&\"\"#F(%\"yGF(F(F(F(*&F,F( ,&*&F+F(F'F(F(F,F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Questio n: why did we need the transpose on the left?" }}{PARA 0 "" 0 "" {TEXT -1 11 "By default " }{TEXT 19 6 "Vector" }{TEXT -1 27 " generate s a column vector!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We choose a scale for the geometric shape defined by the quadratic form:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq:=simplify(QF)=1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*$)%\"xG\"\"#\"\"\"F+*(\"\"%F +F)F+%\"yGF+F+*$)F.F*F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "To graph the equation one can proceed in various ways. In the important \+ " }{TEXT 19 5 "plots" }{TEXT -1 54 " package there is a convenient fun ction for this case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " implicitplot(eq,x=-2..2,y=-2..2,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 789 482 482 {PLOTDATA 2 "6%-%'CURVESG6[p7$7$$!\"#\"\"!$ \"3!H/8R<_c%R!#=7$$!3/H9dG92\"*>!#<$\"3O(G9dG92\"RF-7$7$$!33++++++S=F1 $\"3ZdG9dG92MF-F.7$F57$$!3#*p2Bp2BWommmmmTAF-7$$!3%[G9dG92c \"F1$\"3e'************R#F-7$FLFA7$FG7$$!3;,++++]\"\\\"F1$\"3$\\+++++]6 #F-7$7$$!3K++++++g8F1$\"3B,+++](=i\"F-FS7$7$FZ$\"3'4++++v=i\"F-7$$!3,o mmmmm@7F1$\"3Qsmmmmm;5F-7$7$$!3Q+++++++7F1$\"3mmr&G9dGR*!#>F\\o7$7$$!3 [++++++S5F1$\"3vQYQ:YQ:@Fgo7$$!33**********\\n6F1$\"3[l************zFg o7$F^pFbo7$7$Fjo$\"3OPYQ:YQ:@Fgo7$$!3@9+++++D\"*F-$!3]E**********\\ZFg o7$7$$!3[/++++++))F-$!3$\\qsssssA'FgoFhp7$7$$!3=/++++++sF-$!3C(******* ***\\c\"F-7$$!3w%**********\\Z)F-$!3)[.++++++)Fgo7$Fiq7$$!3O.++++++))F -$!3i0FFFFFFiFgo7$7$$!3'Q++++++g&F-$!3!Rmmmmmmr#F-7$$!3,_G9dG92gF-$!3C .++++++CF-7$7$F[s$!3_.++++++CF-Fdq7$7$$!3c.++++++SF-$!3!p********\\i:% F-7$FgsFes7$FisFer7$7$F]sF[s7$$!3Mjmmmmm;FF-Ffr7$F]tFds7$7$F]s$\"3%oG9 dG92c\"F17$$!3\\C0@%ot%RGF-$\"3q************z;F17$7$Fft$\"3[********** **z;F17$$!3\\(o2Bp2B/$F-$\"3#zwI#p2BWF17$7$$!3oSI\"R<_c% RF-$\"3M**************>F1Fju7$7$F\\r$!3a#**********\\Z)F-7$FgqFeq7$Fiv 7$FasF[s7$7$F\\r$\"3'3+++++v;\"F17$$!3!y7dG9dGR*Fgo$\"3s*************> \"F17$7$$!3?Hr&G9dGR*FgoFcw7$$!3Rimmmmm;5F-$\"3dlmmmmm@7F17$7$$!3Y)*** ****\\(=i\"F-$\"3m************f8F1Fiw7$7$$!3=)*******\\(=i\"F-$\"3U*** *********f8F17$$!36%**********\\6#F-$\"3])*********\\\"\\\"F17$7$$!3ek mmmmmTAF-$\"3d************>:F1Fjx7$F`y7$FasFct7$7$FapF_p7$$\"3WRYQ:YQ: @FgoFjo7$7$$\"3$3k%Q:YQ:@FgoFjo7$$!3==**********\\ZFgo$!3V;+++++D\"*F- 7$7$$!3Q*psssssA'FgoF_qF`z7$7$$!32+FFFFFFiFgoF_qFfv7$7$Fap$\"3'e+++++] Z)F-7$$\"3&otsssssA'Fgo$\"3!y************z)F-7$Fa[l7$$\"3%z/+++++v%Fgo $\"3z*)*********\\7*F-7$7$$!3=2YQ:YQ:@Fgo$\"3\")************R5F1Fg[l7$ F]\\lF]w7$7$FOFM7$FJFH7$Fe\\l7$FVFT7$7$FfnFZFg\\l7$7$FjnFZ7$$\"3brmmmm m;5F-$!3znmmmmm@7F17$7$$\"3!R;dG9dGR*FgoFcoF\\]l7$7$$\"3]ir&G9dGR*FgoF coFhy7$7$FO$\"3*3'G9dG92gF-7$$\"3S,+++++l:F-$\"3_(************>(F-7$F] ^lF^[l7$7$F+F(7$F2F/7$7$F8F6Fe^l7$Fg^l7$F>F<7$7$FDFBFi^l7$F[_lFd\\l7$7 $$\"3)o*************RF-$\"3c.++++DcTF-7$$\"3Mommmmm;FF-$\"3?(********* ***f&F-7$Fc_lFj]l7$7$Ff_lFd_l7$Fa_lF__l7$F[`lF^_l7$7$F`^lF^^l7$F[^lFO7 $F_`lFj_l7$7$Fd[l$\"3SUFFFFFFiFgo7$$\"3)p+++++]Z)F-Fap7$Fe`lF^`l7$Fb`l 7$$\"3o))*********\\7*F-$\"3Ec+++++]ZFgo7$7$F`\\l$!3c3YQ:YQ:@FgoFj`l7$ 7$FcwFaw7$F^wF\\r7$FealF`al7$7$FcwFgw7$$\"3zlmmmmm@7F1Fjw7$7$FbxF`xFia l7$F]bl7$F]yF[y7$7$Fcy$!3'[mmmmm;C#F-F_bl7$7$F\\uFft7$$\"3g'G9dG92c\"F 1Fas7$7$FctFasFabl7$Febl7$FauF_u7$7$FguFeuF\\cl7$7$Fgu$!3qaG9dG92MF-7$ F]v$!3/zUr&G92\"RF-7$7$FcvFavFccl-%'COLOURG6&%$RGBG\"\"\"F*F*-%(SCALIN GG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "Eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'RTABLEG6%\"(OC4\"-%'MATRIXG6#7$7#\"\"$7#!\"\"&%'VectorG6#%'columnG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "A hyperbola is characterized by having one positive and one negative eigenvalue. Let us perform th e transformation to normal form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "S:=Eigenvectors(A)[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'RTABLEG6%\"'Wc(*-%'MATRIXG6#7$7$!\"\"\"\"\"7$F/F/%'Matr ixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Sinv:=MatrixInverse( S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%SinvG-%'RTABLEG6%\"'k=^-%'MA TRIXG6#7$7$#!\"\"\"\"##\"\"\"F07$F1F1%'MatrixG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "Adiag:=Sinv . A . S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AdiagG-%'RTABLEG6%\"'?`s-%'MATRIXG6#7$7$!\"\"\"\"!7$ F/\"\"$%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "QFdiag: =Transpose(Vector([x,y])) . Adiag . 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When \+ calling the " }{TEXT 19 12 "implicitplot" }{TEXT -1 202 " it may be ne eded to adjust the range for the x and y variables so that the entire \+ shape becomes visible. The interpretation of the eigenvalues (their in verses) should become quite obvious in this case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 27 "1.4 Differential equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "L et us explore a few simple equations: relationships between a function and some of its derivatives define an equation that determines the fu nction. For example: the exponential function is defined as the functi on which is equal to its derivative." