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0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 17 "November 11, 1997" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 28 "A Brief Overview of Maple V " }} {PARA 258 "" 0 "" {TEXT -1 47 "with Sample Applications to Elementary \+ Physics " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 490 " Software packages like Maple are called Computer Al gebra Systems because they can manipulate symbolic information in ways that let a user obtain analytic solutions to many kinds of pure and a pplied mathematics problems. Since (some) humans can do this sort of \+ thing themselves, what makes Computer Algebra Systems interesting is t he speed and reliability with which they perform such manipulations. \+ It will surprise no one to hear that this power can be put to good use in physics. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 467 " Today, Computer Algebra Systems are much more \+ than the collections of symbolic manipulation routines they once were, and their new capabilities only increase their value to physics stude nts and professionals. These capabilities include sophisticated tools for numeric computation and graphics as well as a user interface that not only supports computation but also the production of documents re porting computations and their results. This document is a Maple " } {TEXT 256 9 "worksheet" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 18 " Getting Started " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 " When you double-click on the Maple \+ icon (Wmaple32 on PCs, Maple(FPU) on Macs or Maple(PPC) on Power Macs) or enter " }{TEXT 257 6 "xmaple" }{TEXT -1 127 " at the prompt in an \+ XWindows session a window displaying three control bars, a status bar \+ and a blank Maple worksheet opens. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 " The " }{TEXT 258 8 "menu ba r" }{TEXT -1 76 " appears at the top of the window. Menus which drop \+ down when you click on " }{TEXT 259 1 "F" }{TEXT -1 7 "ile or " } {TEXT 260 1 "E" }{TEXT -1 132 "dit provide access to operations like O pening, Saving and Printing worksheets; Copying, Cutting and Pasting a nd Exiting Maple. The " }{TEXT 261 8 "tool bar" }{TEXT -1 166 ", just below the menu bar, has button shortcuts to some of these and to othe r common operations. Just below it, control buttons and the informati on displayed on the " }{TEXT 262 11 "context bar" }{TEXT -1 103 " vary depending on whether you are working on text, Maple input or various \+ types of Maple output. The " }{TEXT 263 10 "status bar" }{TEXT -1 45 " appears at the bottom of the Maple window. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 225 " The blank w orksheet is displayed in the space between the context and status bars . The cursor blinks to the right of the prompt, >, in the worksheet' s upper lefthand corner. This prompt marks the beginning of a Maple \+ " }{TEXT 264 12 "input region" }{TEXT -1 27 ". Maple is waiting for a n " }{TEXT 265 10 "expression" }{TEXT -1 4 " to " }{TEXT 266 8 "evalua te" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 " Maple expressions can represent mathemati cal objects like numbers and functions and can invoke Maple " }{TEXT 267 8 "commands" }{TEXT -1 117 " that manipulate such objects or creat e special types of output from them, graphs for example. They can als o define " }{TEXT 268 10 "procedures" }{TEXT -1 60 " (programs) that a dd new capabilities to the Maple system. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 " When Maple evaluat es an expression the result is stored and can be displayed in an " } {TEXT 270 13 "output region" }{TEXT -1 252 " inserted into the workshe et beneath the input region containing the expression. The trick, of \+ course, is to know what expressions to enter and evaluate so that Mapl e's output provides answers to questions of interest to you (or to you r instructor). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 153 " A lot of analysis and problem solving can be done with fairly simple and intuitive types of Maple expressions. Fo r example, it is quite clear from " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "3*6 + 4^2/(3 - 1); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "that arithmetic expressions ar e entered using the usual computerish symbols, that the usual rules of precedence apply and that those rules can be overridden by using pare ntheses. