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269 "" 0 "" {TEXT 631 9 "in \+ " }}{PARA 266 "" 0 "" {TEXT 630 48 "Electrical, Computer, and Systems \+ Engineering " }}{PARA 271 "" 0 "" {TEXT -1 13 "Using Maple V" } {TEXT 618 9 "Release 4" }}{PARA 19 "" 0 "" {TEXT 617 9 "in " }} {PARA 19 "" 0 "" {TEXT -1 51 "Electrical, Computer, and Systems Engine ering " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 " Index" }}{PARA 17 "" 0 "" {TEXT -1 65 " \+ Page No." }}{PARA 17 "" 0 "" {TEXT 619 10 "I. Basics" }} {PARA 17 "" 0 "" {TEXT -1 20 " 1 Introduction " }{TEXT 642 41 "... ................................... " }{TEXT -1 2 "3 " }}{PARA 17 " " 0 "" {TEXT -1 22 " 2 Getting Started " }}{PARA 17 "" 0 "" {TEXT -1 62 " 3 Maple's user interface ............................ 4 " }}{PARA 17 "" 0 "" {TEXT -1 62 " 4 Help ........................ ...................... 5" }}{PARA 17 "" 0 "" {TEXT -1 62 " 5 Basi c Rules ....................................... 6" }}{PARA 17 "" 0 "" {TEXT -1 62 " 6 Operators and Special Characters .............. .... 7" }}{PARA 17 "" 0 "" {TEXT -1 62 " 7 Constants ........... .............................. 8" }}{PARA 17 "" 0 "" {TEXT -1 14 " \+ 8 Packages" }}{PARA 17 "" 0 "" {TEXT -1 62 " 9 Function Index .. .................................. 9" }}{PARA 17 "" 0 "" {TEXT -1 62 " 10 Plotting .......................................... 10" }} {PARA 17 "" 0 "" {TEXT -1 3 " " }}{PARA 17 "" 0 "" {TEXT 623 19 "II. Math Functions" }}{PARA 17 "" 0 "" {TEXT -1 62 " 1 Symbolic and F loating Point Results ............... 15" }}{PARA 17 "" 0 "" {TEXT -1 62 " 2 Common Maple ...................................... 16 " }}{PARA 17 "" 0 "" {TEXT -1 38 " Function Operator, solve, assi gn" }}{PARA 264 "" 0 "" {TEXT 622 6 " " }{TEXT 627 4 "Trig" } {TEXT -1 48 " .............................................." }{TEXT 628 4 " 18" }}{PARA 17 "" 0 "" {TEXT -1 62 " 3 Complex Variables \+ ................................. 21" }}{PARA 17 "" 0 "" {TEXT -1 62 " 4 Calculus .......................................... 22" }} {PARA 17 "" 0 "" {TEXT 620 6 " " }{TEXT -1 15 "Differentiation" } }{PARA 17 "" 0 "" {TEXT -1 63 " Integration .................... ................... 23 " }}{PARA 17 "" 0 "" {TEXT -1 62 " 5 Matric es .......................................... 24" }}{PARA 17 "" 0 " " {TEXT 621 1 " " }{TEXT -1 61 " 6 Transforms ..................... ................... 27" }}{PARA 17 "" 0 "" {TEXT -1 25 " Laplace and Fourier" }}{PARA 17 "" 0 "" {TEXT -1 62 " Z Transform ..... .................................. 29" }}{PARA 17 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT 626 19 "III. Maple Examples" }}{PARA 17 "" 0 "" {TEXT -1 62 " 1 Laplace analysis of a circuit ..................... 30 " }}{PARA 17 "" 0 "" {TEXT -1 62 " 2 Final and Initial value ..... ...................... 32" }}{PARA 17 "" 0 "" {TEXT -1 26 " 3 Stat istical analysis" }}{PARA 17 "" 0 "" {TEXT -1 62 " Linear Regress ion with Maple ...................... 33" }}{PARA 17 "" 0 "" {TEXT -1 62 " 4 Frequency Responce ................................ 34 " }}{PARA 17 "" 0 "" {TEXT 624 4 " " }{TEXT -1 58 "5 Bode Plots .. ...................................... 37" }}{PARA 17 "" 0 "" {TEXT 625 4 " " }{TEXT -1 58 "6 Lissajous figures ...................... ........... 40" }}{PARA 17 "" 0 "" {TEXT -1 62 " 7 Convolution .. ..................................... 41" }}{PARA 17 "" 0 "" {TEXT -1 62 " 8 State Variable Analysis ........................... 42 " }}{PARA 17 "" 0 "" {TEXT -1 62 " 9 Robotics .................... ...................... 48" }}{PARA 17 "" 0 "" {TEXT -1 62 " 10 Elec tromagnetics .................................. 49" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 294 13 "Using Maple V" } {TEXT 501 9 "Release 4" }}{PARA 19 "" 0 "" {TEXT 499 9 "in " }} {PARA 19 "" 0 "" {TEXT 295 51 "Electrical, Computer, and Systems Engin eering " }}{PARA 19 "" 0 "" {TEXT 500 1 " " }}{PARA 4 "" 0 "" {TEXT 607 10 "I. Basics" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 " 1 I ntroduction" }}{PARA 0 "" 0 "" {TEXT -1 323 "Maple is a tool that can \+ be used to shed light on many of the difficult and mathematically comp lex concepts in ECSE. These notes include specific examples of Maple' s use to analyze and display the data, solve complex problems, and ca rry out transformations. These notes and worksheet are available on t he ECSE web page, " }{TEXT 643 32 "http://www.ecse.rpi.edu/Courses/" } {TEXT -1 474 ".\n\nThe distinctive feature of Maple is its ability to \+ compute, operate and process in symbolic form. Basic and advanced mat hematical operations are performed using symbolic names like x, y, t, \+ amps, volts, etc. Maple is easy to become familiar with and use. Its three dimensional graphic ability allows a problem to be viewed from \+ every side. Its ability to differentiate, integrate, solve systems of equations, and handle a matrix make it an ECSE student's best friend! " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 " 2 Getting Started" }}{PARA 0 "" 0 "" {TEXT -1 144 "\nMaple V \+ release 4.0 is available on the Rensselaer Computing Systems (RCS).\n \nTo start Maple:\n 1. Log into an RCS graphic workstation.\n 2. Type " }{TEXT 261 6 " maple" }{TEXT -1 415 " at the command prompt, or sel ect Maple from the RCS Applications Menu.\n\nA window will appear with a Maple prompt. The controls in the Maple window are accessed via me nu and tool bar using a mouse. Some controls have keyboard equivalent s, which are short cuts to the menu selection, and are noted in the Ma ple menu next to the command.\n\n\nIf you have problems starting Maple , refer to ITS memo 108, on the RCS. \n\n" }{TEXT 278 48 "/caupnm/doc/ its.umca/itslOS.txt or /itslOS.dvi \n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 3 Maple's user interface " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT 523 1 "1" }{TEXT -1 79 ". In this text, \+ a menu command will be referred to as it appears in the menu. " } {TEXT 260 1 "F" }{TEXT 259 4 "ile," }}{PARA 14 "" 0 "" {TEXT 257 1 "H " }{TEXT 258 4 "elp," }{TEXT -1 5 " and " }{TEXT 256 1 "F" }{TEXT 527 5 "ormat" }{TEXT -1 31 " are examples of menu commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT 524 1 "2" }{TEXT -1 54 ". \+ As in all Windows standard applications, typing the " }{TEXT 515 17 "u nderlined letter" }{TEXT -1 20 " while holding the " }{TEXT 406 5 "" }{TEXT -1 52 " key is the non-mouse method of selecting an item. \n" }}{PARA 14 "" 0 "" {TEXT 520 1 "3" }{TEXT 521 285 ". Move around t he worksheet with the arrow keys or jump to a location with the mouse \+ to edit command lines and text. Maple is simular to a word processor a nd conforms to most Windows application standards and has many short c ut commands. These and others are listed in the help file. " }}{PARA 14 "" 0 "" {TEXT 595 39 "Topic search \"editing\" found this list:" }} {PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT 23 2489 "EMACS VI Function \nMo de Mode Description \+ \n \+ \nCtrl-P k Recall Previous line \+ \nCtrl-N j Recall Next line \+ \nCtrl-B h Move cursor Backward \+ \nCtrl-F l Move cursor Forward \+ \nCtrl-A 0 Move cursor to beginning of line \+ \nCtrl-E $ Move cursor to End of line \+ \nCtrl-] % Move cursor to matching parenth esis, bracket, or brace \nCtrl-W w Move cursor to next Word \+ \nCtrl-Y b Move cursor to previous w ord \nCtrl-D x Delete character at cu rsor \nBackSp X Delete character to left of cursor and move left \nCtrl-K C Delete to end of line \nCtrl-G S Delete entire line \nCtrl-V Switch bet ween insert and replace modes \nCtrl-U U Undo ch anges made to line \nCtrl-R ? Sear ch for line containing what has been typed \nEscape E nter VI command mode \n i \+ Insert text before character \n \+ a Append text after character \n \+ I Insert text at beginning of line \n \+ A Append text at end of line \n \+ R Enter Replace mode \+ \n r Replace one character and return to VI mode \+ \n s Delete character and insert text \+ \n D Delete to end of line and remain in VI mode \+ \n f Search forward for character \+ \n F Search backward for character \+ \n ; Repeat last f or F search \+ \n , Repeat last f or F search in oppos ite direction \nCtrl-L Redraw line \+ \nCtrl-Space Complete command \+ \nCtrl-T Display matching commands " }}{PARA 14 "" 0 "" {TEXT -1 1 "\n" }{TEXT 525 1 "4" }{TEXT -1 17 ". \+ Zoom with the \"" }{TEXT 516 1 "X" }{TEXT -1 29 "\" buttons on the too l bar or " }{TEXT 517 7 "Ctrl-1 " }{TEXT -1 7 "through" }{TEXT 611 7 " Ctrl-6" }{TEXT -1 20 " are zoomfactors.