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "DE:=diff(y(x),x)=y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/-%%diffG6$-%\"yG6#%\"xGF,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sol:=dsolve(DE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$solG/-%\"yG6#%\"xG*&%$_C1G\"\"\"-%$expGF(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rhs(sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%$ _C1G\"\"\"-%$expG6#%\"xGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "A \+ special solution is obtained by fixing an initial condition. Let us ge neralize slightly: we formulate a problem where the rate of decay is p roportional to the amount of material present. A positive material con stant " }{TEXT 19 1 "a" }{TEXT -1 30 " controls the proportionality." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "DE:=diff(N(t),t)=-a*N(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/-%%diffG6$-%\"NG6#%\"tGF,,$ *&%\"aG\"\"\"F)F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Suppose that at time zero we have 100 atoms." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IC:=N(0)=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I CG/-%\"NG6#\"\"!\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Equatio n and initial condition are placed into a set:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(\{DE,IC\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"NG6#%\"tG,$*&\"$+\"\"\"\"-%$expG6#,$*&%\"aG F-F)F-!\"\"F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The constant \+ " }{TEXT 19 1 "a" }{TEXT -1 116 " has dimensions of inverse time. Note that functions can only be evaluated for dimensionless argument! The \+ constant " }{TEXT 19 6 "t0=1/a" }{TEXT -1 117 " thus has something to \+ do with the so-called half-life. It describes the time scale at which \+ a population shrinks to " }{TEXT 19 3 "1/e" }{TEXT -1 93 " of its init ial value. We simply measure time in units of this scale for the follo wing graph:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(subs(a= 1,rhs(sol)),t=0..3,N=0..100);" }}{PARA 13 "" 1 "" {GLPLOT2D 889 314 314 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"$+\"F)7$$\"3s******\\i9R l!#>$\"33ch&R32qO*!#;7$$\"3/++vVA)GA\"!#=$\"3yZ*)HRG$*[))F27$$\"3+++]P eui=F6$\"3,QeTmkX+$)F27$$\"3A++]i3&o]#F6$\"3)p=T3?uEy(F27$$\"3%)***\\( oX*y9$F6$\"3)GgbZUD%*H(F27$$\"3z***\\P9CAu$F6$\"3eKL*)G\"R#yoF27$$\"3! )***\\P*zhdVF6$\"3?cuF!pfS*\\F6$\"3q3@z[4\"*ogF27$$ \"3$)***\\(=$f%GcF6$\"3'G@(yFv&ep&F27$$\"3Q+++Dy,\"G'F6$\"3iSj;it.O`F2 7$$\"33++]7Bs%F27$ $\"3!)*****\\7nD:)F6$\"3%4Lq*3rDDWF27$$\"3[+++D!*oy()F6$\"3'4@81(zncTF 27$$\"3))***\\PpnsM*F6$\"3!G-&om9$p#RF27$$\"3,++]siL-5!#<$\"3u'QCux4-n $F27$$\"3-+++!R5'f5Fgp$\"33-.x#>3fY$F27$$\"3)***\\P/QBE6Fgp$\"3%[aqsc@ DC$F27$$\"3!******\\\"o?&=\"Fgp$\"31P!49`Ho0$F27$$\"31+]Pa&4*\\7Fgp$\" 3_k]P6rIlGF27$$\"33+]7j=_68Fgp$\"3knv#\\R(4%p#F27$$\"33++vVy!eP\"Fgp$ \"3,\"G$)RBaj_#F27$$\"34+](=WU[V\"Fgp$\"3)*\\o!)R)G:Q#F27$$\"3)****\\7 B>&)\\\"Fgp$\"3qLjj!43YB#F27$$\"3)***\\P>:mk:Fgp$\"3g&HNR(ze\"4#F27$$ \"3'***\\iv&QAi\"Fgp$\"3/3(oO$=cu>F27$$\"31++vtLU%o\"Fgp$\"37b@>\\R^b= F27$$\"3!******\\Nm'[F gp$\"35;4N-JEP9F27$$\"3z*****\\@80+#Fgp$\"3KL8f\\%eEN\"F27$$\"31++]7,H l?Fgp$\"3qR+LV]\"yE\"F27$$\"3()**\\P4w)R7#Fgp$\"3?+F5B%Rb>\"F27$$\"3;+ +]x%f\")=#Fgp$\"3j,#zeBH77\"F27$$\"3!)**\\P/-a[AFgp$\"3Q0ZY#)>`b5F27$$ \"3/+](=Yb;J#Fgp$\"3O!)4CGkq4**Fgp7$$\"3')****\\i@OtBFgp$\"34!3]`m&p;$ *Fgp7$$\"3')**\\PfL'zV#Fgp$\"3JmFKoZ&Qt)Fgp7$$\"3>+++!*>=+DFgp$\"3!>\" fULh+2#)Fgp7$$\"3-++DE&4Qc#Fgp$\"3xXZeB13,xFgp7$$\"3=+]P%>5pi#Fgp$\"3f [hV4::IsFgp7$$\"39+++bJ*[o#Fgp$\"3z71#zM[G#oFgp7$$\"33++Dr\"[8v#Fgp$\" 3M'**fsNtTQ'Fgp7$$\"3++++Ijy5GFgp$\"3o)*['3+nd,'Fgp7$$\"31+]P/)fT(GFgp $\"3wbJe,gNYcFgp7$$\"31+]i0j\"[$HFgp$\"3$\\99<+[SJ&Fgp7$$\"\"$F)$\"3q% R'yOoqy\\Fgp-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q\" NFe[l-%%VIEWG6$;F(Ffz;F(F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 46.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 15 "Exerc ise 1.4.1:" }}{PARA 0 "" 0 "" {TEXT -1 67 "Define a pair of first-orde r decay equations, where one population " }{TEXT 19 5 "N1(t)" }{TEXT -1 26 " decays with a given rate " }{TEXT 19 4 "1/t0" }{TEXT -1 32 ", \+ and feeds a second population " }{TEXT 19 5 "N2(t)" }{TEXT -1 54 " wit h this same rate which itself decays at a rate of " }{TEXT 19 6 "0.5/t 0" }{TEXT -1 108 ". Solve the pair of equations and graph their solut ions. Explore what happens when the daughter population " }{TEXT 19 2 "N2" }{TEXT -1 42 " decays with twice the rate of the parent." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Define the harmonic oscillator equation for a f rictionless oscillator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r estart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "DE:=m*diff(x(t), t$2)=-k*x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/*&%\"mG\"\"\"- %%diffG6$-%\"xG6#%\"tG-%\"$G6$F/\"\"#F(,$*&%\"kGF(F,F(!