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 320 " Additional examples of computations done by evaluat ing Maple expressions will be discussed shortly but, first, notice the semicolon that marks the end of the expression above. Since almost a ny combination of symbols constitutes a conceivable Maple expression a nd since expressions can run on for several lines you " }{TEXT 269 4 " must" }{TEXT -1 361 " mark their ends. The only way for Maple to know that your expression is complete is for you to mark the end of it wit h a semicolon or a colon. If you use a colon, Maple evaluates the exp ression, stores the result but does not display it. This is useful wh en you need but do not want to see complicated intermediate results wh en problem solving with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 378 " To have Maple evaluate or re evaluate an expression simply hit the key when the cursor is i n the input region containing it. If the worksheet contains input reg ions following that one, Maple will put the cursor in the next one whe n it is ready to accept more input. Otherwise, it will put the cursor in a new input region it creates at the end of your worksheet. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 " \+ To convert an input region into a " }{TEXT 272 11 "text region" } {TEXT -1 253 " in which you can write explanatory comments simply clic k the T button on the tool bar when the cursor is in the region. The \+ usual word processing tools are available to you in text regions. To \+ toggle a text region back into an input region click the " }{XPPEDIT 18 0 "Sigma" "I&SigmaG6\"" }{TEXT -1 134 " button on the tool bar. To insert a new input region beneath the region containing the cursor cl ick the [> button on the tool bar. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 408 " There is space here to de monstrate only a few basic types of computation that can be done by ev aluating Maple expressions. However, the examples in the remainder of this section do illustrate things a new user should try, offer an opp ortunity to suggest ways of avoiding the few pitfalls that can frustra te such users and provide some context for the Maple applications pres ented in the next section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 93 " The output Maple returns afte r evaluating this arithmetic expression may surprise you. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "3 *6 + 4^2/3 - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#n\"\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Ma ple returns this rational representation of the result because it is \+ " }{TEXT 271 5 "exact" }{TEXT -1 164 ". If you want the useful, but u sually approximate, floating-point representation of a number, you mus t ask Maple for it. You can do so explicitly or implicitly. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(3*6 + 4^2/3 - 1); 3*6.0 + 4^2/3 - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLLA!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +LLLLA!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The first of these expressions invokes Maple's " }{TEXT 273 5 "evalf" }{TEXT -1 22 " command which forces " }{TEXT 274 4 "eval " }{TEXT -1 10 "uation to " }{TEXT 275 1 "f" }{TEXT -1 256 "loating po int. The second demonstrates that floating point computation is gener ally \"contagious\". Once a floating point number, in this case 6.0, \+ appears in an expression Maple generally uses floating point arithmeti c to evaluate the entire expression. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 " The following examples i llustrate some of Maple's other arithmetic capabilities. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "sqr t(2); arctan(sqrt(2.0)); Pi; evalf(Pi,50); cos(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"##\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+!=mJb*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"S^P*Rpr>%)G]zKQVEYQKz*e`EfTJ!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalc(ln((1+I)/sqrt(2))); evalf(ln((1+I)/sqrt(2))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"IG\"\"\"%#PiGF&#F&\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$!+\"R!=QE!#>\"\"\"%\"IG$\"+M;)R&y! #5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "You see that Maple knows about familiar functions and certain s pecial constants like " }{XPPEDIT 18 0 "Pi" "I#PiG6\"" }{TEXT -1 5 " a nd " }{TEXT 278 1 "i" }{TEXT -1 117 " and that Maple's default floatin g point accuracy of 10 significant digits can be increased or decrease d as needed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 " Note well that Maple is case sensitive. It " } {TEXT 286 7 "matters" }{TEXT -1 49 ", for example, that the P in Pi is capitalized! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 " To see documentation on a Maple command like " }{TEXT 279 5 "evalf" }{TEXT -1 4 " or " }{TEXT 280 5 "evalc" }{TEXT -1 89 " you need only enter its name after a question mark in an input region and hit . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "?evalf" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "If you scroll to the bot tom of the window that opens, you will usually find examples demonstra ting the command's use. You can also find information on various Mapl e features in this " }{TEXT 281 11 "Help System" }{TEXT -1 22 ". For \+ example, enter " }{TEXT 282 7 "?inifcn" }{TEXT -1 113 " into an input \+ region and hit to see a list of the functions Maple knows abou t when you start it. Click " }{TEXT 284 1 "H" }{TEXT -1 29 "elp on th e menu bar and then " }{TEXT 283 1 "U" }{TEXT -1 31 "sing Help for mor e on what the " }{TEXT 285 11 "Help System" }{TEXT -1 18 " can do for \+ you. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 " Symbolic Maple expressions are often used to repres ent functions and equations involving them. In these examples x and \+ " }{TEXT 277 1 "y" }{TEXT -1 20 " appear as variable " }{TEXT 276 5 "n ames" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "7*x^2*exp(x/2)*sin(x); 27*x*y+x^2/y ^3; tan(y)=sqrt(x-y^2)/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"xG \"\"#-%$expG6#,$F%#\"\"\"F&F,-%$sinG6#F%F,\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"%\"yGF&\"#F*&F%\"\"#F'!\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#%\"yG*&,&%\"xG\"\"\"*$F'\"\"#!\"\" #F+F-F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " Evaluating the following expressions assigns values \+ to the variables " }{TEXT 289 1 "f" }{TEXT -1 5 " and " }{TEXT 288 1 " z" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := 7*x^2*exp(x/2)*sin(x); z := 2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*(%\"xG\"\"#-%$expG6#,$F'#\"\" \"F(F.-%$sinG6#F'F.\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG\"\" #" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "A variable can be assigned any Maple expression as a value, and o nce a value is assigned to a variable its name evaluates to that value . Thus, " }{TEXT 290 1 "f" }{TEXT -1 45 " is now a convenient name fo r the expression " }{XPPEDIT 18 0 "7*x^2*exp(x/2)*sin(x)" "**\"\"(\"\" \"*$%\"xG\"\"#F$-%$expG6#*&F&F$\"\"#!\"\"F$-%$sinG6#F&F$" }{TEXT -1 49 " and the value 2 is automatically plugged in for " }{TEXT 291 1 "z " }{TEXT -1 71 " when expressions involving that variable are evaluate d. For example, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "Int(f,x=0..Pi)=int(f,x=0..Pi); 9*z^4 - 2 *z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*(%\"xG\"\"#-%$expG 6#,$F)#\"\"\"F*F0-%$sinG6#F)F0\"\"(/F);\"\"!%#PiG,**&-F,6#,$F8F/F0F8F* #\"#G\"\"&*&F;F0F8F0#!$C#\"#DF;#FC\"$D\"FEF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$S\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 " Such " }{TEXT 325 15 "full evaluatio n" }{TEXT -1 108 " is obviously convenient, but it does create a pitfa ll. Imagine that you assign a value to a variable, say " }{TEXT 292 1 "A" }{TEXT -1 65 ", as you begin work on a new worksheet. If you us e the variable " }{TEXT 293 1 "A" }{TEXT -1 187 " later in your Maple \+ session having forgotten that it has the value you assigned it earlier , you will be surprised and confused by the results Maple returns for \+ you expressions involving " }{TEXT 294 1 "A" }{TEXT -1 86 ". Note tha t this will be the case even if you have deleted all earlier reference s to " }{TEXT 295 1 "A" }{TEXT -1 28 " by editing you worksheet! " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 " \+ There are a couple of lessons here. First, when using Maple you m ust keep its " }{TEXT 296 14 "internal state" }{TEXT -1 214 " in mind. This includes variable values that you have assigned. Second, you m ust construct a worksheet with care if you want its appearance to refl ect the evolution of Maple's state as it was being constructed. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 " \+ To " }{TEXT 311 8 "unassign" }{TEXT -1 46 " a variable's value eval uate expressions like " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "z := 'z';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "which assign a variable's name to be its value, the normal state of affairs for free variables. Use " }{TEXT 308 7 "?quotes" }{TEXT -1 15 " to access the " }{TEXT 309 11 "Help Sys tem" }{TEXT -1 56 " documentation on the use of quotation marks in Map le. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " The following examples illustrate a few simple symb olic computations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "expand((x+1)^2*(x-1)); expand(sin(a+b)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"xG\"\"$\"\"\"*$F%\"\"#F'F%! \"\"F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"aG\"\"\"-% $cosG6#%\"bGF)F)*&-F+F'F)-F&F,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "factor(x^3+x^2-x-1); combine(sin(a)*cos(b)+cos(a)*si n(b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"F&F&\"\"#,&F% F&!