\n \n" }{TEXT 526 1 "5" } {TEXT -1 2 ". " }{TEXT 518 40 "Clicking and holding on a Toolbar butto n" }{TEXT -1 61 " will identify its function at the bottom of the scre en. To " }{TEXT 519 33 "omit getting the actual operation" }{TEXT -1 60 ", drag the pointer off the button before releasing the hold." }} {PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 588 15 "File Ma nagement" }}{PARA 14 "" 0 "" {TEXT 544 1 "S" }{TEXT 546 3 "ave" } {TEXT 545 35 " a Maple worksheet by clicking the " }{TEXT 547 11 "Disk button" }{TEXT 548 63 " on the tool bar (this button looks like a dis k), or selecting:" }}{PARA 14 "" 0 "" {TEXT 549 1 "S" }{TEXT 550 3 "av e" }{TEXT 555 4 " or " }{TEXT 553 4 "Save" }{TEXT 551 1 "A" }{TEXT 552 1 "s" }{TEXT 554 8 " in the " }{TEXT 556 1 "F" }{TEXT 558 3 "ile" }{TEXT 557 11 " menu.\n \n" }{TEXT 559 1 "O" }{TEXT 567 3 "pen" } {TEXT 560 26 " a worksheet by selecting " }{TEXT 561 1 "O" }{TEXT 566 3 "pen" }{TEXT 562 8 " in the " }{TEXT 563 1 "F" }{TEXT 565 3 "ile" } {TEXT 564 40 " menu and selecting the desired file. \n\n" }{TEXT 568 1 "C" }{TEXT 569 4 "lose" }{TEXT 570 29 " a worksheet by clicking the \+ " }{TEXT 571 8 "X button" }{TEXT 572 111 " in the upper right corner o f its window.\n(If the sheet is not saved, Maple will ask about your \+ intentions.)\n\n" }{TEXT 575 1 "Q" }{TEXT 577 3 "uit" }{TEXT 576 23 " \+ Maple by clicking the " }{TEXT 573 8 "X button" }{TEXT 574 121 " in th e upper right corner of the main Maple window.\n(If the sheet is not s aved, Maple will ask about your intentions.)\n" }}{PARA 4 "" 0 "" {TEXT 589 8 "Printing" }}{PARA 14 "" 0 "" {TEXT -1 37 "Print the works heet by selecting the " }{TEXT 590 14 "printer button" }{TEXT -1 28 " \+ on the tool bar, or select " }{TEXT 578 1 "P" }{TEXT 580 4 "rint" } {TEXT -1 8 " in the " }{TEXT 579 1 "F" }{TEXT 581 3 "ile" }{TEXT -1 11 " menu.\nThe " }{TEXT 583 3 "Pre" }{TEXT 582 1 "v" }{TEXT 584 3 "ie w" }{TEXT -1 18 " selection in the " }{TEXT 585 1 "F" }{TEXT 586 3 "il e" }{TEXT -1 85 " menu lets you see what your printout will look like \+ before you waste time and paper." }}{PARA 0 "" 0 "" {TEXT 587 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " \+ 4 Help" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT 528 1 "1" }{TEXT -1 62 ". For Help with a Maple command, highlight the com mand, select" }{TEXT 280 1 " " }{TEXT 279 1 "H" }{TEXT 281 3 "elp" } {TEXT -1 113 ", and then follow the menu. \n(If Maple knows the word y ou highlighted, help for it will be the second item in the" }}{PARA 14 "" 0 "" {TEXT -1 1 " " }{TEXT 282 1 "H" }{TEXT 283 3 "elp" }{TEXT -1 10 " menu.) \n\n" }{TEXT 529 1 "2" }{TEXT -1 52 ". Help with a comm and can also be obtained using the" }{TEXT 284 7 " \"help\"" }{TEXT -1 4 " or " }{TEXT 285 4 "\"?\" " }{TEXT -1 47 "commands followed by t he command in question:\n>" }{TEXT 419 6 "help (" }{TEXT -1 7 "command " }{TEXT 286 33 " ); Note: the \" ; \" is required!" }{TEXT -1 5 "\no r\n>" }{TEXT 420 1 "?" }{TEXT -1 8 " command" }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 68 "Examples of the command \+ in use can be viewed by:\n>example(command); " }{TEXT 287 30 "Note: th e \" ; \" is required!\n\n" }{TEXT -1 1 "\n" }{TEXT 530 1 "3" }{TEXT -1 47 ". An index of help topics can be viewed under " }{TEXT 288 1 " C" }{TEXT 291 7 "ontents" }{TEXT -1 22 ", or search for a key " } {TEXT 289 1 "T" }{TEXT 290 4 "opic" }{TEXT -1 8 " in the " }{TEXT 421 1 "H" }{TEXT 422 3 "elp" }{TEXT -1 32 " menu at the top of the window. \n" }}{PARA 14 "" 0 "" {TEXT 531 1 "4" }{TEXT -1 56 ". A very good Map le tutorial is available on the Net at:" }}{PARA 14 "" 0 "" {TEXT 522 55 "http://www.math.rpi.edu/www/Materials/maple.links.html." }{TEXT -1 1 "\n" }}{PARA 14 "" 0 "" {TEXT -1 71 "(It will give you a basic fe el for Maple in a short time, well spent.)\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 " 5 Basic Rules" }}{PARA 14 "" 0 "" {TEXT 293 1 "1" } {TEXT 330 43 ". All Maple commands are entered after the " }{TEXT 296 1 ">" }{TEXT 297 51 " prompt and must be terminated with either a colo n " }{TEXT 298 4 "(:) " }{TEXT 299 13 "or semicolon " }{TEXT 300 4 "(; )." }}{PARA 14 "" 0 "" {TEXT 292 2 "* " }{TEXT 301 13 "semicolon (;)" }{TEXT 302 49 " - Maple will display the output of the line.\n* " } {TEXT 303 10 "colon (:) " }{TEXT 304 28 " - No output is displayed. \+ " }{TEXT 305 24 "This can reduce clutter!" }{TEXT 306 2 "\n\n" }{TEXT 538 1 "2" }{TEXT 539 10 ". Use the " }{TEXT 536 14 "" } {TEXT 537 51 " anywhere on a command line to execute that line. " } {TEXT -1 4 "The " }{TEXT 533 6 " " }{TEXT -1 4 "key " }{TEXT 532 67 "is used to jump to the next command prompt, without an execution. \+ " }{TEXT 534 11 "" }{TEXT 535 34 " will jump to the previo us prompt." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT 331 1 "3" }{TEXT 332 87 ". Maple output is centered, in italics, and a ppears as you would have handwritten it.\n\n" }{TEXT 540 1 "4" }{TEXT 541 34 ". Add a new command line with the " }{TEXT 307 1 "[" }{TEXT 309 1 ">" }{TEXT 308 51 " button in the tool bar at the top of the win dow.\n\n" }{TEXT 542 1 "5" }{TEXT 543 158 ". If Maple doesn't like or \+ understand your command line, it will print a cryptic error message. \+ Examples of the two most often seen are:\n> H:=(s*(s-3+I*20))\n" } {TEXT 409 50 "Warning, incomplete statement or missing semicolon" }} {PARA 14 "" 0 "" {TEXT -1 37 "(Yes, the semicolon is missing here.)" } }{PARA 14 "" 0 "" {TEXT 410 23 "or\n> H:=(s*(s-3+I*20);\n" }{TEXT 411 29 "Syntax error, `;` unexpected\n" }{TEXT 412 1 "\n" }{TEXT 311 72 "N ote: This last one has its semicolon but lacks a closing parenthesis! \+ " }}{PARA 14 "" 0 "" {TEXT 310 67 " Maple wants to know why you told it you were finished. \n" }}{PARA 14 "" 0 "" {TEXT 644 1 "6 " }{TEXT 645 41 ". Remaining text on a command line after " }{TEXT 312 4 "\"#\"," }{TEXT 313 60 " is ignored by Maple as user comments.\n > H:=(s*(s-3+I*20)): " }{TEXT 407 50 "# These words will be ignored; \+ they follow the #." }{TEXT 408 1 "\n" }}{PARA 14 "" 0 "" {TEXT 646 1 " 7" }{TEXT 647 110 ". The maple leaf button on the tool bar toggles bet ween Maple command lines and text, which is also ignored..\n" }}{PARA 14 "" 0 "" {TEXT 329 1 "8" }{TEXT 333 121 ". Maple retains a buffer of the outputs from the last three command executions. These result ar e available for use as " }{TEXT 314 1 "\"" }{TEXT 315 2 ", " }{TEXT 316 2 "\"\"" }{TEXT 317 5 " and " }{TEXT 318 3 "\"\"\"" }{TEXT 319 2 " .\n" }{TEXT 322 37 "NOTE: The order in this buffer is as " }{TEXT 320 8 "executed" }{TEXT 321 17 ", not as written!" }{TEXT 323 69 " If you go back and execute a line again, out of order, you may get " }{TEXT 324 5 "burnt" }{TEXT 325 26 ", or at least bad answers." }}{PARA 14 " " 0 "" {TEXT 326 38 "Try it; execute the second line twice!" }{TEXT 327 3 "\n \n" }{TEXT 413 7 "> 5*10;" }}{PARA 14 "" 0 "" {TEXT 328 18 " >(\")/10;\n> (\"\")/2;" }{TEXT 414 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 " 6 Operators and Special Characters" }}{PARA 258 "" 0 " " {TEXT 344 3 "+,-" }{TEXT -1 28 " Add, subtract\n" } {TEXT 343 6 "/,*,&*" }{TEXT -1 54 " Division, Scalar and Mat rix Multiplication\n" }{TEXT 342 8 "^ or ** " }{TEXT -1 26 " R aise to a power\n" }{TEXT 341 6 "=,>,< " }{TEXT -1 42 " Equa l, Greater than, Less than\n" }{TEXT 340 5 ">=,<=" }{TEXT -1 44 " \+ Greater or equal, Less or equal\n" }{TEXT 339 2 "<>" }{TEXT -1 25 " Not equal\n" }{TEXT 338 1 "\"" }{TEXT -1 47 " \+ Re-use the command one line up\n" }{TEXT 337 3 "\"\" " } {TEXT -1 50 " Re-use the command from two lines up" }} {PARA 259 "" 0 "" {TEXT 345 4 "\"\"\" " }{TEXT -1 52 " Re- use the command from three lines up\n" }{TEXT 336 2 "# " }{TEXT -1 41 " Starts an ignored comment\n" }{TEXT 335 3 "\\ " } {TEXT -1 59 " Continuation symbol for multiple input line s\n" }{TEXT 347 1 "!" }{TEXT -1 25 " Factorial" }} {PARA 260 "" 0 "" {TEXT 610 2 "->" }{TEXT -1 71 " Makes \+ a general procedure of the function it is used in." }}{PARA 261 "" 0 " " {MPLTEXT 0 21 35 " f:= x -> 3*x + 5; " }{TEXT -1 25 "see examples section II-2" }}{PARA 262 "" 0 "" {TEXT 334 2 ":=" } {TEXT -1 95 " Assigns the preceding variable name to the stuff that follows\n\n " }{TEXT 612 18 "H:=(s*(s-3+I* 20)):" }{TEXT -1 47 " # Here \"H\" is the variable.