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "A second-order ordinary different ial equation requires us to specify two initial conditions. These corr espond to specifying intial displacement from equilibrium, as well as \+ intial speed." }}{PARA 0 "" 0 "" {TEXT -1 90 "We choose a displacement of one unit (would be 1 meter in SI units), and no initial speed." }} {PARA 0 "" 0 "" {TEXT -1 61 "The formulation of this initial condition is somewhat tricky:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IC: =x(0)=1,D(x)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"xG6 #\"\"!\"\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(\{DE,IC\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$so lG/-%\"xG6#%\"tG-%$cosG6#*(%\"mG#!\"\"\"\"#%\"kG#\"\"\"F1F)F4" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "We recognize the circular frequenc y " }{TEXT 19 15 "omega=sqrt(k/m)" }{TEXT -1 25 ". The period is given as " }{TEXT 19 12 "T=2*Pi/omega" }{TEXT -1 37 ", and we measure time \+ in units where " }{TEXT 19 3 "T=1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "sol1:=subs(k=1,m=1,rhs(sol));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%sol1G-%$cosG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot([sol1,diff(sol1,t)],t=0..3*Pi,color=[red,b lue],title=\"position(red) and velocity(blue) in a harmonic oscillator \");" }}{PARA 13 "" 1 "" {GLPLOT2D 1082 354 354 {PLOTDATA 2 "6'-%'CURV ESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3_*=rYWLe8&!#>$\"3c:W01X\"o)**!#=7$$\" 3\"zBM*)omr-\"F2$\"3Czxm&z#HZ**F27$$\"3'oN,M.]2a\"F2$\"3M%p%\\\"4R:))* F27$$\"3#eZoyPLV0#F2$\"3i>)[DzE(*y*F27$$\"3?:(z;il![HF2$\"3%)301XOeo&* F27$$\"39b4\\lyzTQF2$\"3'>iBafh5F*F27$$\"3g2t*oi))>&eF2$\"390=[vD-O$)F 27$$\"3AKF`PU]vyF2$\"3/i\"4tmKe0(F27$$\"3c#3#pJCS*))*F2$\"3Q&oZaObd\\& F27$$\"3%f4faQac<\"!#<$\"3e*emH'>R\\QF27$$\"3a&3x^A?BF\"Fhn$\"3$)*[jNG 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approximately g=10 m/sec^2, and the pendulum arm is \+ of 1 m length:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g:=10: l: =1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "DE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F*\"\"#,$*&\"#5\" \"\"-%$sinG6#F'F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Start t he pendulum near the top from rest and keep in mind that the angle is \+ measured in radians:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "IC: =theta(0)=3.13,D(theta)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IC G6$/-%&thetaG6#\"\"!$\"$8$!\"#/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "sol:=dsolve(\{DE,IC\},numeric,output=listproce dure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Theta:=eval(theta (t),sol):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Theta(1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3#f'H_d+Z/I!#<" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "plot(Theta(t),t=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 1182 428 428 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=\"p=(F,$\"+ielHdF07$Fg\\l$\"+[!4y5%F07$$\"+q%*\\%R(F,$\"+!=Y'oHF07$F \\]l$\"+E_&\\;#F07$Fa]l$\"+)zRE4\"F07$Ff]l$\"+:w>5`Fdbl7$F[^l$\"+.mn'* =Fdbl7$F`^l$!+#[ir7&Fjhl7$Fe^l$!+RZY@KFdbl7$Fj^l$!+[3g)Q(Fdbl7$F__l$!+ U9#QT\"F07$Fd_l$!+9ej!)GF07$F^`l$!+VcW(R&F07$Fh`l$!+?O:[5F,7$$\"+]oi\" o*F,$!++N[]9F,7$F]al$!+jG&>&>F,7$$\"+50O\"*)*F,$!+4Gd:FF,7$Fbal$!+niq> OF,-Fgal6&FialF(F(Fjal-%+AXESLABELSG6$Q\"t6\"Q!Febm-%%VIEWG6$;F(Fbal%( DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 15 "Exercise 1.4. 3:" }}{PARA 0 "" 0 "" {TEXT -1 379 "Explore the mathematical pendulum \+ solution for different initial conditions: (a) a smaller intial displa cement: observe how the period of oscillation depends on the intial an gle, except when it is small enough such that the linearized regime is reached when sin(x(t)) can be replaced by x(t); (b) the same initial \+ displacement as above, but with additional initial angular speed." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Note that inaccuracies in the numerical solution may occur for intial angles ra ther close to " }{TEXT 19 2 "Pi" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "Some explanation s are in order to understand how the initial condition on the velocity is specified. We have worked so far only with expressions, and differ entiated them. The subtle point about the initial condition is that fi rst we have to differentiate the map x(t), and then evaluate it at t=0 , the initial point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 277 38 "Maps (functions, procedures) in Maple:" }}{PARA 0 "" 0 "" {TEXT -1 300 "If we would like the convenience of true maps that \+ we enjoy when calling sin(s), exp(t), etc., from built-in functions, w e need to define our own maps. This is done in a way that reminds us o f the formal mathematical definition of a map. The arrow is formed usi ng the hyphen and 'greater than' signs." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x->x^2*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&)9$\"\"#\"\"\"-%$sinG6#F.F0 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"sG\"\"#\"\"\"-%$sinG6#F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG\"\"#\"\"\"-%$sinG6#F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\" \"-%$sinG6#\"\"#F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Maps are \+ differentiated with the " }{TEXT 19 1 "D" }{TEXT -1 10 " operator:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g:=D(f);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*(\"\"#\" \"\"9$F/-%$sinG6#F0F/F/*&)F0F.F/-%$cosGF3F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*(\"\"#\"\"\"%\"sGF&-%$sinG6#F'F&F&*&)F'F%F&-%$cosGF*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(s),s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#\"\"\"%\"sGF&-%$sinG6#F'F&F&*&)F'F%F&-%$cosGF* F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The conversion of a map i nto an expression has been done above by calling the map with argument " }{TEXT 19 1 "x" }{TEXT -1 42 " or some other independent variable n ame. " }{TEXT 19 10 "f(x), g(x)" }{TEXT -1 28 " have become expression s in " }{TEXT 19 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 " To convert the other way around one uses the " }{TEXT 19 7 "unapply" } {TEXT -1 24 " procedure. For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h_expr:=diff(f(t),t$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'h_exprG,(*&\"\"#\"\"\"-%$sinG6#%\"tGF(F(*(\"\"%F(F,F(-%$cosGF +F(F(*&)F,F'F(F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " h:=unapply(h_expr,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\" tG6\"6$%)operatorG%&arrowGF(,(*&\"\"#\"\"\"-%$sinG6#9$F/F/*(\"\"%F/F3F /-%$cosGF2F/F/*&)F3F.F/F0F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Maps can be graphed directly, but one is not allowed to e nter a name for the independent variable." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "plot([f,g,h],-2..3,color=[red,blue,green]);" }} {PARA 13 "" 1 "" {GLPLOT2D 891 397 397 {PLOTDATA 2 "6'-%'CURVESG6$7Y7$ $!\"#\"\"!$!3%os-tq*=PO!#<7$$!3smm;HU,\"*=F-$!39X$[nyeTR$F-7$$!3SL$3FH '='z\"F-$!3TW2SgGoWJF-7$$!3gmmTgBa*o\"F-$!3Bz$3!Q:XMGF-7$$!3amm\"H_\"> #e\"F-$!3E&oFw[nJ]#F-7$$!3ML$3_!4Nv9F-$!3C*eSoJ`n;#F-7$$!3km;/wfHw8F-$ !3]iD*=Wu%e=F-7$$!3;+]PM.tt7F-$!3SBJ^5uK^:F-7$$!3em;/,oln6F-$!3)*\\&)G /@7a7F-7$$!3%)**\\(oWB>1\"F-$!3Qus\\\"=wz%)*!#=7$$!3eJLLepjJ&*Fen$!3qT XuWJt1uFen7$$!3Ulm;z/ot&)Fen$!3E*4AR#p5ebFen7$$!3u)****\\P[_\\(Fen$!3t dGY@wSFQFen7$$!3A*****\\7)Q7kFen$!3Q,rWi^nfCFen7$$!3e*****\\i^)o`Fen$! 3#H/\\%enEu9Fen7$$!3vlmT50A@WFen$!3yBh#=WIMO)!#>7$$!3OKLLeaR%H$Fen$!35 **QVMu46NFdp7$$!3kJLLLo#)RBFen$!3Ed&*Q)=\\$p7Fdp7$$!3f***\\PfO%H7Fen$! 3c*>.j8LO&=!#?7$$!3MSLLL3`lCFdp$!3Et`=pag)\\\"!#A7$$\"3+L+]i!f#=$)Fdp$ \"3?Q96yP0\\d!#@7$$\"3+-+v=xpe=Fen$\"3/&)e#z[RWQ'Fdq7$$\"3pxgFen$\"3!eH0m'zJ4@Fen7$$\"3 w++v$f4t.(Fen$\"3/Shbre^/KFen7$$\"3OPL$e*Gst!)Fen$\"3#)HX?G&R%4ZFen7$$ \"3Y+++]#RW9*Fen$\"3m2L$)*zKYi'Fen7$$\"3:++DJE>>5F-$\"3%)*[,-z5p%))Fen 7$$\"3F+]i!RU07\"F-$\"3$>?t#*Qw/8\"F-7$$\"3+++v=S2L7F-$\"3()fO&f$GeM9F -7$$\"3Jmmm\"p)=M8F-$\"3`Ym!yzk/t\"F-7$$\"3B++](=]@W\"F-$\"3S$fiF-$\"3fiD`&=6Za$F-7$$\"3w**\\ilAFj?F-$\"39E.U(\\+7v$F-7$ $\"3ym\"zW7@^6#F-$\"3]EJy>+=FQF-7$$\"3yLLL$)*pp;#F-$\"3'*Q$zI$Rp&)QF-7 $$\"3)QL3-$H**>AF-$\"3n8dNL$*yDRF-7$$\"3)RL$3xe,tAF-$\"3wEjTJsBWRF-7$$ \"3h+v=n(*fDBF-$\"3'\\#H[\\E_RRF-7$$\"3Cn;HdO=yBF-$\"3N@X;P^K5RF-7$$\" 3*Q$e9\"z-lU#F-$\"3C%4t'\\'>1'QF-7$$\"3a+++D>#[Z#F-$\"3Q+86Ot&yy$F-7$$ \"3SnmT&G!e&e#F-$\"3782S%)\\[GNF-7$$\"3#RLLL)Qk%o#F-$\"3_M'45w_*zJF-7$ $\"37+]iSjE!z#F-$\"3i8ge$=e$zEF-7$$\"3L+++DM\"3%GF-$\"3w)\\;kI?4R#F-7$ $\"3a+]P40O\"*GF-$\"3)p]mr\"*o,2#F-7$$\"3G+voa-oXHF-$\"3KU**oDY3*o\"F- 7$$\"\"$F*$\"3%[!)QD2!3q7F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fb]l-F $6$7[o7$F($\"3AdT6hBgs>F-7$F/$\"3))3Ub.p;kCF-7$F4$\"3w$\\#R#oz/y#F-7$$ !37+DcEV'Gu\"F-$\"36z7yj)fT\"HF-7$F9$\"3>q^Df#=r,$F-7$$!3YmmmTp'ej\"F- $\"3%o$\"36B6d!G_c8$F-7$$!30+D197xG:F-$\"31Y$z2dKI:$F-7 $FC$\"3W?R^)y3Z9$F-7$$!3))**\\iSM#eU\"F-$\"3Tv*R=>Fa6$F-7$FH$\"35ds*yG #zmIF-7$FM$\"3'*G'*3<'z2\"HF-7$FR$\"3K9fD9A)Ho#F-7$FW$\"3\\O/A;>9/CF-7 $Fgn$\"3i4piPtE!3#F-7$F\\o$\"3C_\"R#G#4wx\"F-7$Fao$\"3t<)GyiCDV\"F-7$F fo$\"37K^A)fqm4\"F-7$F[p$\"3efq3)=Q)ozFen7$F`p$\"3s\"*G\"=_x+b&Fen7$Ff p$\"3W&yQ&z&)\\eJFen7$F[q$\"3a'p/'yYb<;Fen7$F`q$\"3)Q&z\"f=Cb^%Fdp7$Ff q$\"3\"eL3TwWL#=Fdq7$F\\r$\"3a&p44<:=2#Fdp7$Fbr$\"3Z`cnn]]E5Fen7$Fgr$ \"3Ys))y')*RY^#Fen7$F\\s$\"3a5N>fQU-WFen7$Fas$\"3ahb^y$*HCpFen7$Ffs$\" 3MkkI$ePN(**Fen7$F[t$\"3GLn6fsI)G\"F-7$F`t$\"3R%HF%yxH<;F-7$Fet$\"3=#R WHtn\"f>F-7$Fjt$\"3kd\\?v_U!G#F-7$F_u$\"37'='oSr;kDF-7$Fdu$\"3Z2yhx6jI GF-7$Fiu$\"3PLT1I**G6IF-7$$\"3GLLeR%p\")Q\"F-$\"39eS$3-S,3$F-7$F^v$\"3 FUk;9kGFJF-7$$\"3mm\"H#oZ1\"\\\"F-$\"3k`HRLIt\\JF-7$Fcv$\"3CcL(RLo::$F -7$$\"3%o;Hd!fX$f\"F-$\"3YH8rBOcGJF-7$Fhv$\"3qgV_q:,yIF-7$$\"3_Le*[t\\ sp\"F-$\"3)=&*=,)p4/IF-7$F]w$\"3-!)*Q;^>O!HF-7$Fbw$\"3@;IcOw1/EF-7$Fgw $\"3cB\")>]Re*=#F-7$F\\x$\"3#zu\"=WWPB;F-7$Ffx$\"3MCcaT3;(\\*Fen7$F`y$ \"3_(o.F`&4L8Fen7$Fjy$!3M'***y[+$p(zFen7$F_z$!3cybdYU`j7F-7$Fdz$!3CP1H J([=v\"F-7$$\"3(RL3_5,-`#F-$!3;kh8R=]PBF-7$Fiz$!3)pZejX])[HF-7$$\"3m+] P%37^j#F-$!33+'\\D()pa^$F-7$F^[l$!3McT'ePp))4%F-7$$\"3-n\"z>6but#F-$!3 juT)*[!>Ai4! 