\"\"F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,&%\"aG\"\"\"% \"bGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "solve(x^2+7*x-2,x ); subs(x=y*sin(phi), x^2+7*x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ ,&#!\"(\"\"#\"\"\"*$\"#d#F'F&F*,&F$F'F(#!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"yG\"\"#-%$sinG6#%$phiGF&\"\"\"*&F%F+F'F+\"\"(!\" #F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "diff(f,x); series(t an(sin(x)),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%\"xG\"\"\"-%$ expG6#,$F%#F&\"\"#F&-%$sinG6#F%F&\"#9*(F%F,F'F&F-F&#\"\"(F,*(F%F,F'F&- %$cosGF/F&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"\"#F% \"\"'\"\"$#!\"\"\"#S\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(f,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,( *$%\"xG\"\"##!\"%\"\"&F'#\"#K\"#D#F-\"$D\"\"\"\"F1-%$expG6#,$F'#F1F(F1 -%$cosG6#F'F1\"\"(*(,(F&#F(F+F'#\"#CF.#!$w\"F0F1F1F2F1-%$sinGF9F1F:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DE := t*diff(y(t),t)+2*y(t )=sin(t); dsolve(DE,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/, &*&%\"tG\"\"\"-%%diffG6$-%\"yG6#F(F(F)F)F-\"\"#-%$sinGF/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*&,(-%$sinGF&!\"\"*&F'\"\"\"-%$c osGF&F/F/%$_C1GF-F/F'!\"#F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 297 11 "Help System" }{TEXT -1 214 " includes documentation on and examples demonstrating the use \+ of the commands that appear in these expressions. It also includes do cumentation on and examples illustrating the use of these other useful commands: " }{TEXT 298 7 "collect" }{TEXT -1 2 ", " }{TEXT 299 7 "co nvert" }{TEXT -1 2 ", " }{TEXT 300 5 "limit" }{TEXT -1 2 ", " }{TEXT 301 8 "simplify" }{TEXT -1 2 ", " }{TEXT 310 6 "normal" }{TEXT -1 2 ", " }{TEXT 302 3 "rhs" }{TEXT -1 2 ", " }{TEXT 303 3 "lhs" }{TEXT -1 2 ", " }{TEXT 304 5 "numer" }{TEXT -1 2 ", " }{TEXT 305 5 "denom" } {TEXT -1 2 ", " }{TEXT 306 6 "fsolve" }{TEXT -1 5 " and " }{TEXT 307 15 "dsolve[numeric]" }{TEXT -1 147 ". Note that the last two of these are designed to find approximate answers to problems that do not admi t analytic solutions. Here is an example, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(cos(x)=x,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " This over view of basic Maple use concludes with a brief look at Maple graphics. Only the basic " }{TEXT 326 4 "plot" }{TEXT -1 129 " command is demo nstrated here. For information on the many commands that create 2- an d 3-dimensional plots and animations click " }{TEXT 327 1 "H" }{TEXT -1 27 "elp on the menu bar, click " }{TEXT 328 1 "C" }{TEXT -1 86 "ont ents, click the + button by Graphics and follow the hypertext links th at appear. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 " In its simplest form the " }{TEXT 312 4 "plot" } {TEXT -1 210 " command takes two arguments, an expression representing the function of a single variable whose graph is to be drawn and an e quation defining the range of the independent variable which the graph is to show. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "plot(tanh(x), x=-5..5, color=black); " }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#7W7$$!\"&\"\"!$!1^fiU?4****!# ;7$$!1LLLe%G?y%!#:$!1iJo]ff)***F-7$$!1mmT&esBf%F1$!1W7Mf$[z***F-7$$!1L L$3s%3zVF1$!1oG#G0do***F-7$$!1ML$e/$QkTF1$!1qphp<<&***F-7$$!1nmT5=q]RF 1$!1v<&)p#)f#***F-7$$!1LL3_>f_PF1$!1r\\ce:+*)**F-7$$!1++vo1YZNF1$!1C\" RLiFM)**F-7$$!1LL3-OJNLF1$!1VmdE+ou**F-7$$!1++v$*o%Q7$F1$!1%p2o]w8'**F -7$$!1mmm\"RFj!HF1$!1f*oaa)QS**F-7$$!1LL$e4OZr#F1$!1t:,gZn7**F-7$$!1++ +v'\\!*\\#F1$!1Vb%=.!*e')*F-7$$!1+++DwZ#G#F1$!1M7]s#QRz*F-7$$!1+++D.xt ?F1$!1_FN\\C&))o*F-7$$!1LL3-TC%)=F1$!1[lJ'y!o[&*F-7$$!1mmm\"4z)e;F1$!1 ;.[N/m+$*F-7$$!1mmmm`'zY\"F1$!1o5#)\\h)=**)F-7$$!1++v=t)eC\"F1$!1FKm?N Er%)F-7$$!1nmm;1J\\5F1$!1#RJ[&>Q:yF-7$$!1$***\\(=[jL)F-$!1'**4t1HU#oF- 7$$!1'***\\iXg#G'F-$!1YI!yTK&obF-7$$!1emmT&Q(RTF-$!13F3vhT=RF-7$$!1hm \"HdGe:$F-$!1\"og&[Z2bIF-7$$!1lm;/'=><#F-$!1W`.k%*RQ@F-7$$!1++D\"yQ16 \"F-$!1A4_cX416F-7$$!1EMLLe*e$\\!#=$!1UE?\\d&e$\\Fis7$$\"1pmT5D,`5F-$ \"1;_ePw8\\5F-7$$\"1sm;zRQb@F-$\"1PO(*f`hA@F-7$$\"1PLL$e,]6$F-$\"1qGs; B,=IF-7$$\"1-+](=>Y2%F-$\"1Yd66]:jQF-7$$\"1vmm\"zXu9'F-$\"1BoZ]IeuaF-7 $$\"1,+++&y))G)F-$\"1$*GlCQy)z'F-7$$\"1++]i_QQ5F1$\"1lPZ7i\\sxF-7$$\"1 ,+D\"y%3T7F1$\"10=wppkd%)F-7$$\"1++]P![hY\"F1$\"1Vd3.5S))*)F-7$$\"1LLL $Qx$o;F1$\"1WeD:$oLJ*F-7$$\"1+++v.I%)=F1$\"12$G:VI([&*F-7$$\"1mm\"zpe* z?F1$\"1***H\"*)3i#p*F-7$$\"1,++D\\'QH#F1$\"1#ze*o9`)z*F-7$$\"1LLe9S8& \\#F1$\"1].