\n\n \+ " }{TEXT 613 11 "Fred:=(2I):" }{TEXT -1 67 " # Fred is assigned 2*(s quareroot of (-1)).\n\n Thus " }{TEXT 614 6 "H*Fred" } {TEXT -1 4 " is " }{TEXT 615 19 "(s*(s-3+I*20))*(2I)" }{TEXT -1 4 ". \+ \n\n" }{TEXT 346 46 "Unassign a variable by assigning it to itself:" } {TEXT -1 24 "\n >Fred " }{TEXT 371 4 ":= '" }{TEXT -1 4 "Fred" }{TEXT 372 1 "'" }{TEXT -1 4 "; # " }{TEXT 424 37 "Note the s ingle quotes around 'Fred'." }{TEXT -1 89 "\n Maple fo rgets the value of Fred, and you may now use Fred as you like.\n" } {TEXT 423 33 " " }}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 13 " 7 Constants" }}{PARA 263 "" 0 "" {TEXT 348 11 "A ll Reals " }{TEXT -1 50 " All Real numbers are considered const ants.\n" }{TEXT 349 2 "Pi" }{TEXT -1 29 " 3.14159..... \+ " }}{PARA 263 "" 0 "" {TEXT -1 18 " " }{TEXT 596 5 "N ote:" }{TEXT -1 22 " The non-capitalized \"" }{TEXT 597 2 "pi" }{TEXT -1 23 "\" is the Greek letter \"" }{XPPEDIT 598 0 "pi" "I#piG6\"" } {TEXT -1 3 ".\"\n" }}{PARA 263 "" 0 "" {TEXT 350 11 "infinity " } {TEXT -1 47 " The upper limit of Maple's imagination \n" }{TEXT 351 1 "E" }{TEXT -1 41 " Base of the natural log\n" } {TEXT 352 1 "I" }{TEXT -1 102 " Squareroot of (-1)\n \+ (\"j\" is not \"I\", but can be made so by > j:= ;)\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 " 8 Packages" }}{PARA 0 "" 0 "" {TEXT -1 320 "Basic arithmetic and calculus commands are directly available in Maple. In order to use commands not built in, a Package , Library, or the specific command from a package or library has to be loaded.\n\nA command or commands from a Package are loaded by specify ing the package and the desired commands as arguments to the " }{TEXT 616 15 "with() command." }{TEXT -1 90 " Or, similarly, the whole pack age can be loaded by neglecting to specify any commands.\n\n>" }{TEXT 416 15 " with(inttrans)" }{TEXT -1 63 "; Once the package is loaded t he command is available for use." }}{PARA 0 "" 0 "" {TEXT -1 83 "(Mapl e will list all the functions loaded from the package, unless suppress ed by a " }{TEXT 415 4 "(:) " }{TEXT -1 5 ")\n\n> " }{TEXT 353 7 "lapl ace" }{TEXT -1 77 "(exp(-t),t,s); Now that inttrans is loaded, the la place command can be used," }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 354 2 "or" }{TEXT -1 2 "\n\n" }{TEXT 355 26 "Load commands individuall y" }{TEXT -1 119 " as needed. Doing this conserves memory, and is hel pful when using only a few commands from that package.\n\n> inttrans[ " }{TEXT 356 7 "laplace" }{TEXT -1 47 "](exp(-t),t,s);\n\nPackages you will be using: \n\n" }{TEXT 357 6 "linalg" }{TEXT -1 60 " \+ Linear Algebra (matrix and vector operations)\n" }{TEXT 358 7 "stude nt" }{TEXT -1 63 " more calculus commands (integration by pa rts, etc.)\n" }{TEXT 359 8 "inttrans" }{TEXT -1 40 " Laplace \+ and Fourier transforms" }}{PARA 0 "" 0 "" {TEXT 502 5 "stats" }{TEXT -1 32 " Linear regression" }}{PARA 0 "" 0 "" {TEXT -1 74 " \nTo get a complete list of packages:\n\n> ? packages;\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " 9 Function Ind ex" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 460 18 "Built in Function s" }}{PARA 17 "" 0 "" {TEXT 450 21 "Function " }{TEXT -1 7 " " }{TEXT 599 11 "Description" }}{PARA 17 "" 0 "" {TEXT 468 5 "abs()" }{TEXT -1 36 " Absolute Value" }}{PARA 17 "" 0 "" {TEXT 467 20 "array() " }{TEXT -1 26 " De fine a data array" }}{PARA 17 "" 0 "" {TEXT 482 9 "collect()" }{TEXT -1 64 " Collects like powers of the specified variabl e" }}{PARA 17 "" 0 "" {TEXT 481 9 "combine()" }{TEXT -1 47 " \+ Merges terms of an expression" }}{PARA 17 "" 0 "" {TEXT 478 7 "evalf()" }{TEXT -1 68 " Evaluates a function for a floating point number" }}{PARA 17 "" 0 "" {TEXT 479 7 "evalm()" } {TEXT -1 63 " Evaluates an expression involving mat rices " }}{PARA 17 "" 0 "" {TEXT 451 5 "exp()" }{TEXT -1 26 " \+ e^x\n" }{TEXT 485 8 "expand()" }{TEXT -1 77 " \+ Distributes multiplication, expands trig and exponentials\n" } {TEXT 486 8 "factor()" }{TEXT -1 40 " Factors a poly nomial " }}{PARA 17 "" 0 "" {TEXT 469 6 "frac()" }{TEXT -1 36 " \+ Fractional Part" }}{PARA 17 "" 0 "" {TEXT 471 5 "gcd()" }{TEXT -1 46 " Greatest Common Devisor\n" }{TEXT 470 5 "lcm()" }{TEXT -1 43 " Least Common Multipl e" }}{PARA 17 "" 0 "" {TEXT 452 13 "ln() or log()" }{TEXT -1 19 " \+ ln x\n" }{TEXT 453 8 "logl0() " }{TEXT -1 28 " \+ log10(x)\n" }{TEXT 454 8 "1og[b]()" }{TEXT -1 28 " \+ logb (x)\n" }{TEXT 483 9 "normal() " }{TEXT -1 70 " \+ Converts an expression to the \"factored normal form\"" }}{PARA 17 " " 0 "" {TEXT 455 7 "round()" }{TEXT -1 45 " Round t o nearest Integer\n" }{TEXT 484 6 "sort()" }{TEXT -1 60 " \+ Arranges a polynomial in order of power" }}{PARA 17 "" 0 "" {TEXT 456 7 "trunc()" }{TEXT -1 44 " Remove Fractio nal Part\n\n" }{TEXT 600 27 "sin(),cos(),tan()... " }{TEXT -1 4 "Trig" }}{PARA 17 "" 0 "" {TEXT 457 19 "arcsin(), arccos()," }}{PARA 17 "" 0 "" {TEXT 591 11 "arctan()..." }}{PARA 17 "" 0 "" {TEXT 592 8 " Inverse " }{TEXT -1 7 " \n" }}{PARA 17 "" 0 "" {TEXT 458 11 "Heav iside()" }{TEXT -1 31 " Unit Step U(t)\n" }{TEXT 459 7 "Dirac()" }{TEXT -1 32 " Unit impulse" }}{PARA 17 " " 0 "" {TEXT 472 7 "ztran()" }{TEXT -1 54 " Find th e Z transform of a function" }}{PARA 17 "" 0 "" {TEXT 473 11 "invztran ()" }{TEXT -1 54 " Find the function from its Z transf orm" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT -1 1 " \n" }{TEXT 461 23 "Linear Algebra (linalg)" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{TEXT 463 12 "leastsqrs() \+ " }{TEXT -1 67 " Find the leastsquare, actually does line ar regression" }}{PARA 17 "" 0 "" {TEXT -1 39 " \+ \"best line\" " }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{TEXT 464 10 "l insolve()" }{TEXT -1 43 " Solve a system of equations" }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{TEXT 465 26 "matrix() \+ " }{TEXT -1 15 "Define a matrix" }}{PARA 17 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 466 26 "vector() " }{TEXT -1 29 "Define a ro w or colomb vector" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 " " {TEXT 462 21 "Transforms (inttrans)" }}{PARA 17 "" 0 "" {TEXT 474 10 "fourier( )" }{TEXT -1 57 " Find the fourier transf orm of a function" }}{PARA 17 "" 0 "" {TEXT 475 13 "invfourier( )" } {TEXT -1 58 " Find the function from its Fourier transfor m" }}{PARA 17 "" 0 "" {TEXT 476 10 "laplace( )" }{TEXT -1 57 " \+ Find the Laplace transform of a function" }}{PARA 17 "" 0 "" {TEXT 477 13 "invlaplace( )" }{TEXT -1 58 " Find the func tion from its Laplace transform" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }} {PARA 17 "" 0 "" {TEXT 480 16 "Statistic (stat)" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT 593 8 "describe" }{TEXT -1 72 " Sub package: mean, standarddeviation, variance, ec t." }}{PARA 17 "" 0 "" {TEXT -1 1 " " }{TEXT 594 4 "fit " }{TEXT -1 52 " Used for \"best curve\" analysis" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "10 Plotting" }}{PARA 0 "" 0 "" {TEXT -1 107 "Maple offers a wide variety of plot output options ranging fro m simple 2d x-y plots to complex 3d figures.\n" }}{PARA 0 "" 0 "" {TEXT 275 5 "NOTE:" }{TEXT -1 17 " Maple help for " }{TEXT 360 4 "plo t" }{TEXT -1 5 " and " }{TEXT 361 6 "plot3d" }{TEXT -1 36 " is excelle nt, with many examples. " }}{PARA 0 "" 0 "" {TEXT -1 127 "\nThe plot \+ will appear in either a separate graphic window or in the worksheet it self (default), depending on your selection in " }{TEXT 364 1 "P" } {TEXT 365 11 "lot display" }{TEXT -1 11 " under the " }{TEXT 362 1 "O " }{TEXT 363 6 "ptions" }{TEXT -1 252 " menu. The plot can be modifie d at this point in a number of ways. Scale, axis, point of view, and \+ shading, are all changeable either from the original command line or g raphics menu and tool bar, both of which appear after highlighting the plot. \n\n\n" }{TEXT 276 5 "NOTE:" }}{PARA 14 "" 0 "" {TEXT 601 1 " \+ " }{TEXT -1 1 "\n" }{TEXT 366 13 "Point of view" }{TEXT -1 93 " (3d p lots) can be changed by clicking in the plot area and dragging the sha pe of the plot.