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Now we introduce multi-line do loops, and ultimately also multi-line procedures." }}{PARA 0 "" 0 "" {TEXT -1 190 "To have \+ an example to work on we consider the Taylor series expansion of a kno wn function. We begin by using the Maple built-in procedure, and then \+ construct the Taylor polynomail ourselvcs." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(sin(x),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG\" \"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(-%\"OG6#F%\"\")" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "In order to work with the result w e have to subtract the order term:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sinT8:=convert(taylor(sin(x),x=0,8),polynom);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sinT8G,*%\"xG\"\"\"*&#F'\"\"'F'*$)F &\"\"$F'F'!\"\"*&#F'\"$?\"F'*$)F&\"\"&F'F'F'*&#F'\"%S]F'*$)F&\"\"(F'F' F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([sin(x),sinT8],x =0..2*Pi,-1.5..1.5,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1123 345 345 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$\"\"!F)F(7$$\"3i]cC&eb&p 8!#=$\"3+9&o'z\"y_O\"F-7$$\"3YqR*pd)>hDF-$\"3K!y=V&*)GLDF-7$$\"3Zs[E^d K,RF-$\"3R]F<<.6.QF-7$$\"3ab^NehL]_F-$\"3I,*y(H4U7]F-7$$\"35*QhW&\\$Hf 'F-$\"3%=pD*eceDhF-7$$\"3zS11.fpPyF-$\"3w>82*oU&fqF-7$$\"3upr'fwtl7*F- $\"3@)*QjP!>8\"zF-7$$\"3!QRa&4L&f/\"!#<$\"3[[%3w7ESl)F-7$$\"3YjXwg<#)y 6FQ$\"3Pi.Z,bcT#*F-7$$\"36qGW%H$\\:8FQ$\"3!e$pSF\"oen*F-7$$\"3SH22vMov 8FQ$\"35_*\\T'zD5)*F-7$$\"3$*)e)pbO(eV\"FQ$\"3#ffI'ft64**F-7$$\"3/]+3V Nj.:FQ$\"3)\\\"3MzUXx**F-7$$\"396:YIMRr:FQ$\"2-6Ot@)******FQ7$$\"3Z'z] kaJ%R;FQ$\"3Ay8:y_Xw**F-7$$\"3!=3SCmpuq\"FQ$\"3s;Nd#HZn!**F-7$$\"33u?N %*p.tFQ$\"3cBCb^l'3E*F-7$$\"3#=uye<)G*4#FQ$\"3;2*>c4&oN')F-7$$\"3Ua[# o!GC>AFQ$\"3Se*Rp1I-(zF-7$$\"3-YwJf&y(eBFQ$\"3c:s(f4sF0(F-7$$\"3oNrpV8 H#[#FQ$\"3(y]$4EukDhF-7$$\"3EzY)QW/yh#FQ$\"3?Pawd/k,]F-7$$\"35v)>8'\\% ou#FQ$\"3i\"*R%R(HvXQF-7$$\"3w$GCS?&[\")GFQ$\"3!\\o6f)Q%=d#F-7$$\"3Z%) ='p(p70IFQ$\"3Kh5Xo]Ug8F-7$$\"3sA%)\\K8\\QJFQ$\"3^)pb)>hJ,J!#?7$$\"3c& fJlT>qF$FQ$!3)[9aqyJ,N\"F-7$$\"39l2\"4_3wR$FQ$!3w\"*=GeHGKDF-7$$\"3MOn Gd![y_$FQ$!39*)zsmMAnPF-7$$\"3#*=4JU#)RiOFQ$!3#35.)=3zv\\F-7$$\"3%G,f% [$HSz$FQ$!3!fgI[?W72'F-7$$\"3KoE+7#*Q@RFQ$!3#=$4p_xMJqF-7$$\"3y(f0ii+G 1%FQ$!3WWqZN&GL'zF-7$$\"3;fORr]')*=%FQ$!33*3&3nLil')F-7$$\"3R&p%*z[LbK %FQ$!3]$[$Q0***4E*F-7$$\"3M'pExBp%[WFQ$!3'R4g7n[Pl*F-7$$\"3,z&[2')pc^% FQ$!30-Py@682)*F-7$$\"3oh/x$[qGe%FQ$!3$euj/)>C;**F-7$$\"3qQq'H.,hk%FQ$ !3Ief(ReP!y**F-7$$\"3e;O;#eJ$4ZFQ$!3;'*phhK&*****F-7$$\"3sU'G^sDax%FQ$ !311%\\AUQ,)**F-7$$\"3&)oO4o)>:%[FQ$!32^6Oe=u;**F-7$$\"3;@9vt*Qh!\\FQ$ !31K)*)[7\"*G\")*F-7$$\"3Ou\"4%z!e2(\\FQ$!3_#49llz!o'*F-7$$\"3*zX#[4&e g5&FQ$!3]_&yQBx]B*F-7$$\"3n*e![/*ojB&FQ$!3o\"*oMmwMe')F-7$$\"3#ob8X5I' p`FQ$!3Froqht!o\"zF-7$$\"3PA\"GX$yy,bFQ$!3x5TwV@sUqF-7$$\"39L1/jqABcFQ $!3)\\'4LY'Q38'F-7$$\"3mweKB,TidFQ$!3iCF-zq_v\\F-7$$\"3'oIe;6(*o)eFQ$! 3s)y_Bpo*fQF-7$$\"3nKcu0ii>gFQ$!3dChVjR=0EF-7$$\"3w)*zj%)[mYhFQ$!3]'*= #eVn4O\"F-7$$\"3)****>YH&=$G'FQ$!3/UE[]'efD\"!#D-%'COLOURG6&%$RGBG$\"* ++++\"!\")F(F(-F$6$7WF'7$F+$\"3)o/o'z\"y_O\"F-7$F1$\"3ot\")=a*)GLDF-7$ F6$\"3A46T6.6.QF-7$F;$\"31,GWY3U7]F-7$F@$\"32Gu#H,&eDhF-7$FE$\"3ck)f0j R&fqF-7$FJ$\"3W:PkAqI6zF-7$FO$\"3=Au8X_)Rl)F-7$FU$\"3^_c/$)eWT#*F-7$FZ $\"3Gf&RD2[bn*F-7$F^o$\"3@#Q;Fi:%3**F-7$Fho$\"3'z3G?)QU)***F-7$Fbp$\"3 =Sz0TiV.**F-7$F\\q$\"3)[\">)=,:rj*F-7$Faq$\"3[r8b-Ii\\#*F-7$Ffq$\"3.@0 ::Or9')F-7$F[r$\"3_]p&Rz4e$zF-7$F`r$\"3]VANeG_$*pF-7$Fer$\"3*)4c[K7NKg F-7$Fjr$\"33,41R,._[F-7$F_s$\"3Zo.vV'okh$F-7$Fds$\"3w;K%Ge'[@AF-7$Fis$ \"3@N&z%*\\fK_)!#>7$F^t$!3%p8*>NTjYrF_bl7$Fdt$!3Mbb$z6q:W#F-7$Fit$!3T; si8pmKSF-7$F^u$!37:]xZ&*pbeF-7$Fcu$!3%*4S&R?Tk(yF-7$Fhu$!3u\\z./IM-5FQ 7$F]v$!3f[\"\\WuR1B\"FQ7$Fbv$!3^v-6N0'\\^\"FQ7$Fgv$!3)ePyS;!G1=FQ7$F\\ w$!3%Qzr)okvl@FQ7$Faw$!3#4$Hh,&4ia#FQ7$F[x$!3*RJ.B5Kr.$FQ7$Fex$!3!e!* \\ja!z(e$FQ7$F_y$!3K]'*G63%pF%FQ7$Fiy$!3?d^;-\"*f(3&FQ7$F^z$!3>Lx>a!)R 7hFQ7$Fcz$!3enOVrHI/tFQ7$Fhz$!3;SF>l4'=x)FQ7$F][l$!3Ec(p?Jq@0\"!#;7$Fb [l$!3_!RX\"[WHV7Fiel7$Fg[l$!3W4!fe@CT]\"Fiel7$$\"3w\"4#\\Y7y\"Fiel7$$\"3yp>qe;E`fFQ$!3Mq2BS<4[>Fiel7 $Fa\\l$!3wXb9$3'FQ$!3PD=zc(>yJ#Fiel7$Ff\\l$!3]id $[\"oW@DFiel7$$\"3P***G'*3D\\@'FQ$!3aWHB([[&eFFiel7$F[]l$!3wWfr\"p7f,$ Fiel-Fa]l6&Fc]lF(F(Fd]l-%+AXESLABELSG6$Q\"x6\"Q!Ffhl-%%VIEWG6$;F($\"+3 `=$G'!\"*;$!#:!\"\"$\"#:Fbil" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 55 "How are the terms generated? We choose a maximum order \+ " }{TEXT 19 1 "N" }{TEXT -1 176 "; we need the derivatives of the func tion and evaluate it at the expansion point, namely the origin. For si mplicity we work with expressions at first, and only later with maps. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=7;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"NG\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$sinG6#%\" xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "fT:=eval(f,x=0): for \+ n from 1 to N do: fd:=diff(f,x$n); fT:= fT + eval(fd,x=0)*x^n/n!; od; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG-%$cosG6#%\"xG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#fTG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#fdG,$-%$sinG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG% \"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG,$-%$cosG6#%\"xG!