\"exU[')*F-7$$\"1,+D1#=bq#F1$\"10&*[Xr06**F-7$$\"1LLL3s?6HF 1$\"1$[,Fzl4%**F-7$$\"1++DJXaEJF1$\"1w9&3$Reh**F-7$$\"1ommm*RRL$F1$\"1 m1ae/hu**F-7$$\"1om;a<.YNF1$\"1>6*RB!Q$)**F-7$$\"1NLe9tOcPF1$\"1u[(oC% 3*)**F-7$$\"1,++]Qk\\RF1$\"1h.h(f#e#***F-7$$\"1NL$3dg6<%F1$\"19)o%fnB& ***F-7$$\"1ommmxGpVF1$\"1e$))G([z'***F-7$$\"1++D\"oK0e%F1$\"1C(oA?**y* **F-7$$\"1,+v=5s#y%F1$\"1A'yR*yf)***F-7$$\"\"&F*$\"1^fiU?4****F--%'COL OURG6&%$RGBGF*F*F*-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(Fj[l%(DEFAULTG " 2 166 169 169 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Notice how " } {TEXT 313 4 "a..b" }{TEXT -1 44 " is used to indicate a range in Maple . Use " }{TEXT 314 14 "?plot[options]" }{TEXT -1 15 " to access the \+ " }{TEXT 315 11 "Help System" }{TEXT -1 229 " documentation on optiona l arguments, like the third one in the example above. These are used \+ to control the appearance of plots and can be a necessity when plottin g a function on a domain in which it is singular. 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You will probably find those defined by the " }{TEXT 330 5 "plots " }{TEXT -1 2 ", " }{TEXT 331 7 "DEtools" }{TEXT -1 2 ", " }{TEXT 332 6 "linalg" }{TEXT -1 5 " and " }{TEXT 333 9 "orthopoly" }{TEXT -1 33 " packages most useful at first. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 22 " \+ Sample Applications " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 67 " Energies of the Particle States in a Fi nite Square-Well Potential" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 " This simply illustrates the power o f the strategy of plotting expressions, visually searching for solutio ns to an equation or system of equations and then using " }{TEXT 334 6 "fsolve" }{TEXT -1 44 " to refine your estimate of the solutions. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 " In " }{TEXT 335 15 "Quantum Physics" }{TEXT -1 81 " Gasiorowic z shows that the energies of even parity states of a particle of mass \+ " }{TEXT 336 1 "m" }{TEXT -1 37 " in a square-well potential of depth \+ " }{XPPEDIT 18 0 "-V[0]" ",$&%\"VG6#\"\"!!\"\"" }{TEXT -1 19 " that ex tends from " }{XPPEDIT 18 0 "x=-a" "/%\"xG,$%\"aG!\"\"" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "x=a" "/%\"xG%\"aG" }{TEXT -1 43 " can be determin ed by solving the equation " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Eeq := tan(y) = sqrt(lambda \+ - y^2)/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EeqG/-%$tanG6#%\"yG*&, &%'lambdaG\"\"\"*$F)\"\"#!\"\"#F-F/F)F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "y= q*a" "/%\"yG*&%\"qG\"\"\"%\"aGF&" }{TEXT -1 74 " is the rescaled magni tude of the particle's momentum within the well and " }{XPPEDIT 18 0 " lambda=2*m*V[0]*a^2/hbar^2" "/%'lambdaG*,\"\"#\"\"\"%\"mGF&&%\"VG6#\" \"!F&%\"aG\"\"#*$%%hbarG\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 " The follow ing plot shows that there are two even parity states when " }{XPPEDIT 18 0 "lambda" "I'lambdaG6\"" }{TEXT -1 20 " has the value 25. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "P1 := plot(sqrt(25-y^2)/y, y=0..2*Pi, 0..10, color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "P2 := plot(tan(y), y=0..2*Pi , 0..10, color=blue, discont=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](\{P1,P2\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7R7$$\"+)eb&p8!#5$\"+2&\\%\\O!\")7$$\"1+++j4^= :!#;$\"1m9m[/=\"H$!#97$$\"1+++PjYn;F1$\"1tXZ`L*o*HF47$$\"1+++6F*$\"+A\\2UDF-7$$\"1+++MyGjAF1$\"13MLP5\"p?# F47$$\"1+++#e)>hDF1$\"1N[d,\"['\\>F47$$\"1+++x.B'*GF1$\"1#>Y$RL[B#[W)R\"F47$$\" 1+++fdK,RF1$\"1&R&3I$3xF\"F47$$\"1+++l4$ed%F1$\"1\"H]SD7\")3\"F47$$\"1 ,++phL]_F1$\"1:MQO3bq%*!#:7$$\"1+++o\\$Hf'F1$\"1\"*QvktlfpPyF1$\"1F?%RUh0I'Ffo7$$\"1,++%ytl7*F1$\"14Tp:$okQ&Ffo7$$\"1+++7L&f/ \"Ffo$\"10JVCFcuYFfo7$$\"1+++j<#)y6Ffo$\"1%Qss&e&>7%Ffo7$$\"1+++(H$\\: 8Ffo$\"11S\"pNZpm$Ffo7$$\"1+++fO(eV\"Ffo$\"1`Yb9S_NLFfo7$$\"1+++MMRr:F fo$\"12&H2Wl1-$Ffo7$$\"1+++m'puq\"Ffo$\"1=B+!QrAv#Ffo7$$\"1+++IVgQ=Ffo $\"1G2Z6)=*GDFfo7$$\"1+++orod>Ffo$\"1D^\\(oD,N#Ffo7$$\"1+++!=)G*4#Ffo$ \"1x3NO7mh@Ffo7$$\"1+++6GC>AFfo$\"1A-gIf$*=?Ffo7$$\"1+++k&y(eBFfo$\"1o DRsy.p=Ffo7$$\"1+++\\8H#[#Ffo$\"16U&el1&[qF$Ffo$\"1Cyd,lQ_6Ffo7$$\"1+++G&3wR$Ffo$\"1,jo8Wm z5Ffo7$$\"1+++k![y_$Ffo$\"1#*44:DN/5Ffo7$$\"1+++]#)RiOFfo$\"1ZpW()\\I% H*F17$$\"1+++c$HSz$Ffo$\"1ua()f^W$e)F17$$\"1+++?#*Q@RFfo$\"1pgzpie5zF1 7$$\"1+++M1!G1%Ffo$\"1\"eu]gWL<(F17$$\"1+++!3l)*=%Ffo$\"1#G7w5#G7lF17$ $\"1+++(\\LbK%Ffo$\"1K,&3f))zz&F17$$\"1+++Z#p%[WFfo$\"1-Dr\\giJ^F17$$ \"1+++$\\qGe%Ffo$\"1uI$4(GgiVF17$$\"1+++#fJ$4ZFfo$\"1@&=WGcsc$F17$$\"1 +++y)>:%[Ffo$\"1'[j*Gt]zDF17$$\"1+++*3e2(\\Ffo$\"1cZZ(Q'G'3\"F17$%%FAI LGFbz-%'COLOURG6&%$RGBG\"\"!FgzFgz-F$6$7gn7$FgzFgz7$$\"1Hh'Q'*))QU$!