\n" }}{PARA 14 "" 0 "" {TEXT -1 1 " " }{TEXT 367 9 "Plo t size" }{TEXT -1 175 " can be changed by dragging the edge of the pl ot frame, which appears when you click on the plot. Right clicking on the plot brings up a quick menu for modifying the plot.\n\n" }}{PARA 4 "" 0 "" {TEXT 270 8 "plot( ) " }}{PARA 0 "" 0 "" {TEXT -1 5 "\nThe \+ " }{TEXT 369 12 "general form" }{TEXT -1 34 " of the command for a 2d \+ plot is:\n" }{TEXT 417 45 ">plot(\{f(x),g(x),h(x)\},x=x1..x2,style=Sty le);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "T he above command line produces multiple plots of the functions f,g,h.. . in the range xl to x2." }}{PARA 0 "" 0 "" {TEXT 370 5 "Style" } {TEXT -1 26 " = Line, Patch, or Point.\n" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 274 19 "Examples of 2dplots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 271 7 "xy plot" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(cos(x),x=-Pi..Pi); " }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7en7$$!1+++aEfTJ!#:$!\"\"\"\"!7$$!1%*=fu[ 6tIF*$!1z^UHJcw**!#;7$$!1*y$=&4PY+$F*$!1Q6_uBO1**F37$$!1%\\pa%\\0XHF*$ !1JnEMb[2)*F37$$!1)>bdzsa)GF*$!1&Gi)*4-Qn*F37$$!1GF/y+Y^FF*$!1'4@/['e[ #*F37$$!1Hk/P!flh#F*$!1!*H&3#42`')F37$$!1+*\\r:*H#[#F*$!19yQN?D/zF37$$ !1/\\-iI#yN#F*$!1A_`rTd#3(F37$$!1'yfaFN*GAF*$!10Yc\\dX;hF37$$!1c#=AMRc 4#F*$!1pt&)f:x5]F37$$!1W^s!*3xi>F*$!1y !G'QDDF37$$!1GPC&**=dq\"F*$!16*zm#o8X8F37$$!1?#*>q:F*$\"1JX(>nl5(f! #>7$$!1tQ#z)H7M9F*$\"1#ox^N#[i8F37$$!1Q1#QK))HI\"F*$\"178SY@=YEF37$$!1 l&*)e[0R=\"F*$\"13bEF*3Jx$F37$$!1R3ltWIU5F*$\"1!3!3q_JU]F37$$!1LG\\C%) \\B#*F3$\"1?m3t%*[RgF37$$!17Nf'*39GyF3$\"1\\6(QM;$*3(F37$$!1@%o,08If'F 3$\"1+)RI?/U!zF37$$!1TsSX?)yB&F3$\"1u&ext1$f')F37$$!1utIooZZRF3$\"1)*) e:FO4B*F37$$!1KhRQW2,EF3$\"1?*[$3Nij'*F37$$!1f;Rsb'G)>F3$\"1'e(>va0/)* F37$$!1'=(Q1nlk8F3$\"1@*e)=+.2**F37$$!1>]%GF\\$yp!#<$\"1D!z+?hc(**F37$ $!1m<==[J,J!#=$\"1Q0F4>&*****F37$$\"1,-fMFF;mF_t$\"1qqy]/7y**F37$$\"1e )*4&pnUN\"F3$\"1&)oF>!Q%3**F37$$\"1u'z#=K@d>F3$\"1)=%3jk24)*F37$$\"1\" \\f9ue,c#F3$\"17BHoa1u'*F37$$\"1YE*z5aD'QF3$\"1%\\yX@iKE*F37$$\"10(*4h f03_F3$\"1z$RT8'=u')F37$$\"1APQDqOClF3$\"1!RqPxng%zF37$$\"1'RKNmlzz(F3 $\"1Lc(*Rfc5rF37$$\"1GId3)z?@*F3$\"1bWQLke[gF37$$\"1[fJECF[5F*$\"1>*3= U)o!*\\F37$$\"1*)e?V3%R=\"F*$\"1E&>b[\"ysPF37$$\"1!o*>$fwoI\"F*$\"17DD &el'3EF37$$\"1_'H&RyFT9F*$\"1A*3$znc\"H\"F37$$\"11@>Q*Qxc\"F*$\"1>GdRQ QdIFet7$$\"1_OSCs#**p\"F*$!1:Vc\\Ks(G\"F37$$\"1pV*fVl\"H=F*$!1>!QF*$!1#fpYP!GAF*$!1?2C'R]$4hF37$$\"1UTC#>&>gBF*$!1K'yNQ+$*4(F37$$\"1c_,@Wj \"[#F*$!1Zp?\"Ry,!zF37$$\"1*)of\"[<3i#F*$!1X\"pT^PVn)F37$$\"18W>qWIXFF *$!1ZCj^4+D#*F37$$\"1&QkXcL!yGF*$!1-U:o))oa'*F37$$\"1Ke9/HbTHF*$!14br+ oe+)*F37$$\"1zssVA20IF*$!1,tSNb&p!**F37$$\"1SO')[CLtIF*$!1_8!pw6n(**F3 7$$\"1+++aEfTJF*F+-%'COLOURG6&%$RGBG$\"#5F,F-F--%+AXESLABELSG6$%\"xG%! G-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFc^l%(DEFAULTG" 2 364 251 251 2 0 1 0 2 9 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 0 186 227 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 272 11 "polar plots" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "s := t->100/(100+(t-Pi/2)^8):\nr := t->s (t)*(2-sin(7*t)-cos(30*t)/2):\nplot([r(t),t,t=-Pi/2..3/2*Pi],numpoints =2000,coords=polar);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7\\hr 7$$!1&otrIu&3K!#F$!14!y&G*oVc\"!#<7$$\"1-tN+w()z^!#?$!1(R&3Qi5v:F-7$$ \"1@&f\"oO)=s*F1$!1T*QHm'y!e\"F-7$$\"1WTO3+/$[\"!#>$!1][i%znIe\"F-7$$ \"1xN+(\\iK*>F<$!1WV02%y4e\"F-7$$\"1!*G7pF#H\\#F<$!1GL&f5#eu:F-7$$\"1^ ,q=LXXHF<$!1Tf2G2*[c\"F-7$$\"1f([/!*)))*R$F<$!1'y)=I&o6b\"F-7$$\"1I:Ey \"H:&QF<$!1]3>o\"3K`\"F-7$$\"1]wXIK>!G%F<$!1'\\O_)zr6:F-7$$\"1&pT?x%*f p%F<$!1%y=gF<$!1/[88`j i8F-7$$\"1e283-8fiF<$!1zyX-C_I8F-7$$\"1QFY7>'\\^'F<$!1$3dg1[8H\"F-7$$ \"1#*RGalo3nF<$!1#pd\"=eud7F-7$$\"1()G0#of+\"pF<$!1s(GH(Hr=7F-7$$\"1'y 24d^'pqF<$!1ou%Qy#o%=\"F-7$$\"1]GWdk_GsF<$!1LM;9([%[6F-7$$\"1Yd\\heWot F<$!1Om7&>Mb6\"F-7$$\"1#eZwrN!3vF<$!1L6NXrR$3\"F-7$$\"1(3U0eydj(F<$!1f CNM)[j0\"F-7$$\"1U_%G5Z%zxF<$!1Fd_.%H..\"F-7$$\"1bS'4a1G%zF<$!1aO-*HBt +\"F-7$$\"1&zMdX![.\")F<$!1D*pp=52\"**!#=7$$\"1Mn#*\\&=PI)F<$!1B*y0>%) *y(*F\\t7$$\"1T5$>+\\![&)F<$!1)ea&Q+3&p*F\\t7$$\"1j&z*)*=jK))F<$!1-a\" >:d%o'*F\\t7$$\"1)Qel(*y\"f\"*F<$!136]t>\"F\\t$!1rc y;-n$3\"F-7$$\"1^Gbc[8y7F\\t$!1'Gzjvla7\"F-7$$\"1]aa73ht8F\\t$!1F3hZhA w6F-7$$\"1L/(=#H\\y9F\\t$!1`n5.z\"GB\"F-7$$\"1qg*o'*G6g\"F\\t$!1i%GkHc $*H\"F-7$$\"1A$\\=5g@t\"F\\t$!1$eizC_.P\"F-7$$\"1Rc6&)Qpz=F\\t$!1Ok9Lt w\\9F-7$$\"18!R$>Z%)R?F\\t$!1@nWI\"p]`\"F-7$$\"1;Q3S;M*>#F\\t$!1`u6c)H *=;F-7$$\"1;#)H(fwnR#F\\t$!1`D(o0>6s\"F-7$$\"1![#pcFf'e#F\\t$!1Y6p]ij< =F-7$$\"1M07RHg-GF\\t$!1b\\w!\\9a#>F-7$$\"1(G$)Gb'GAIF\\t$!1.FZ1c\"G.# F-7$$\"11bV'o<1G$F\\t$!1U-\"\\(RLc@F-7$$\"1BS`^K4;NF\\t$!1h&>*yoQmAF-7 $$\"1$3sA5Sfz$F\\t$!1n0w%RKTR#F-7$$\"1,/d73$>1%F\\t$!1T\\W3Ae7DF-7$$\" 1_d4@&**QO%F\\t$!1EYAB9lVEF-7$$\"1$3YFVfGj%F\\t$!1K;narUdFF-7$$\"1(3&G mWfZ\\F\\t$!1!)eqF25()GF-7$$\"1zr$*4I(*e_F\\t$!1%**e6*4y6IF-7$$\"1Z(eL pAhd&F\\t$!1Xxfj<8NJF-7$$\"1*pd,kYn*eF\\t$!1!*Q_B!HhD$F-7$$\"1$zZ6Wh/!*=r$F-7$$\"1\"eWIjiB^(F\\ t$!1u)eA%4W3QF-7$$\"1G;I8&[[%yF\\t$!1=QWrHk4RF-7$$\"1U9Jdd)z:)F\\t$!1x \"\\n'*>2+%F-7$$\"1]n_Bi#QY)F\\t$!1\\V,bR[&3%F-7$$\"1#e]3!R&\\x)F\\t$! 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The first is a collection of data points, and the second is a line that could have been calculated as \"" }{TEXT 425 9 "best line" }{TEXT -1 107 "\". T he style of each plot in this graph is set in this command line as is \+ the color assigned to the data. " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 268 9 "plot3d()\n" }{TEXT -1 4 "The " }{TEXT 368 12 "general for m" }{TEXT -1 35 " of the command for a 3d plot is:\n\n" }{TEXT 418 50 ">plot3d(\{f(x,y),g(x,y),h(x,y)\},x=x1..x2,y=y1..y2);" }{TEXT -1 84 " \+ \nThis produces multiple plots of the functions f,g,h... in the range \+ xl-x2,y1-y2. " }{TEXT 269 34 "Style, color, axis, and projection" } {TEXT -1 170 " options for 3d plots can be entered in the command line , or selected from the main graphic menu and graphics tool bar that ap pear after clicking on the completed graph.\n" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 277 19 "Examples of 3dplots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "H:=(s*(s-2+I*5)*(s-2-I*5)/((s+5)*(s-1+I*15)* (s-1-I*15)));" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s:= sigma+I*omega;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot3d(abs(H), si gma=-10..5,omega=-25..25, view=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG*.%\"sG\"\"\",(F&F'!\"#F'%\"IG\"\"&F',(F&F'F)F'F*!\"&F',&F&F' F+F'!\"\",(F&F'F/F'F*\"#:F/,(F&F'F/F'F*!#:F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&%&sigmaG\"\"\"*&%\"IGF'%&omegaGF'F'" }}{PARA 13 "" 1 "" {INLPLOT "6%-%%GRIDG6%;$!#5\"\"!$\"\"&F);$!#DF)$\"#DF)7;7;$ \"+M=b!H\"!\"*$\"+-&\\aI\"F5$\"+1Y$yI\"F5$\"+9u1*G\"F5$\"+B?]T7F5$\"+% *>Tk6F5$\"+P2go5F5$\"+EDzR(*F($\"+b#z'=!*F($\"+U`c$p)F($\"+Xjbh))F($\" +3P%QT*F($\"+Ehyo(*F($\"+@P%QT*F($\"+^jbh))F(FF$\"+X#z'=!*F($\"+6DzR(* F($\"+M2go5F5$\"+$*>Tk6F5$\"+A?]T7F5F:F8$\"+/&\\aI\"F5$\"+N=b!H\"F57;$ \"+3A?/8F5$\"++tXB8F5$\"+XK0I8F5$\"+)HONJ\"F5$\"+WGrj7F5$\"+Q%>$y6F5$ \"+)3c'p5F5$\"+$H\\ch*F($\"+w^c\"z)F($\"+`)zYU)F($\"+$)**fa')F($\"+`l8 =%*F($\"+y,yX**F($\"+ul8=%*F($\"+*)**fa')F($\"+[)zYU)F($\"+m^c\"z)F($ \"+x#\\ch*F($\"+'3c'p5F5$\"+P%>$y6F5$\"+UGrj7F5$\"+(HONJ\"F5$\"+VK0I8F 5F]oF[o7;$\"+-n[=8F5$\"+)GaGM\"F5$\"+^?wa8F5$\"+?>mT8F5$\"+Yg0!H\"F5$ \"+\\CS&>\"F5$\"+mPur5F5$\"+@w\"*y%*F($\"++1jL&)F($\"+=m@6\")F($\"+kem -%)F($\"+sc\"F5$\"+Wg0!H\"F5F`r$\"+_?wa8F5F\\r$ \"+.n[=8F57;$\"+Z&RLL\"F5$\"+!=gOO\"F5$\"+Zk?#Q\"F5$\"+#)p1u8F5$\"+U$p 9K\"F5$\"+#Q0l@\"F5$\"+m%f_2\"F5$\"+r$='H$*F($\"+#H5FC)F($\"+LIXZxF($ \"+cJV!4)F($\"+!exL\\*F($\"+nI]v5F5$\"+9wP$\\*F($\"+xJV!4)F(Fiu$\"+y-r U#)F($\"+X$='H$*F($\"+k%f_2\"F5$\"+y`];7F5$\"+R$p9K\"F5$\"+zp1u8F5$\"+ Yk?#Q\"F5Fit$\"+[&RLL\"F57;$\"+)Rj'[8F5$\"+Yp&eQ\"F5$\"+Iei79F5$\"+^?[ 69F5$\"+Bp9f8F5$\"+_5tU7F5$\"+7tr!3\"F5$\"+YPCo\"*F($\"+kFc!y)3\"F5$\"+*>Ce**)F($\"+!y)3\"F5$\"+D*QbF\"F5$\"+\"3FYS\"F5$ \"+v&eZX\"F5$\"++ZBY9F5$\"+#4u$49F5Faz7;$\"+C(R,Q\"F5$\"+%oqSV\"F5$\"+ ?N>$[\"F5$\"+1q([]\"F5$\"+jv\"*f9F5$\"+'y#*oJ\"F5$\"+IvH+6F5$\"+c^%R\" ))F($\"+8([k;(F($\"+%Q=rK'F($\"+T[O5mF($\"+*)f'fQ*F($\"+GUUw;F5$\"+%4m fQ*F($\"+k[O5mF($\"+\"Q=rK'F($\"+(o[k;(F($\"+9^%R\"))F($\"+EvH+6F5$\"+ #y#*oJ\"F5$\"+hv\"*f9F5Fh]lFf]l$\"+&oqSV\"F5$\"+D(R,Q\"F57;$\"+&e%)eR \"F5$\"+YVrf9F5$\"+!QlN_\"F5$\"+OC%Hc\"F5$\"+(3yw_\"F5$\"+*G!