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG,&%\"xG\"\"\"*&#F'\"\"'F'*$)F& \"\"$F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG-%$sinG6#%\"xG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG,&%\"xG\"\"\"*&#F'\"\"'F'*$) F&\"\"$F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG-%$cosG6#%\" xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG,(%\"xG\"\"\"*&#F'\"\"'F'* $)F&\"\"$F'F'!\"\"*&#F'\"$?\"F'*$)F&\"\"&F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG,$-%$sinG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG,(%\"xG\"\"\"*&#F'\"\"'F'*$)F&\"\"$F'F'!\"\"*&#F' \"$?\"F'*$)F&\"\"&F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG,$-% $cosG6#%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fTG,*%\"xG\"\" \"*&#F'\"\"'F'*$)F&\"\"$F'F'!\"\"*&#F'\"$?\"F'*$)F&\"\"&F'F'F'*&#F'\"% S]F'*$)F&\"\"(F'F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "One can s uppress the output during the do-loop by terminating the loop with " } {TEXT 19 3 "od:" }{TEXT -1 60 " - that is the usually preferred mode, \+ and then one can put " }{TEXT 19 5 "print" }{TEXT -1 40 " statements i nto the loop for debugging." }}{PARA 0 "" 0 "" {TEXT -1 41 "The loop c an also be terminated with the " }{TEXT 19 7 "end do:" }{TEXT -1 95 " \+ statement. We will stick, however, with the historical terminator that represents an inverted " }{TEXT 19 2 "do" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 69 "This loop is simple enough that we could have prog rammed a one-liner:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eval (f,x=0)+add(eval(diff(f,x$n),x=0)*x^n/n!,n=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&#F%\"\"'F%*$)F$\"\"$F%F%!\"\"*&#F%\"$?\" F%*$)F$\"\"&F%F%F%*&#F%\"%S]F%*$)F$\"\"(F%F%F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Keep in mind, however, that the explicit do loop is \+ easier to read and debug." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 279 15 "Exercise 1.5.1:" }}{PARA 0 "" 0 "" {TEXT -1 93 "T est the loop with different functions and compare your results with Ma ple's result from the " }{TEXT 19 6 "taylor" }{TEXT -1 70 " procedure. Change the order of expansion. Change the expansion point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Note that we ha ve separated out the first term in the Taylor polynomial, because:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "add(eval(diff(f,x$n),x=0)*x ^n/n!,n=0..N);" }}{PARA 8 "" 1 "" {TEXT -1 61 "Error, wrong number (or type) of parameters in function diff\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We can incorporate the first term by avoiding the call to " }{TEXT 19 4 "diff" }{TEXT -1 10 " using an " }{TEXT 19 15 "if-then- else-fi" }{TEXT -1 18 " branch construct:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 111 "for n from 0 to N do: if n=0 then fT:=eval(f,x=0); else fd:=diff(f,x$n); fT:= fT + eval(fd,x=0)*x^n/n!; fi; od;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fT;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&#F%\"\"'F%*$)F$\"\"$F%F%!\"\"*&#F%\"$?\" F%*$)F$\"\"&F%F%F%*&#F%\"%S]F%*$)F$\"\"(F%F%F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "The branching causes the output to be suppressed (t hat may not have been intentional by the software developers)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "An " } {TEXT 280 49 "important comment concerns the recursive updating" } {TEXT -1 19 " of the expression " }{TEXT 19 2 "fT" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 37 "The only reason why a statement like " } {TEXT 19 15 "fT:= fT + blah;" }{TEXT -1 16 " works is that " }{TEXT 19 2 "fT" }{TEXT -1 194 " as been initialized to some value (expressio n) before the first appearance of the statement. Otherwise an infinite loop might be generated, although Maple usually catches attempts of t his sort:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "y1:=y1+5;" }}{PARA 8 "" 1 "" {TEXT -1 28 "Error, rec ursive assignment\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The above \+ statement has no meaning if " }{TEXT 19 2 "y1" }{TEXT -1 197 " has not been previously declared. Undeclared symbols are not initialized (For tran would initialize a previously unused variable to zero). We can't \+ add anything to an unitialized symbolic variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "There are more loop progr amming elements that control the flow, such as " }{TEXT 19 5 "while" } {TEXT -1 87 " - it is well worth looking at the examples in the help p ages for this type of control." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 366 "Let us now implement a procedure \+ that uses the differentiation operator, i.e., it works on maps to achi eve the same goal. It will be an example that shows how to pass argume nts into a procedure. Note that one cannot pass answers back from the \+ procedure, except via the final executed statement. If a procedure nee ds to update several variables, it is done through a " }{TEXT 19 6 "gl obal" }{TEXT -1 22 " declaration of these." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The differentiation operator " }{TEXT 19 1 "D" }{TEXT -1 33 " is applied repeatedly using the " } {TEXT 19 4 "D@@n" }{TEXT -1 31 " syntax. Consult the help page " } {TEXT 19 2 "?D" }{TEXT -1 26 " and look at the examples!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "MyTaylor:=proc(fmap,point,maxorder: :nonnegint) local order,res,x; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " res:=fmap(point); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "for order fro m 1 to maxorder do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "res:=res + ( D@@order)(fmap)(point)*(x-point)^order/order!; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "unapply(res ,x); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "A note on writing (typing) procedures:" }}{PARA 0 " " 0 "" {TEXT -1 101 "as a beginner keep to writing one statement per c ommand prompt, unless they are very straightforward;" }}{PARA 0 "" 0 " " {TEXT -1 93 "write the statements at isolated prompts (execution gro ups), and do not Enter the statements;" }}{PARA 0 "" 0 "" {TEXT -1 87 "after finishsing with the end statement review the program flow; chec k for uncompleted " }{TEXT 19 2 "do" }{TEXT -1 12 "-loops, and " } {TEXT 19 15 "if-then-else-fi" }{TEXT -1 22 " branching constructs." }} {PARA 0 "" 0 "" {TEXT -1 101 "join the execution statements (prompts) \+ into a single execution group for the entire procedure (from " }{TEXT 19 4 "proc" }{TEXT -1 4 " to " }{TEXT 19 4 "end:" }{TEXT -1 2 ");" }} {PARA 0 "" 0 "" {TEXT -1 71 "now submit the procedure to the Maple eng ine; if you terminate it with " }{TEXT 19 4 "end;" }{TEXT -1 53 " the \+ syntax will be displayed as understood by Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MyTaylor(ex p,2,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%& arrowGF&,.-%$expG6#\"\"#\"\"\"*&F+F/,&9$F/F.!\"\"F/F/*&#F/F.F/*&F+F/)F 1F.F/F/F/*&#F/\"\"'F/*&F+F/)F1\"\"$F/F/F/*&#F/\"#CF/*&F+F/)F1\"\"%F/F/ F/*&#F/\"$?\"F/*&F+F/)F1\"\"&F/F/F/F&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "unapply(simplify(taylor(exp(x),x=2,6)),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&+1, &9$\"\"\"\"\"#!\"\"-%$expG6#F.\"\"!F0F-,$*&#F-F.F-F0F-F-F.,$*&#F-\"\"' F-F0F-F-\"\"$,$*&#F-\"#CF-F0F-F-\"\"%,$*&#F-\"$?\"F-F0F-F-\"\"&-%\"OG6 #F-F:F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cosT6:=MyTay lor(cos,0,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&cosT6Gf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,*\"\"\"F-*&#F-\"\"#F-*$)9$F0F-F-!\"\"*&#F-\"#C F-*$)F3\"\"%F-F-F-*&#F-\"$?(F-*$)F3\"\"'F-F-F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([cos(x),cosT6(x)],x=0..2*Pi,-1.5..1. 5,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1123 345 345 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$\"\"!F)$\"\"\"F)7$$\"3;`#Gi#zxZo!#>$ \"3!eFW#HJcw**!#=7$$\"3i]cC&eb&p8F2$\"3#*p#fVPij!**F27$$\"3q5)>63x`'>F 2$\"3wuHGMb[2)*F27$$\"3YqR*pd)>hDF2$\"3[#y/-5-Qn*F27$$\"3Zs[E^dK,RF2$ \"3'4g??['e[#*F27$$\"3ab^NehL]_F2$\"35D5SC42`')F27$$\"35*QhW&\\$Hf'F2$ \"3FQ^ZT?D/zF27$$\"3zS11.fpPyF2$\"3=_>U!=uD3(F27$$\"3upr'fwtl7*F2$\"3' )e\\phdX;hF27$$\"3!QRa&4L&f/\"!#<$\"3[yHev:x5]F27$$\"3YjXwg<#)y6Fhn$\" 39g)pymR,#QF27$$\"36qGW%H$\\:8Fhn$\"39.oK.jQDDF27$$\"3$*)e)pbO(eV\"Fhn $\"3KlS\"H&o8X8F27$$\"396:YIMRr:Fhn$!3)R^L8JO5(f!#@7$$\"3!=3SCmpuq\"Fh n$!33@,ABB[i8F27$$\"3LmSEEVgQ=Fhn$!3!p()*f7@=YEF27$$\"3S`:%R;(od>Fhn$! 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To plot without converting to expressions change the specification of the independent variable range:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([cos,MyTaylor(cos,Pi,6)],0..2* Pi,-1.5..1.5,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1123 345 345 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$\"\"!F)$\"\"\"F)7$$\"3;`#Gi#z xZo!#>$\"3!eFW#HJcw**!#=7$$\"3i]cC&eb&p8F2$\"3#*p#fVPij!**F27$$\"3q5)> 63x`'>F2$\"3wuHGMb[2)*F27$$\"3YqR*pd)>hDF2$\"3[#y/-5-Qn*F27$$\"3Zs[E^d K,RF2$\"3'4g??['e[#*F27$$\"3ab^NehL]_F2$\"35D5SC42`')F27$$\"35*QhW&\\$ Hf'F2$\"3FQ^ZT?D/zF27$$\"3zS11.fpPyF2$\"3=_>U!=uD3(F27$$\"3upr'fwtl7*F 2$\"3')e\\phdX;hF27$$\"3!QRa&4L&f/\"!#<$\"3[yHev:x5]F27$$\"3YjXwg<#)y6 Fhn$\"39g)pymR,#QF27$$\"36qGW%H$\\:8Fhn$\"39.oK.jQDDF27$$\"3$*)e)pbO(e V\"Fhn$\"3KlS\"H&o8X8F27$$\"396:YIMRr:Fhn$!3)R^L8JO5(f!#@7$$\"3!=3SCmp uq\"Fhn$!33@,ABB[i8F27$$\"3LmSEEVgQ=Fhn$!3!p()*f7@=YEF27$$\"3S`:%R;(od >Fhn$!3%R#pX#*)3Jx$F27$$\"3#=uye<)G*4#Fhn$!3nht,N_JU]F27$$\"3Ua[#o!GC> AFhn$!3L_)*oQ%*[RgF27$$\"3-YwJf&y(eBFhn$!3k\\\"Q8J;$*3(F27$$\"3oNrpV8H #[#Fhn$!3UtA:G>r1$f')F27$$\"35v)>8'\\% ou#Fhn$!3(H:T1DO4B*F27$$\"3w$GCS?&[\")GFhn$!3E3Gi$\\BOm*F27$$\"3*Q3$\\ !41L%HFhn$!33D)eOYbS!)*F27$$\"3Z%)='p(p70IFhn$!3Zw5r5+.2**F27$$\"3g`,t a\"4=2$Fhn$!3iPR!e>hc(**F27$$\"3sA%)\\K8\\QJFhn$!3'p)f24>&*****F27$$\" 3#*3]^u`v2KFhn$!3Ms?.b/7y**F27$$\"3c&fJlT>qF$Fhn$!38)zW\"G!Q%3**F27$$ \"3e!=@(oRJPLFhn$!3\\.t5wk24)*F27$$\"39l2\"4_3wR$Fhn$!3;&3vb[lSn*F27$$ \"3MOnGd![y_$Fhn$!3eDXLTAEj#*F27$$\"3#*=4JU#)RiOFhn$!35CZ!4<'=u')F27$$ \"3%G,f%[$HSz$Fhn$!37)oJ.#y1YzF27$$\"3KoE+7#*Q@RFhn$!3e'42e*fc5rF27$$ \"3y(f0ii+G1%Fhn$!3k9U,*\\'e[gF27$$\"3;fORr]')*=%Fhn$!3![Trb\\)o!