#< $\"1J`,NvADMF_[l7$$\"1S7*QW'*HS'F_[l$\"1FFOI7w6kF_[l7$$\"1QVI!Q9Lv*F_[ l$\"1(zY\\MfVy*F_[l7$$\"1xq()RSe78F1$\"1wRUf>F1$\"1'3#>$\\\"*[)>F17$$\"1>H*>WV;G#F1$\"1m $R))**y?K#F17$$\"1-.YuK)[h#F1$\"1l>Cn#ehn#F17$$\"1\"oeDSaq%HF1$\"13s03 \\WNIF17$$\"1#\\HoBL()G$F1$\"1v)*R4\\m7MF17$$\"1*)[.STo*e$F1$\"1!e]'F17$$\"1& G*e*RYp*eF1$\"1=ji;[8\"p'F17$$\"1Fegg$Gd?'F1$\"1!*f@<:uZrF17$$\"1&4\\6 66Xa'F1$\"1VMa\"[)F17$$\"1*)Gk6Ir. sF1$\"1(ofyP]sx)F17$$\"1!yaSW)F1$\"1Q%f>+9,2\"Ffo7$$\"1([UTI@S\\)F1$ \"1nxwc5'p8\"Ffo7$$\"1GR:X,i>))F1$\"1Y8lI'3X@\"Ffo7$$\"1j#)y2c*f:*F1$ \"1p9&\\!GM,8Ffo7$$\"1w2BtL2&[*F1$\"1)*)[&\\-)RR\"Ffo7$$\"1X)f@.tM!)*F 1$\"1$44#*=y?\\\"Ffo7$$\"1\"[un:+d,\"Ffo$\"1:\"=WaTDh\"Ffo7$$\"1u%y!oi YZ5Ffo$\"18Ajvh7L@wrX6Ffo$\"1y@&3&e24AFfo7$$\"1'[*z&*GL x6Ffo$\"1HQMrQ+4CFfo7$$\"1d#*Gn*z.@\"Ffo$\"1![/*4mP`EFfo7$$\"1I_7?&*oU 7Ffo$\"1f')HJgiPHFfo7$$\"1!)4lFY^w7Ffo$\"1wk]j'R%*H$Ffo7$$\"1pwSEA448F fo$\"13(eMytMt$Ffo7$$\"1FKUEvSU8Ffo$\"1nfibF5-VFfo7$$\"1DT$*epWv8Ffo$ \"17#p4*4v`]Ffo7$$\"1@*ogw1eS\"Ffo$\"1#yBN%H*e+'Ffo7$$\"1uy9JDgS9Ffo$ \"10)*H60TPwFfo7$$\"1j'Q'y?:Ffo$\"1sv\\'>Nz*>F47$$\"1/q\\@imO:Ffo$\"1! )RX:Ffo$\"1M;rdgw0RF47$$\"1-&[FFJPb\"Ffo$\"1\")H Pb)f$feF47$$\"1w8cNv*zb\"Ffo$\"1^T:67\"G\"yF47$$\"1^UP)zjAc\"Ffo$\"1HV Y1r&><\"!#87$$\"1)o!yHpRk:Ffo$\"1>w\"e*eii:Fa[m7$$\"1Dr=h+`m:Ffo$\"1:! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 189 " To illustrate the slick way \+ in which Maple can handle the Lagrangian approach to mechanics begin b y representing the bob's motion in a vertical plane using Cartesian co ordinates. Let " }{TEXT 339 1 "x" }{TEXT -1 87 " denote horizontal di splacement of the bob from the pendulum's point of suspension and " } {TEXT 340 1 "y" }{TEXT -1 43 " denote its vertical displacement measur ed " }{TEXT 338 4 "down" }{TEXT -1 29 " from the suspension point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 " The following defines a few commands that facilitate Lagrangian \+ computations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "read `a:fdiff.mpl`; " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Once loaded, you can in dicate that the particle's " }{TEXT 341 1 "x" }{TEXT -1 5 " and " } {TEXT 342 1 "y" }{TEXT -1 39 " coordinates depend on time as follows \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "depend(\{x,y\}, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%\"IG%\"xG%\"yG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 89 "and can enter first and second derivatives of thes e functions using the concise notation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dx, ddx, dy, ddy;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&-%%diffG6$%\"xG%\"tG-F$6$F#F'-F$6$%\"y GF'-F$6$F*F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 78 "This makes entering the expression for the particle's k inetic energy a snap. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "T := (m/2)*(dx^2 + dy^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$*&%\"mG\"\"\",&*$-%%diffG6$%\"xG%\"t G\"\"#F(*$-F,6$%\"yGF/F0F(F(#F(F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The corresponding potential energy and Lagrangian are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V := -m*g*y; L := T - V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*(%\"mG\"\"\"%\"gGF(%\"yGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*&%\"mG\"\"\",&*$-%%diffG6$%\"xG%\"t G\"\"#F(*$-F,6$%\"yGF/F0F(F(#F(F0*(F'F(%\"gGF(F4F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " If the fun ctions " }{TEXT 343 1 "x" }{TEXT -1 5 " and " }{TEXT 344 1 "y" }{TEXT -1 439 " are treated as independent generalized coordinates, this Lagr angian produces the equations of motion that govern projectile motion. It leads to the pendulum equation of motion only when the holonomic \+ constraint imposed by the massless rod that suspends the bob is accoun ted for. This is most easily done when using polar coordinates as gen eralized coordinates in the plane. The transformation to a convenient set of polar coordinates (" }{XPPEDIT 18 0 "theta = 0" "/%&thetaG\"\" !" }{TEXT -1 80 " when the bob is directly below the suspension point) is easily accomplished by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "depend(\{r,theta\},t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'%\"IG%\"xG%\"yG%\"rG%&thetaG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x := r*sin(theta): y := \+ r*cos(theta): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "With these assingments made Maple's full evaluation automatically makes the coordinate transformation for you. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L := simplify(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*&% \"mG\"\"\"-%%diffG6$%\"rG%\"tG\"\"##F(F.*(F'F(F,F.