\\p8F5$\" +\\>N;6F5$\"+-ooC')F($\"+bLlXnF($\"+![^Wv&F($\"+Btj+fF($\"+PZD4))F($\" +x!)y#y#F5$\"+l[D4))F($\"+Stj+fF($\"+p9XadF($\"+FLlXnF($\"+lnoC')F($\" +V>N;6F5$\"+%G!\\p8F5$\"+&3yw_\"F5$\"+PC%Hc\"F5Fe`l$\"+ZVrf9F5$\"+'e%) eR\"F57;$\"+JcF69F5$\"++#ef[\"F5$\"+.RDn:F5$\"+eI:I;F5$\"+m2\\6;F5$\"+ cO8P9F5$\"+*R:$Q6F5$\"+P/YI%)F($\"+8T8,jF($\"+H$*Q]^F($\"+xiS1^F($\"+V G=9wF($\"+0E7)3(!\"#$\"+bH=9wF($\"+*G1k5&F($\"+>$*Q]^F($\"+#3M6I'F($\" +(Qg/V)F($\"+$R:$Q6F5$\"+]O8P9F5FjclFhcl$\"+0RDn:F5$\"+-#ef[\"F5Fbcl7; 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1'R19-3ii\"F^y$!1rt`>4Z/CF^y$!1z(*e`EfTJF^y$!1F_Km3'\\#QF^y$!1_$e\"QH) GW%F^y$!1@(>60'y%)\\F^y$!1SEq#4)RTaF^y$!1])yUI1\\!eF^y$!1lZcf44pgF^y$! 1=E'eyJ%HiF^y$!1!ezrI&=$G'F^y-%+AXESLABELSG6%%\"xG%\"yG%!G" 3 292 156 156 2 0 1 0 2 5 4 1 2 1.000000 79.000000 25.000000 14010 14091 14086 14104 14112 15050 15100 20040 0 17060 0 0 0 0 0 0 0 1 0 370 8681 0 142 -7287 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT 262 34 "Styles Description" }}{PARA 17 "" 0 "" {TEXT 263 5 "Patch" }{TEXT -1 41 " solid shape w ith lines\n" }{TEXT 264 23 "Patch w/o grid " }{TEXT -1 21 "sol id shape no lines\n" }{TEXT 265 19 "Patch and contour " }{TEXT -1 43 " solid shape with constant height lines\n" }{TEXT 503 11 "Hidden l ine" }{TEXT -1 1 "\n" }{TEXT 602 8 "Contour " }{TEXT -1 46 " \+ just the constant height lines\n" }{TEXT 266 9 "Wireframe" } {TEXT -1 1 "\n" }{TEXT 267 5 "Point" }{TEXT -1 30 " p oints alone" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 19 "II. Math Functions" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 39 " 1 Symbolic and Floating Point Results" }}{PARA 0 "" 0 "" {TEXT -1 186 " \nFor symbolic results, Maple uses the default compact notation. Some times it becomes too complicated to understand, because Maple retains \+ real numbers in compact symbolic form (such as " }{XPPEDIT 18 0 "Pi" " I#PiG%*protectedG" }{TEXT -1 82 "), and does not evaluate their floati ng point values unless commanded explicitly.\n" }}{PARA 0 "" 0 "" {TEXT 427 36 "Consider this initial value problem:" }{TEXT -1 1 "\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f:=(Pi*(exp(-s)+2))/s;# The function f is d efined." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(%#PiG\"\"\",&-%$exp G6#,$%\"sG!\"\"F'\"\"#F'F'F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Fo:=limit(s*f,s=infinity);# Initial Value Formula" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FoG,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "evalf(\");# The floating point value of the above l imit." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3`=$G'!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 52 "\nSometimes Maple uses unusual symbols to retain t he " }{TEXT 373 20 "symbolic closed form" }{TEXT -1 112 " notation. C onsider the solution to this equation involving a combination of linea r and exponential functions.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(exp(-2*x)-x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%)La mbertWG6#\"\"##\"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 51 "\nLambertW(a) /a in this example is the solution of " }{XPPEDIT 378 0 "e^(-ax)- x = 0" "/,&)%\"eG,$%#axG!\"\"\"\"\"%\"xGF(\"\"!" }{TEXT -1 75 ". This e xpression does not have a numerical value unless evaluated using " } {TEXT 377 9 "evalf ( )" }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(\",4); # \"4\" is the number of desired digits. \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%jU!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 374 53 "This command yields the \+ same result as the two above:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(solve(exp(-2*x)-x,x),4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%jU!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 641 16 " 2 Common Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Function Operator ( " }{TEXT 426 1 "-" }{TEXT -1 3 "> )" }}{PARA 0 "" 0 "" {TEXT -1 62 "The Maple F unctional operator, (the arrow) is used to make a " }{TEXT 434 18 "ge neral proceedure" }{TEXT -1 240 " of a given command line. It makes t he line portable, in that whatever variable names are used remain unas signed. Thus, in the example below, \"x\" is used twice, once as \"2 \" and again as \"4\", yet at the end of the day it is still just \"x \"." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 433 42 "If it was \"6\"pr eviously, it is \"6\" still." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> 3*x + 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\" xG6\"6$%)operatorG%&arrowGF(,&9$\"\"$\"\"&\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h:=x->x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG:6#%\"xG6\"6$%)operatorG%&arrowGF(*$9$ \"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f(2);h(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 428 13 "solve(f = a )" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 435 8 "solve( )" }{TEXT -1 81 " command can solve equations, systems o f equations and multi-variable equations.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(x^2+x=-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#!\"\"\"\"# \"\"\"*&%\"IGF'\"\"(#F'F&F+,&F$F'F(F$" }}}{PARA 0 "" 0 "" {TEXT 436 77 "NOTE: if the \"=-2\" were omitted, the equation would be assumed e qual to zero." }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "solve(q^3-k,q); # Here zero is assumed, and the solved values fo r q are the roots of the function.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%*$%\"kG#\"\"\"\"\"$,&F##!\"\"\"\"#*(%\"IGF&F'#F&F+F$F%F.,&F#F)F,F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 429 45 "solve(\{f1,f2\},\{x,y\}) Multi-variate equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "s:=(solve (\{x+y=b,a*x-2*y/3=k\},\{x,y\})); # Again the braces contain the set o f equations.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG<$/%\"xG*&,&% \"kG\"\"$%\"bG\"\"#\"\"\",&%\"aGF+F-F.!\"\"/%\"yG,$*&,&*&F0F.F,F.F.F*F 1F.F/F1F+" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 437 194 "NOTE: The solution set abov e was assigned the name \"s\". The variables x and y can assume the v alues of the elements of the solution set \"s\" only through the execu tion of the assign( ) function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT 430 9 "assign( )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "assign(s);# assigns the solution set to the resp ective variables." }}}{PARA 0 "" 0 "" {TEXT -1 60 " No output is displ ayed. In order to verify the assignment:\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "x,y; # show the values of x and y.\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$*&,&%\"kG\"\"$%\"bG\"\"#\"\"\",&%\"aGF&F(F)!\"\" ,$*&,&*&F+F)F'F)F)F%F,F)F*F,F&" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "Or don't assign the variable names to the solution set. The elements can be refered to as: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&% \"kG\"\"$%\"bG\"\"#\"\"\",&%\"aGF'F)F*!\"\"F$" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&,&*&%\"a G\"\"\"%\"bGF)F)%\"kG!\"\"F),&F(\"\"$\"\"#F)F,F.F$" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "y:='y'; x:='x';# unassigns the variables." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"xGF$" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 635 4 "Trig" }}{PARA 0 "" 0 "" {TEXT -1 121 "Maple can solve equations involving trig functions. As a general rule, Maple works with these functions in the form: \+ " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x) = C*g(x)*cos(a*x)-S*g(x)*sin(a x) " "/-%\"fG6#%\"xG,&*(%\"CG\"\"\"-%\"gG6#F&F*-%$cosG6#*&%\"aGF*F&F*F *F**(%\"SGF*-F,6#F&F*-%$sinG6#%#axGF*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The sinusoidal form is often mo re convenient:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) \+ =R*g(x)*cos(a*x+phi) " "/-%\"fG6#%\"xG*(%\"RG\"\"\"-%\"gG6#F&F)-%$cosG 6#,&*&%\"aGF)F&F)F)%$phiGF)F)" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "R >=0" "1\"\"!