*\\F2 7$$\"3R&p%*z[LbK%Fhn$!3'QYDqc\"ysPF27$$\"3M'pExBp%[WFhn$!3L'GUFnl'3EF2 7$$\"3oh/x$[qGe%Fhn$!3=4d,soc\"H\"F27$$\"3e;O;#eJ$4ZFhn$!35e,\"yX$RdI! #?7$$\"3&)oO4o)>:%[Fhn$\"3o*y!R^Js(G\"F27$$\"3Ou\"4%z!e2(\\Fhn$\"3/CaO V7/bDF27$$\"3*zX#[4&eg5&Fhn$\"3(p'>9&G)zNQF27$$\"3n*e![/*ojB&Fhn$\"3!p +]&z/I.]F27$$\"3#ob8X5I'p`Fhn$\"32-G'*3.N4hF27$$\"3PA\"GX$yy,bFhn$\"3E O\\&RI+$*4(F27$$\"39L1/jqABcFhn$\"3KytI?$y,!zF27$$\"3mweKB,TidFhn$\"3C %Qb^XPVn)F27$$\"3'oIe;6(*o)eFhn$\"3`bo#[!4+D#*F27$$\"3nKcu0ii>gFhn$\"3 Yfw#e$))oa'*F27$$\"3G:=>Xb9$3'Fhn$\"3**=**yvne+)*F27$$\"3w)*zj%)[mYhFh n$\"3n&\\^\"=b&p!**F27$$\"3P***G'*3D\\@'Fhn$\"37jT:eYH&=$G'Fhn$\"2))***************Fhn-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F (-F$6$7W7$F($\"3o+D)H%GN67Fhn7$F4$\"3'*>&))f:L*R6Fhn7$F>$\"3IFpO*[-i2 \"Fhn7$FC$\"3SyP)=l1&)***F27$FH$\"3[$QJ5J*fe\"*F27$FM$\"3\"y+[#3)3%Q#) F27$FR$\"3!GRE@!3W0tF27$FW$\"31>?(*p&=&fiF27$Ffn$\"3!=(\\l&\\v')4&F27$ F\\o$\"3!>V'=m')\\sQF27$Fao$\"3S(*)3@YX\\b#F27$Ffo$\"3[\"3UH\"pMi8F27$ F[p$\"3%R\"**Qm6@ZHF_p7$Fap$!3?$HpaMW\"e8F27$Ffp$!3O'foB%)fTk#F27$F[q$ !3ME7<9k;sPF27$F`q$!3B7Mp!)Q(>/&F27$Feq$!3'*\\95[2ORgF27$Fjq$!3gD(3Td \"G*3(F27$F_r$!3!))zh:O&>/zF27$Fdr$!3u6X/6`If')F27$Fir$!3'e5kY5O4B*F27 $F^s$!3#=KI%)[BOm*F27$Fcs$!3'yRmIYbS!)*F27$Fhs$!3m`7o5+.2**F27$F]t$!3% ))z.e>hc(**F2Fat7$Fgt$!3I\")>.b/7y**F27$F\\u$!3oTn6G!Q%3**F27$Fau$!3S+x6m<#F\\y7$F^y$\"3JVLHi[Z/8F27$Fcy$\"3))=Y'z5'* \\e#F27$Fhy$\"3!ek)))fi^))QF27$F]z$\"3!=k#yZ'=44&F27$Fbz$\"3!z#Q:o*e>D 'F27$Fgz$\"3U]F8#pRRK(F27$F\\[l$\"31>!=;zJOB)F27$Fa[l$\"3K%QXh/ij=*F27 $Ff[l$\"3!QxTAGs='**F27$F[\\l$\"3m.A=!*e6s5Fhn7$Fe\\l$\"3'HU]gMh,9\"Fh n7$F_]l$\"3xD2SOGN67Fhn-Fd]l6&Ff]lF(F(Fg]l-%+AXESLABELSG6$Q!6\"F^hl-%% VIEWG6$;F($\"+3`=$G'!\"*;$!#:!\"\"$\"#:Fjhl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 15 "Exercise 1.5.2:" }}{PARA 0 "" 0 "" {TEXT -1 71 "Explore local Taylor expansions by making use of the defined proce dure " }{TEXT 19 8 "MyTaylor" }{TEXT -1 97 ". In particular, consider \+ effects from a finite radius of convergence by looking at the function " }{TEXT 19 9 "1/(1+x^2)" }{TEXT -1 46 " or similar expressions while expanding about " }{TEXT 19 3 "x=0" }{TEXT -1 64 ". Generate expansio ns about other points and graph your results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 15 "Exercise 1.5.3: " }}{PARA 0 "" 0 "" {TEXT -1 50 "Design a procedure of your own which \+ uses Maple's " }{TEXT 19 6 "taylor" }{TEXT -1 23 " procedure (or our o wn " }{TEXT 19 8 "MyTaylor" }{TEXT -1 69 ") to investigate the converg ence behavior of the Taylor series about " }{TEXT 19 3 "x=0" }{TEXT -1 77 " (McLaurin series) of a function provided by the user up to som e fixed order " }{TEXT 19 1 "N" }{TEXT -1 131 ". The last statement ex ecuted by the procedure should be a plot which displays the function t ogether with the polynomials of order " }{TEXT 19 6 "n=1..N" }{TEXT -1 21 " in different colors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "135 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 36473060 36640136 36725496 36894180 37309508 36265248 37448020 37503164 13314220 13056500 36461500 36846240 35671064 }{RTABLE M7R0 I5RTABLE_SAVE/36473060X*%)anythingG6"6"[gl!#%!!!"$"$"""""#""$F& } {RTABLE M7R0 I5RTABLE_SAVE/36640136X,%)anythingG6"6"[gl!"%!!!#*"$"$%$a11G%$a21G%$a31G%$a12G% $a22G%$a32G%$a13G%$a23G%$a33GF& } {RTABLE M7R0 I5RTABLE_SAVE/36725496X*%)anythingG6"6"[gl!#%!!!"$"$#""*""#!"(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/36894180X*%)anythingG6"6"[gl!#%!!!"$"$,(%$a11G#""*""#%$a12G!"(%$a 13GF),(%$a21GF)%$a22GF-%$a23GF),(%$a31GF)%$a32GF-%$a33GF)F& } {RTABLE M7R0 I5RTABLE_SAVE/37309508X,%)anythingG6"6"[gl!"%!!!#*"$"$#!#6"#;#""*"")#!"*F)#"#PF )#!#JF,#"#RF)#""$F)#!""F,#"""F)F& } {RTABLE M7R0 I5RTABLE_SAVE/36265248X,%)anythingG6"6"[gl!"%!!!#*"$"$,(%$a11G#!#6"#;%$a21G#"#P F+%$a31G#""$F+,(F(#""*"")F,#!#JF5F/#!""F5,(F(#!"*F+F,#"#RF+F/#"""F+,(%$a12GF)%$ a22GF-%$a32GF0,(FBF3FCF6FDF8,(FBF;FCF=FDF?,(%$a13GF)%$a23GF-%$a33GF0,(FHF3FIF6F JF8,(FHF;FIF=FJF?F& } {RTABLE M7R0 I5RTABLE_SAVE/37448020X,%)anythingG6"6"[gl!"%!!!#*"$"$*&,&*&%$a22G"""%$a33GF+F+ *&%$a23GF+%$a32GF+!""F+,.*(%$a31GF+%$a12GF+F.F+F+*(F3F+%$a13GF+F*F+F0*(%$a21GF+ F4F+F,F+F0*(F8F+F6F+F/F+F+*(%$a11GF+F*F+F,F+F+*(F;F+F.F+F/F+F0F0,$*&,&*&F3F+F.F +F0*&F8F+F,F+F+F+F1F0F0*&,&*&F3F+F*F+F0*&F8F+F/F+F+F+F1F0,$*&,&*&F4F+F,F+F+*&F6 F+F/F+F0F+F1F0F0*&,&*&F3F+F6F+F0*&F;F+F,F+F+F+F1F0,$*&,&*&F3F+F4F+F0*&F;F+F/F+F +F+F1F0F0*&,&*&F4F+F.F+F+*&F6F+F*F+F0F+F1F0,$*&,&*&F8F+F6F+F0*&F;F+F.F+F+F+F1F0 F0*&,&*&F8F+F4F+F0*&F;F+F*F+F+F+F1F0F& } {RTABLE M7R0 I5RTABLE_SAVE/37503164X,%)anythingG6"6"[gl!"%!!!#:"&"&%$a11G%$a21G%$a31G%$a41G% $a51G%$a12G%$a22G%$a32G%$a42G%$a52G%$a13G%$a23G%$a33G%$a43G%$a53G%$a14G%$a24G%$ a34G%$a44G%$a54G%$a15G%$a25G%$a35G%$a45G%$a55GF& } {RTABLE M7R0 I5RTABLE_SAVE/13314220X,%)anythingG6"6"[gl!"%!!!#*"$"$"""F'""!""#""%F'""$""*F(F & } {RTABLE M7R0 I5RTABLE_SAVE/13056500X*%)anythingG6"6"[gl!#%!!!"$"$"""F'F'F& } {RTABLE M7R0 I5RTABLE_SAVE/36461500X*%)anythingG6"6"[gl!#%!!!"$"$""'"#9"""F& } {RTABLE M7R0 I5RTABLE_SAVE/36846240X,%)anythingG6#%)identityG6"[gl!""!!!#!"$"$F' } {RTABLE M7R0 I5RTABLE_SAVE/35671064X,%)anythingG6"6"[gl!"%!!!#*"$"$,&"""F(%'lambdaG!""F(""!" 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