-F*6$%&thetaGF-F.F/* *F'F(%\"gGF(F,F(-%$cosG6#F3F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The assignment " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r := l; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%\"lG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "imposes the constraint \+ in the same automatic way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"mG\"\"\"%\"lG\"\"#-%%diffG6$%&thetaG%\"tGF(#F&F( **F%F&%\"gGF&F'F&-%$cosG6#F,F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " The pendulum equation of moti on is this Lagrangian's Euler-Lagrange equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DE := diff( fdiff(L,dtheta),t) = fdiff(L,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#DEG/*(%\"mG\"\"\"%\"lG\"\"#-%%diffG6$-F,6$%&thetaG%\"tGF1F(,$**F' F(%\"gGF(F)F(-%$sinG6#F0F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "When values for the parameters appe aring in this equation are assigned and initial conditions specified y ou are ready to use Maple's numerical capabilities to simulate pendulu m motion. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "m := 1: l := 1: g := 9.8:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ICs := ic(theta,t=0,1), ic(dtheta,t =0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ICsG6$/-%&thetaG6#\"\"!\" \"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The following Maple command assigns the name " }{TEXT 345 3 "Sol" }{TEXT -1 100 " to a procedure using an adaptive Ru nge-Kutta algorithm that Maple creates to estimate this motion. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sol := dsolve(\{DE,ICs\},theta,numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SolG:6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%# r2G6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~r eserved.G6\"C&>8&-%&evalfG6#9$@$52-%$absG6#,$F3!\"\"-F<6#,&&%,loc_cont rolG6#\"\"#\"\"\"F3F?4-%'memberG6$&FD6#\"\"'<*F?FG!\"#FF$FG\"\"!$F?FR$ FPFR$FFFRC%>FD-%%copyG6#=F06#;FG\"#LE\\[lBFhnFR\"#=FR\"\"$FR\"#>FR\"\" %$FG!\")\"#?FQ\"\"&F^o\"#@FRFNFG\"#AFR\"\"($FG!\"*\"#BFR\"\")\"&++$\"# CFR\"\"*\"%+5\"#DFR\"#5FR\"#EFR\"#6FR\"#FFR\"#7FR\"#GFR\"#8FR\"#HFR\"# 9FR\"#IFR\"#:FR\"#JFR\"#;FRFGFF\"#KFR\"#%'loc_y0G-FY6#=F06#;FG FFE\\[l#FGFQFFFR>%'loc_y1G-FY6#=F0FbqE\\[l!@$0F;FRC$>&FD6#F[oF3@%1%'Di gitsG-%'evalhfG6#FcrC$>8%-%*traperrorG6#-Fer6#-%=dsolve/numeric_solnal l_rkf45G6,%&loc_FG-%$varG6#FD-Fds6#F^q-Fds6#Ffq-Fds6#%'loc_F1G-Fds6#%' loc_F2G-Fds6#%'loc_F3G-Fds6#%'loc_F4G-Fds6#%'loc_F5G-Fds6#%)loc_workG@ $/Fir%*lasterrorGC%>8'-%+searchtextG6$.Fer-%(convertG6$-%#opG6$FG7#Fir %%nameG>8(-Fcu6$.%)hardwareGFfu@%50FauFR0F_vFR-F`s6,FbsFDF^qFfqF\\tF_t FbtFetFhtF[u-%&ERRORG6#FirFhv7$/%\"tGF7-%$seqG6$/&%$ordG6#,&8$FGFGFG&F ^q6#Fhw/FhwFcqF06%FDF^qFfq" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Such procedures return estimates in an od d format, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "Sol(0); Sol(0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"tG\"\"!/%&thetaG$\"\"\"F&/-%%diffG6$F(F%F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7%/%\"tG$\"\"&!\"\"/%&thetaG$\"1k$Gc!R>]5!#;/-%%diffG 6$F*F%$!1Y&y0%pj$)H!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "but Maple provides special plotting comma nds to handle this sort of output. The following plot " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p lots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "odeplot(Sol, [t, \+ theta], 0..3, color=black); " }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVES G6#7T7$\"\"!$\"\"\"F(7$$\"+\")*[C7'!#6$\"1to!fz+d%)*!#;7$$\"+'z*[C7!#5 $\"1I>%)zK$fQ*F17$$\"+%pMn$=F5$\"1B,SzdWI')F17$$\"+#fz*[CF5$\"1aA'Hd\" e'f(F17$$\"+!\\C71$F5$\"1)Hrj%3e5jF17$$\"+)QpMn$F5$\"1Scs6?t3[F17$$\"+ 'G9dG%F5$\"1>n2$H#yPJF17$$\"+%=fz*[F5$\"1Wf7X/1a8F17$$\"+#3/-^&F5$!1\" y>lqa'*y%!#<7$$\"+!)*[C7'F5$!1Hs1Te]%H#F17$$\"+yQpMnF5$!1,^PIL'p-%F17$ $\"+w(QpM(F5$!1])4:*G>;cF17$$\"+uO=fzF5$!1K#[2Ge1,(F17$$\"+s&G9d)F5$!1 R\"Re1\\!p\")F17$$\"+qMn$=*F5$!1(>Ct.H/1*F17$$\"+o$=fz*F5$!1A`)>gjLm*F 17$$\"+Fj\"3/\"!\"*$!1Ll?t4ik**F17$$\"+<3/-6F\\q$!1,$>^D\"*z&**F17$$\" +2`Ej6F\\q$!1x!y0<3Ok*F17$$\"+(z*[C7F\\q$!1KZe#fkz-*F17$$\"+(G9dG\"F\\ q$!1E@NUxhC\")F17$$\"+x(QpM\"F\\q$!1\"pzmS%QbpF17$$\"+nK;39F\\q$!1$p%) =]L;b&F17$$\"+dxQp9F\\q$!1=4(4#o7bRF17$$\"+ZAhI:F\\q$!1\\#*)R&[\"y@#F1 7$$\"+Pn$=f\"F\\q$!1!o\"Hxq^,SFen7$$\"+F71`;F\\q$\"16Sx!Q@@V\"F17$$\"+ F\\q$\"103,_< vo')F17$$\"+n\"3/-#F\\q$\"1K)*F17$$ \"+FhIlAF\\q$\"1kg#eSz%f$*F17$$\"+<1`EBF\\q$\"1t\"4GzD;f)F17$$\"+2^v(Q #F\\q$\"1;&z/mHja(F17$$\"+(fz*[CF\\q$\"13Gmr?C]iF17$$\"+(3/-^#F\\q$\"1 M,-6&*4SZF17$$\"+x&G9d#F\\q$\"10[lQ$*3jIF17$$\"+nIlKEF\\q$\"1Rl.B#3fF \"F17$$\"+dv(Qp#F\\q$!10F)z%yYxbFen7$$\"+Z?5bFF\\q$!1-g'=)G/rBF17$$\"+ PlK;GF\\q$!1Ll()=Y`)4%F17$$\"+F5bxGF\\q$!1(GtK*=R!o&F17$$\"+ \+ " 0 "" {MPLTEXT 1 0 71 "odeplot(Sol, [theta, dtheta], 0..3, scaling=co nstrained, color=black); " }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6# 7T7$$\"\"\"\"\"!F*7$$\"1to!fz+d%)*!