%\"RG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\" " }{TEXT -1 15 " is in degrees." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "To convert a Maple result to the Sinusoid al format, use:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "R=sqrt(C^2+S^2)" "/%\"RG-%%sq rtG6#,&*$%\"CG\"\"#\"\"\"*$%\"SGF*F+" }{TEXT -1 15 " , " } {XPPEDIT 18 0 " phi=(180/Pi)*arctan(S/C)" "/%$phiG*(\"$!=\"\"\"%#PiG! \"\"-%'arctanG6#*&%\"SGF&%\"CGF(F&" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Example (Conversion to Sinus oidal)" }}{PARA 0 "" 0 "" {TEXT -1 45 "In the following example: C=3, \+ S=4, a=2, and " }{XPPEDIT 18 0 "g(x)= x^2" "/-%\"gG6#%\"xG*$F&\"\"#" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=3*x^2*cos(2*x)-4*x^2*sin(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&%\"xG\"\"#-%$cosG6#,$F'F(\"\"\"\"\"$*&F'F(-%$sinGF+F-! \"%" }}}{PARA 0 "" 0 "" {TEXT -1 43 "By observation, and the conversi ons above:" }{MPLTEXT 0 21 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "R:=sqrt(3^2+4^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG\"\"& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "phi:=((180/Pi)*arctan(4 /3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&%#PiG!\"\"-%'arcta nG6##\"\"%\"\"$\"\"\"\"$!=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(phi,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$H&!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 14 "This yields: " }{XPPEDIT 18 0 "f=5*x^2*c os(2*x+52.9)" "/%\"fG*(\"\"&\"\"\"*$%\"xG\"\"#F&-%$cosG6#,&*&F)F&F(F&F &$\"$H&!\"\"F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "The following, homemade conversion will \+ satisfy Maple and preserve the look and feel of Sinusoidal form:" }} {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "degrees:=(Pi/180):" }}}{PARA 0 "" 0 "" {MPLTEXT 0 21 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "plot([5*x^2*cos(2*x +(phi*degrees)),(f+1)],x=0..5); #The +1 was used to provide some offse t so the two curves could be seen" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'C URVESG6$7ho7$\"\"!F(7$$\"1LLL3x&)*3\"!#;$\"1(zdQ3<;X#!#<7$$\"1nm\"H2P \"Q?F,$\"1S=H0!)z`[F/7$$\"1LL$eRwX5$F,$\"13b=`JO)3\"F/7$$\"1ML$3x%3yTF ,$!1M^7e*Glm\"F,7$$\"1nm\"z%4\\Y_F,$!1$eY*fb*GV&F,7$$\"1MLeR-/PiF,$!1U 7$f)>^/6!#:7$$\"1***\\il'pisF,$!1,3&zOS%3>FI7$$\"1MLe*)>VB$)F,$!1oH\"* 4&GQ&HFI7$$\"1++DJbw!Q*F,$!1\"HwM*yx]TFI7$$\"1nm;/j$o/\"FI$!1_%e&fk^Ra FI7$$\"1LL3_>jU6FI$!1\"*QB.ug6lFI7$$\"1++]i^Z]7FI$!11WpMST*\\(FI7$$\"1 ++](=h(e8FI$!1-1v>.z'3)FI7$$\"1++]P[6j9FI$!1y@9\"=\\O5)FI7$$\"1L$e*[z( yb\"FI$!1\"RKls9$HvFI7$$\"1nm;a/cq;FI$!1(GYS_4O*fFI7$$\"1nmm;t,mFI$\"1-@bWT,;K FI7$$\"1+]i!f#=$3#FI$\"10ur6w$R2)FI7$$\"1+](=xpe=#FI$\"1w6(3)y[A8!#97$ $\"1nm\"H28IH#FI$\"1h*G$\\Mg()=F_r7$$\"1n;zpSS\"R#FI$\"1-bDT.d-CF_r7$$ \"1LL3_?`(\\#FI$\"1dR-qv*z\"HF_r7$$\"1M$e*)>pxg#FI$\"1YJ8b^zmLF_r7$$\" 1nm;z+vbEFI$\"11hp$QII_$F_r7$$\"1+]Pf4t.FFI$\"1t?Y.5A]OF_r7$$\"1n\"zWi ^bv#FI$\"1*\\>fu#*3v$F_r7$$\"1MLe*Gst!GFI$\"1>olx:Q4QF_r7$$\"1+vV)>ST$ GFI$\"1tl\"p8*z@QF_r7$$\"1n;H2\"34'GFI$\"1OrFM8X@QF_r7$$\"1Le9;gn()GFI $\"1[wzP\"*)y!QF_r7$$\"1+++DRW9HFI$\"1*pM_?+2y$F_r7$$\"1+]7y#=o'HFI$\" 1b!e3W-no$F_r7$$\"1++DJE>>IFI$\"1/VQJ&Ho`$F_r7$$\"1+v$4^n)pIFI$\"1*HzK '*)3PLF_r7$$\"1+]i!RU07$FI$\"1&4`^w%o#3$F_r7$$\"1++v=S2LKFI$\"19R=[TwA BF_r7$$\"1mmm\"p)=MLFI$\"1t!)y[J()>9F_r7$$\"1++](=]@W$FI$\"1DiO(e\"G5D FI7$$\"1n\"H#oZ1\"\\$FI$!1iqK1P%oP$FI7$$\"1L$e*[$z*RNFI$!1ufuCZ1m&*FI7 $$\"1n\"Hd!fX$f$FI$!10mU\"HP7m\"F_r7$$\"1,+]iC$pk$FI$!1vFb&ygsQ#F_r7$$ \"1Le*[t\\sp$FI$!1pL5ad1#3$F_r7$$\"1m;H2qcZPFI$!1z[NCP=!y$F_r7$$\"1L$3 _0j,!QFI$!1[4.=>%[Y8(F_r7$$\"1+]ilAFjSFI$!1%pzEuUOp(F_r7$$\"1MLL$)*pp;%FI$!1&4UFI$!1%))>'4hX*)))F_r7$$\"1ML3xe,tUFI$!1?v53b d=\"*F_r7$$\"1*F_r7$$\"1,v=n(*fDVFI$!1ybj!Q&4X# *F_r7$$\"1UNrRduQVFI$!1p/XQOFg#*F_r7$$\"1%eRAr\"*=N%FI$!1*3$4+Soo#*F_r 7$$\"1Ecw%oP]O%FI$!1nU+T0Cq#*F_r7$$\"1n;HdO=yVFI$!1(zyb\">'[E*F_r7$$\" 1](=UAVBS%FI$!1m\"o6QSmB*F_r7$$\"1Me9\"z-lU%FI$!1=#p9r1V=*F_r7$$\"1

#[Z%FI$!1fWX^!ee+*F_r7$$\"1M$3_5,-` %FI$!11,&\\US!y')F_r7$$\"1nmT&G!e&e%FI$!17)30Tbo@)F_r7$$\"1,]P%37^j%FI $!1!p[o\">O\"p(F_r7$$\"1MLL$)Qk%o%FI$!1kiV]b\\gqF_r7$$\"1n\"z>6but%FI$ !1$))3&emruiF_r7$$\"1+]iSjE!z%FI$!1n)3e[;oP&F_r7$$\"1+++DM\"3%[FI$!1\\ ymvKE=WF_r7$$\"1,]P40O\"*[FI$!1Ht`(H4)pLF_r7$$\"1,voa-oX\\FI$!1t'eZ[AC :#F_r7$$\"\"&F($!1Vpz\"f`#G&)FI-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7h o7$F($\"\"\"F(7$F*$\"1y&Q3<;X-\"FI7$F1$\"1=H0!)z`[5FI7$F6$\"1b=`JO)3, \"FI7$F;$\"1n[(=/rML)F,7$F@$\"1>M0SW5nXF,7$FE$!1ACJf)>^/\"F,7$FK$!1:!3 &zOS%3*F,7$FP$!1oH\"*4&GQ&>FI7$FU$!1\"HwM*yx]JFI7$FZ$!1^%e&fk^RWFI7$Fi n$!1\"*QB.ug6bFI7$F^o$!12WpMST*\\'FI7$Fco$!1-1v>.z'3(FI7$Fho$!1z@9\"= \\O5(FI7$F]p$!1(QKls9$HlFI7$Fbp$!1#GYS_4O*\\FI7$Fgp$!1l3/cF_r7$Ffr$\"1-bDT.d-DF_r7$F[s$\"1dR-qv*z,$F_r 7$F`s$\"1YJ8b^zmMF_r7$Fes$\"11hp$QIIi$F_r7$Fjs$\"1u?Y.5A]PF_r7$F_t$\"1 *\\>fu#*3&QF_r7$Fdt$\"1>olx:Q4RF_r7$Fit$\"1sl\"p8*z@RF_r7$F^u$\"1OrFM8 X@RF_r7$Fcu$\"1[wzP\"*)y!RF_r7$Fhu$\"1*pM_?+2)QF_r7$F]v$\"1b!e3W-ny$F_ r7$Fbv$\"1/VQJ&Hoj$F_r7$Fgv$\"1*HzK'*)3PMF_r7$F\\w$\"1&4`^w%o#=$F_r7$F aw$\"19R=[TwACF_r7$Ffw$\"1t!)y[J()>:F_r7$F[x$\"1HiO(e\"G5NFI7$F`x$!1eq K1P%oP#FI7$Fex$!1ufuCZ1m&)FI7$Fjx$!1/mU\"HP7c\"F_r7$F_y$!1uFb&ygsG#F_r 7$Fdy$!1pL5ad1#)HF_r7$Fiy$!1&)[NCP=!o$F_r7$F^z$!1Z4% [Y.(F_r7$Fg[l$!1%pzEuUOf(F_r7$F\\\\l$!1&4'4h X*y)F_r7$Ff\\l$!1?v53bd=!*F_r7$F[]l$!1@,\"Q1,[4*F_r7$F`]l$!1ybj!Q&4X\" *F_r7$Fe]l$!1o/XQOFg\"*F_r7$Fj]l$!1*3$4+Soo\"*F_r7$F_^l$!1mU+T0Cq\"*F_ r7$Fd^l$!1(zyb\">'[;*F_r7$Fi^l$!1m\"o6QSm8*F_r7$F^_l$!1;#p9r1V3*F_r7$F c_l$!1%eBU)[[2!*F_r7$Fh_l$!1fWX^!ee!*)F_r7$F]`l$!12,&\\US!y&)F_r7$Fb`l $!18)30Tbo6)F_r7$Fg`l$!1!p[o\">O\"f(F_r7$F\\al$!1miV]b\\gpF_r7$Faal$!1 &))3&emruhF_r7$Ffal$!1o)3e[;oF&F_r7$F[bl$!1]ymvKE=VF_r7$F`bl$!1Ht`(H4) pKF_r7$Febl$!1t'eZ[AC0#F_r7$Fjbl$!1apz\"f`#GvFI-F_cl6&FaclF(FbclF(-%+A XESLABELSG6$%\"xG%!G-%%VIEWG6$;F(Fjbl%(DEFAULTG" 2 349 234 234 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 167 151 0 0 0 0 0 0 }}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 4 "" 1 "" {TEXT -1 11 "Derivation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The abo ve conversion is based on the complex plane as shown below:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "e^(j*90)=j" "/)%\"eG*&%\"jG\" \"\"\"#!*F'F&" }{TEXT -1 14 " and " }{XPPEDIT 18 0 "-sin(a*x) =cos(a*x+90)" "/,$-%$sinG6#*&%\"aG\"\"\"%\"xGF)!\"\"-%$cosG6#,&*&F(F)F *F)F)\"#!*F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = C*g(x)*cos(a*x)-S*g(x)*sin(ax) " "/-% \"fG6#%\"xG,&*(%\"CG\"\"\"-%\"gG6#F&F*-%$cosG6#*&%\"aGF*F&F*F*F**(%\"S GF*-F,6#F&F*-%$sinG6#%#axGF*!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "g(x)*[C*cos(a*x)-S*cos(ax+90) ]" "*&-%\"gG6#%\"xG\"\"\"7#,&*&%\"CGF'- %$cosG6#*&%\"aGF'F&F'F'F'*&%\"SGF'-F-6#,&%#axGF'\"#!*F'F'!\"\"F'" }} {PARA 0 "" 0 "" {TEXT -1 68 " \+ = " }{XPPEDIT 18 0 "g(x)*Re " "*&-%\"gG6#%\"x G\"\"\"%#ReGF'" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "C*e^(j*a*x)*cos(a*x)+S *e^(j*(a*x+90))" ",&*(%\"CG\"\"\")%\"eG*(%\"jGF%%\"aGF%%\"xGF%F%-%$cos G6#*&F*F%F+F%F%F%*&%\"SGF%)F'*&F)F%,&*&F*F%F+F%F%\"#!*F%F%F%F%" } {TEXT -1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 68 " \+ = " }{XPPEDIT 18 0 "g(x)*Re \+ " "*&-%\"gG6#%\"xG\"\"\"%#ReGF'" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "(C+j* S)*e^(j*a*x)" "*&,&%\"CG\"\"\"*&%\"jGF%%\"SGF%F%F%)%\"eG*(F'F%%\"aGF%% \"xGF%F%" }{TEXT -1 3 "] " }}{PARA 0 "" 0 "" {TEXT -1 7 "Whereas" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "R*g(x)*cos(a*x+phi) = g(x)" "/*(%\"RG\" \"\"-%\"gG6#%\"xGF%-%$cosG6#,&*&%\"aGF%F)F%F%%$phiGF%F%-F'6#F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Re" "I#ReG%*protectedG" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "R*e^(j*phi)*e^(j*a*x)] " "7#*(%\"RG\"\"\")%\"eG*&% \"jGF%%$phiGF%F%)F'*(F)F%%\"aGF%%\"xGF%F%" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "By comparison," }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R*e^(j*phi) = C+j*S" "/ *&%\"RG\"\"\")%\"eG*&%\"jGF%%$phiGF%F%,&%\"CGF%*&F)F%%\"SGF%F%" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "R, " }{XPPEDIT 18 0 "phi " "I$phiG6\"" }{TEXT -1 99 " are related to C and S by the complex-pla ne triangle. " }}{PARA 0 " " 0 "" {MPLTEXT 0 21 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 431 9 "dsol ve( )" }{TEXT -1 45 " Maple's solution to differential equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "My_equation:= diff(y(x),x)*(x^2)+y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,My_equationG/,&*&-%%diffG6$-%\"yG6#%\"xGF.\"\"\"F.\" \"#F/F+F/\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 29 "To solve the above equ ation:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(My_equati on, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&-%$expG6 #*$F'!\"\"\"\"\"%$_C1GF." }}}{PARA 0 "" 0 "" {TEXT -1 46 "C1 is an und etermined constant of integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 159 "To evaluate the unknown constants of int egration, we must specify initial conditions. Consider the differenti al equation:\n\n " }{XPPEDIT 18 0 "x *diff(y(x),x,x)+5*diff(y(x),x)=0" "/,&*&%\"xG\"\"\"-%%diffG6%-%\"yG6#F %F%F%F&F&*&\"\"&F&-F(6$-F+6#F%F%F&F&\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The solution to this equation, wit hout initial conditions, is as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dsolve(x*diff(y(x),x,x) +5*diff(y(x),x)=0,y(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6 #%\"xG,&%$_C1G\"\"\"*&%$_C2GF*F'!\"%F*" }}}{PARA 0 "" 0 "" {TEXT 438 5 "Note:" }{TEXT -1 79 " This is a second-order equation and thus two \+ unknown constants and two initial" }}{PARA 0 "" 0 "" {TEXT -1 44 " \+ conditions have to be specified:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dsolve(\{x*diff(y(x),x,x)+5*diff(y(x),x)=0,y(1)=2, y( 2)=1\},y(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&,&* $F'\"\"%#\"#9\"#:#\"#;F.\"\"\"F1F'!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 432 11 "Simplify( )" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 64 "Simplify ( ) uses basic algebraic ru les to simplify an equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " This one, for example: " }{XPPEDIT 18 0 " x^2-2*x-3*x^2-9*x+88+37*x^2" ",.*$%\"xG\"\"#\"\"\"*&F%F&F$F&!\"\"*&\" \"$F&*$F$F%F&F(*&\"\"*F&F$F&F(\"#))F&*&\"#PF&*$F$F%F&F&" }{TEXT -1 1 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "simplify(x^2-2*x-3*x^2-9*x+88+37*x^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\"#\"#NF%!#6\"#))\"\"\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 439 26 "simplif y(expression,rule);" }{TEXT -1 211 "\nRules of simplification can be s pecified. They are Ei, GAMMA, RootOf, atsign, hypergeom, ln, polar, p ower, radical, sqrt and trig. When a rule or rules are mentioned, Map le performs only the mentioned rules.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simplify(3^2*(cos(2*x)+sin(x)^2), trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%$cosG6#%\"xG\"\"#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x:=(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(\"\",3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$]%!\"# " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 " 3 Complex Variables" }}{PARA 0 "" 0 "" {TEXT -1 80 "\nComplex variable may be constructed by making use of Maple's built-in constan t " }{TEXT 376 8 "sqrt(-1)" }{TEXT -1 14 ", denoted by " }{TEXT 375 1 "I" }{TEXT -1 85 ".\n\n\nThe following example illustrates basic Map le operations with complex variables:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F:=1/(s^2+2*s+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"FG*$,(*$%\"sG\"\"#\"\"\"F(F)\"\"$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "s:= sigma+I*omega; # assigns a complex value to \+ s." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&%&sigmaG\"\"\"*&%\"IGF'% &omegaGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F; # view F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,**$,&%&sigmaG\"\"\"*&%\"IGF(%&o megaGF(F(\"\"#F(F'F,F)F,\"\"$F(!\"\"" }}}{PARA 11 "" 1 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Now a ssign some constant values to omega and sigma.\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "assign(sigma=5, omega=7):\n\nF;# see how F loo ks now" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#!#6\"%xr\"\"\"%\"IG#!#%)F &" }}}{PARA 17 "" 0 "" {TEXT -1 72 "\nSome of the basic complex operat ions performed on F(s) are as follows:\n" }}{PARA 17 "" 0 "" {TEXT 603 7 "Command" }{TEXT -1 19 " " }{TEXT 604 11 "Desc ription" }{TEXT -1 1 "\n" }{TEXT 379 6 "abs(F)" }{TEXT -1 36 " \+ Magnitude of F.\n" }{TEXT 380 13 "conjugate(F) " }{TEXT -1 29 " Conjugate of F.\n" }{TEXT 381 17 "convert(F,polar) " }{TEXT -1 29 " Convert F to polar.\n" }{TEXT 382 8 "evalc(F )" }{TEXT -1 47 " Convert to a complex number.\n" } {TEXT 383 5 "Re(F)" }{TEXT -1 37 " Real part of F. \n" }{TEXT 384 5 "Im(F)" }{TEXT -1 42 " Imaginary \+ part of F.\n" }{TEXT 385 11 "argument(F)" }{TEXT -1 40 " \+ Angle of F (in radians).\n" }{TEXT 386 26 "180*(\")/(Pi) \+ " }{TEXT -1 27 "Converts radians to degrees" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 387 84 "NOTE: The inverse trigonometric, and hyperbol ic functions are calculated in radians." }}{PARA 0 "" 0 "" {TEXT 605 29 "(1 radian = 180/Pi degrees) \n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 " 4 Calculus" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 388 15 " Differentiation" }{TEXT -1 6 "\n\nThe " }{TEXT 389 4 "D( )" }{TEXT -1 39 " command is the differential operator.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(tan);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$%$tanG\"\" #F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D(U*V);# Chain rule ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%\"DG6#%\"UG\"\"\"%\"VGF)F)* &F(F)-F&6#F*F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 122 "\nThe variable of d ifferentiation is implicit in the previous example. If an equation in volves more than one unknown, the " }{TEXT 390 6 "diff()" }{TEXT -1 63 " command allows you to specify the variable of differentiation." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 391 9 "diff(f,x )" }{TEXT -1 69 " (f) is the function, and (x) is the varriable of di fferentiation. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "diff(ex p(a*x)*cos(b*x)+exp(b*x)* sin(a*x),x);# differentiation with respect t o x.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(%\"aG\"\"\"-%$expG6#*&F% F&%\"xGF&F&-%$cosG6#*&%\"bGF&F+F&F&F&*(F'F&-%$sinGF.F&F0F&!\"\"*(F0F&- F(F.F&-F3F)F&F&*(F6F&-F-F)F&F%F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 3 "" 0 "" {TEXT 504 11 "Integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 505 9 "int(f,x) " }{TEXT -1 35 "\nEval uate the indefinite integral:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int(exp(a*x)*cos(b*x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &**%\"aG\"\"\",&*$F%\"\"#F&*$%\"bGF)F&!\"\"-%$expG6#*&F%F&%\"xGF&F&-%$ cosG6#*&F+F&F1F&F&F&**F+F&F'F,F-F&-%$sinGF4F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The probability integral, " }{XPPEDIT 18 0 " int((e^((-x^ 2)/2),x))" "-%$intG6$)%\"eG*&,$*$%\"xG\"\"#!\"\"\"\"\"F+F,F*" }{TEXT -1 44 " , is evaluated in terms of erf(x), Where: " }{XPPEDIT 18 0 "e rf(x)=(2/sqrt(Pi))*int( e^(-t^2),t=0..x)" "/-%$erfG6#%\"xG*(\"\"#\"\" \"-%%sqrtG6#%#PiG!\"\"-%$intG6$)%\"eG,$*$%\"tGF(F./F6;\"\"!F&F)" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(exp(-x ^2/2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%#PiG#\"\"\"\"\"#F(F& -%$erfG6#,$*&F(F&%\"xGF'F&F'F&" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" } {TEXT 392 13 "int(f,x=a..b)" }{TEXT 393 21 " Definite Integration" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(exp(-x ^2),x=0..l00);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$erfG6#%$l00G \"\"\"%#PiG#F)\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ev alf(\",4);# Again, the numerical value via evalf().\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$erfG6#%$l00G$\"%l))!