#;$!1ka;>6-K]F.7$$\"1I>%)zK$fQ*F.$! 1'e$*GW\"zg**F.7$$\"1B,SzdWI')F.$!1^xg4G;n9!#:7$$\"1aA'Hd\"e'f(F.$!1w- D:v,.>F;7$$\"1)Hrj%3e5jF.$!1w\"HE9`!)G#F;7$$\"1Scs6?t3[F.$!1h&zf3Q_g#F ;7$$\"1>n2$H#yPJF.$!1NVe?lzPGF;7$$\"1Wf7X/1a8F.$!1Q0sNSkrHF;7$$!1\"y>l qa'*y%!#<$!1u'fR;Kz*HF;7$$!1Hs1Te]%H#F.$!1J*)4)\\c[\"HF;7$$!1,^PIL'p-% F.$!1&3Z*QT/GFF;7$$!1])4:*G>;cF.$!1[D]ehO\\CF;7$$!1K#[2Ge1,(F.$!1<7\\1 (RY4#F;7$$!1R\"Re1\\!p\")F.$!17K'z`T5o\"F;7$$!1(>Ct.H/1*F.$!1#G>>E-]A \"F;7$$!1A`)>gjLm*F.$!1Y=v2k:5uF.7$$!1Ll?t4ik**F.$!1\\^D\"*z&**F.$\"1Rm6%4;/j#F.7$$!1x!y0<3Ok*F.$\"1)oNN0c>i(F.7$$!1KZe#fk z-*F.$\"1G&*3h-DX7F;7$$!1E@NUxhC\")F.$\"1F1Kh6y*p\"F;7$$!1\"pzmS%QbpF. $\"1(>\\jS+76#F;7$$!1$p%)=]L;b&F.$\"1B*>un8IY#F;7$$!1=4(4#o7bRF.$\"1nr Q)z`!QFF;7$$!1\\#*)R&[\"y@#F.$\"1n'R1\"*=1#HF;7$$!1!o\"Hxq^,SFX$\"1r`* e2k!**HF;7$$\"16Sx!Q@@V\"F.$\"1D:=as1oHF;7$$\"1A`%yXmA@$F.$\"1#ftS%[vH GF;7$$\"1l,$\\f^q([F.$\"1*R$[LaC$f#F;7$$\"17Oby/_qjF.$\"1K)*F.$!1sb4hK]Y_F.7$$\"1kg#eSz%f$*F .$!1+JnA#Ro,\"F;7$$\"1t\"4GzD;f)F.$!1l#)RqBq'[\"F;7$$\"1;&z/mHja(F.$!1 n`0q)42#>F;7$$\"13Gmr?C]iF.$!15.xhz=.BF;7$$\"1M,-6&*4SZF.$!1*f%)fXuqh# F;7$$\"10[lQ$*3jIF.$!1c%)4^blXGF;7$$\"1Rl.B#3fF\"F.$!1&)GGs:-vHF;7$$!1 0F)z%yYxbFX$!1MnM;wf'*HF;7$$!1-g'=)G/rBF.$!10h81?!*3HF;7$$!1Ll()=Y`)4% F.$!1g2ZPY'yr#F;7$$!1(GtK*=R!o&F.$!1N0'HMybV#F;7$$!1%oqOm&\\lqF.$!1AV( >oqz2#F;7$$!1:Dldq)H@)F.$!1DdKSSAi;F;-%'COLOURG6&%$RGBGF*F*F*-%(SCALIN GG6#%,CONSTRAINEDG" 2 239 286 286 2 0 1 0 2 6 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 177 237 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "shows its trajectory in phase s pace. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 " Here is one way to define functions that directly ret urn the values of the pendulum's angular displacement and its angular \+ velocity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "TSol := a -> subs(Sol(a), theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TSolG:6#%\"aG6\"6$%)operatorG%&arrowGF(-%%subsG 6$-%$SolG6#9$%&thetaGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dTSol := a -> subs(Sol(a), dtheta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dTSolG:6#%\"aG6\"6$%)operatorG%&arrowGF(-%%subsG6$-%$SolG6#9$ -%%diffG6$%&thetaG%\"tGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "TSol(0), dTSol(0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"\"\"\" \"!$!1Y&y0%pj$)H!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "These can also be used to produce graphs like t hose above and many others. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 " Here, " }{TEXT 346 4 "TSol" } {TEXT -1 128 " is used to estimate the pendulum's period as an alterna tive to evaluating the usual integral representation of this quantity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "4*fsolve('TSol'(t), t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I(G-9#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "Remember that this depends on the pen dulum's angular amplitude, in this case 1 " }{TEXT 347 6 "radian" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 " The following uses this brute-force method of \+ determining the period to produce a plot of period " }{TEXT 349 6 "ver sus" }{TEXT -1 61 " angular amplitude for the range of amplitudes from 0 to 2.5 " }{TEXT 348 7 "radians" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "for i fr om 1 to 25 do \n ICs := ic(theta,t=0,i/10),ic(dtheta,t=0,0):\n Sol : = dsolve(\{DE,ICs\},theta,numeric):\n P.i := [i/10, 4*fsolve('TSol'(t ), t=0.41..1.0)]:\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "P 0 := [0,P1[2]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot([se q(P.i, i=0..25)], color=black, view=[0..3, 0..4], title=`Period vs. Am plitude`);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#7<7$\"\"!$\"1++ +!3X$3?!#:7$$\"1+++++++5!#;F)7$$\"1+++++++?F/$\"1+++\"=>@,#F+7$$\"1+++ ++++IF/$\"1+++m$Q%=?F+7$$\"1+++++++SF/$\"1+++hnMF?F+7$$\"1+++++++]F/$ \"1+++#Q2*Q?F+7$$\"1+++++++gF/$\"1+++TM?`?F+7$$\"1+++++++qF/$\"1+++:,M q?F+7$$\"1+++++++!)F/$\"1+++voW!4#F+7$$\"1+++++++!*F/$\"1+++m/o8@F+7$$ \"\"\"F($\"1+++B(G-9#F+7$$\"1+++++++6F+$\"1+++-aJq@F+7$$\"1+++++++7F+$ \"1+++Hj?/AF+7$$\"1+++++++8F+$\"1+++ss@UAF+7$$\"1+++++++9F+$\"1+++jSs% G#F+7$$\"1+++++++:F+$\"1+++od5DF+7$$\"1+++++++>F +$\"1+++fE8%e#F+7$$\"\"#F($\"1+++]2BnEF+7$$\"1+++++++@F+$\"1+++s52hFF+ 7$$\"1+++++++AF+$\"1+++2'4x'GF+7$$\"1+++++++BF+$\"1+++c-*)*)HF+7$$\"1+ ++++++CF+$\"1+++l)*QJJF+7$$\"1+++++++DF+$\"1+++?Mh(H$F+-%&TITLEG6#%5Pe riod~vs.~AmplitudeG-%'COLOURG6&%$RGBGF(F(F(-%%VIEWG6$;F($\"\"$F(;F($\" \"%F(" 2 199 188 188 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 -20773 48 0 0 0 0 0 0 }}}}{MARK "123 0 4" 3 }{VIEWOPTS 1 1 0 1 1 1803 }