\"%" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 648 12 " 5 Matrices" } }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 506 9 "matrix( )" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Special matrix com mands are made available after loading the linalg package:\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "restart: with(linalg): # Th e commands norm and trace are not new, but have changed with Release 4 ; thus the warnings." }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new defi nition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definitio n for trace" }}}{PARA 0 "" 0 "" {TEXT -1 53 "\nThe following is a 4 ro w by 3 col matrix called \"B\";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "B:=matrix(4,3,[11,12,13,21,2 2,23,31,32,33, 41,42,43]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-% 'MATRIXG6#7&7%\"#6\"#7\"#87%\"#@\"#A\"#B7%\"#J\"#K\"#L7%\"#T\"#U\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "B:=matrix([[11,12,13],[21,22,23],[31,32,33], [41,42,43]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7&7%\"#6\"#7\"#87%\" #@\"#A\"#B7%\"#J\"#K\"#L7%\"#T\"#U\"#V" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "A Maple vector is a one row matrix, \+ and can be defined by the " }{TEXT 508 9 "matrix( )" }{TEXT -1 62 " co mmand. However, there is a separate command for vectors:\n\n" }{TEXT 394 9 "vector( )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "V:=vecto r([i,j,k]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'VECTORG6#7%% \"iG%\"jG%\"kG" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 395 8 "arr ay( )" }{TEXT 507 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The array( ) command can produce a column vector.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "W:=array([[i],[j],[k]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG-%'MATRIXG6#7%7#%\"iG7#%\"jG 7#%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "i:=array(identit y, 1..3,1..3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eval(i);# to see \+ the matrix\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\" \"\"!F)7%F)F(F)7%F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 124 "A matrix may contain a symbolic parameter and as \+ such is a function of that parameter. Below, matrix A is a function o f R.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A:=R-> matrix(3,3, [-R/3,-1/3,0,60,-3,3,0,1,-1]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"AG:6#%\"RG6\"6$%)operatorG%&arrowGF(-%'matrixG6%\"\"$F/7+,$9$#!\" \"F/F3\"\"!\"#g!\"$F/F5\"\"\"F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A(R);# to see this matrix." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,$%\"RG#!\"\"\"\"$F*\"\"!7%\"#g!\"$F,7% F-\"\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Matrix equations of the form " }{TEXT 396 6 "AX = B" }{TEXT -1 19 " can be solved for " }{TEXT 397 1 "X" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " The following sta tements solve the equation via: " }{XPPEDIT 649 0 "X = A^(-1)*B" "/ %\"XG*&)%\"AG,$\"\"\"!\"\"F(%\"BGF(" }{TEXT 650 1 ":" }{TEXT -1 1 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A:=matrix(3,3,[3,2,3,6,5 ,4,9,8,7]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7 %\"\"$\"\"#F*7%\"\"'\"\"&\"\"%7%\"\"*\"\")\"\"(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "B:=matrix(3,3,[6,0,0,0,-3,0,0,0,6]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7%7%\"\"'\"\"!F+7%F+ !\"$F+7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "X:=evalm( inverse(A)&*B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'MATRIXG6#7 %7%\"\"$!\"&!\"(7%!\"'F*\"\"'7%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 398 8 "e valm( )" }{TEXT -1 35 " function used above is similar to " }{TEXT 399 8 "evalf( )" }{TEXT -1 66 ", except it is used to generate a numer ical value for a matrix.\n\n\n" }{TEXT 400 35 "Command \+ Description\n" }{TEXT -1 1 "\n" }{TEXT 401 2 "&*" }{TEXT -1 43 " \+ Matrix multiplication\n" }{TEXT 402 10 "inverse( )" } {TEXT -1 34 " Inverse of a matrix \n" }{TEXT 403 7 "det( ) " }{TEXT -1 55 " Computes the determinant of the matri x\n" }{TEXT 404 11 "adjoint( ) " }{TEXT -1 46 " Returns the adjoint of the matrix\n" }{TEXT 405 13 "transpose( ) " }{TEXT -1 57 " [j,i]-th element becomes the [i,j]-th element. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 640 36 " 6 Transforms (Laplace, Fourier, Z)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 490 100 "Maple o ffers most of the transforms used in signal processing, controls and o ther applications. The" }{TEXT 651 9 " Laplace " }{TEXT -1 4 "and " } {TEXT 498 13 "Fourier Trans" }{TEXT -1 6 "forms " }{TEXT 609 16 "are p art of the " }{TEXT -1 8 "inttrans" }{TEXT 653 32 " package and must b e loaded with" }{TEXT -1 1 " " }{TEXT 489 7 "with( )" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }{TEXT 487 18 "laplace(f(t),t,s); " }}{PARA 0 "" 0 "" {TEXT 488 21 "invlaplace(f(s),s,t);" }{MPLTEXT 0 21 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0 "" 0 " " {MPLTEXT 0 21 10 "\nExample:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "laplace(t*exp(-t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$, &%\"sG\"\"\"F&F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 491 25 "inverse Laplace transform" }{TEXT -1 40 " can be obtained in a similar fashion: \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "invlaplace(( 5*s-33)/(s^2+6*s+25),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$e xpG6#,$%\"tG!\"$\"\"\"-%$cosG6#,$F)\"\"%F+\"\"&*&F%F+-%$sinGF.F+!#7" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 492 18 "Fourier Transforms" }}{PARA 0 "" 0 "" {TEXT -1 150 "The frequency domain variable in Maple is angular freque ncy in radians/sec. The name of this variable is specified by the user (usually omega or v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {MPLTEXT 0 21 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "assume(a>0):# The variable restriction is noted by the \"~\"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fourier(cos(a*t),t,omega);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#PiG\"\"\"-%&DiracG6#,&%&omegaGF& %#a|irG!\"\"F&F&*&F%F&-F(6#,&F+F&F,F&F&F&" }}}{PARA 256 "" 1 "" {TEXT -1 74 "Note: The Dirac( ) function is defined as zero except at t = 0, such that:" }}{PARA 256 "" 0 "" {TEXT 23 26 " \+ " }{XPPEDIT 18 0 "Int(Dirac(t),t = -infinity..infinity) = 1" "/-%$In tG6$-%&DiracG6#%\"tG/F);,$%)infinityG!\"\"F-\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 27 "It is the impulse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 493 26 "Inverse Fourier transforms" } {TEXT -1 40 " of a function are obtained as follows:\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "invfour ier(sin(a*omega)/(a*omega),omega,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&,*-%*HeavisideG6#,&%\"tG\"\"\"%#a|irGF+F+-F'6#,&F*!\"\"F,F0F0-F' 6#,&F*F+F,F0F0-F'6#,&F*F0F,F+F+F+F,F0#F+\"\"%" }}}{PARA 0 "" 1 "" {TEXT -1 56 "The above result thus represents a rectangular function. " }}{PARA 0 "" 0 "" {TEXT 494 5 "Note:" }{TEXT -1 5 " The " }{TEXT 495 12 "Heaviside( )" }{TEXT -1 83 " function is defined as zero for \+ t < 0, and 1 for t > 0. At zero it is undefined." }}{PARA 0 "" 0 "" {TEXT -1 75 "\nAnother example using the Heaviside function, Given a r ectangle of width (" }{XPPEDIT 18 0 "tau" "I$tauG6\"" }{TEXT -1 21 ") \+ centered at origin:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x:=t->Heaviside(t+tau/2)-Heaviside(t-tau/2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operatorG%&a rrowGF(,&-%*HeavisideG6#,&9$\"\"\"%$tauG#F2\"\"#F2-F.6#,&F1F2F3#!\"\"F 5F:F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume(omega>0): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourier(x(t),t,omega);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"IG\"\"\"-%$expG6#,$*(F%F&%$tauG F&%'omega|irGF&#F&\"\"#F&F-!\"\"F0*(F%F&-F(6#,$F+#F0F/F&F-F0F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "A more fa miliar form of this expression is obtained as follows:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%'omega|irG!\"\"-%$sinG6#,$*&%$tauG\"\"\"F%F-#F-\" \"#F-F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "which is:" }}{PARA 275 "" 0 "" {TEXT -1 1 "\n" }{XPPEDIT 18 0 "tau* sin(omega*tau/2)\n" "*&%$tauG\"\"\"-%$sinG6#*(%&omegaGF$F#F$\"\"#!\"\" F$" }}{PARA 3 "" 0 "" {TEXT 496 11 "Z-Transform" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The Z-Transform is a basi c Maple function. F(z) of a discrete time function f(n) is defined as :\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "F(z)=sum(f(n)* z^(-n),n=0..infinity)" "/-%\"FG6#%\"zG-%$sumG6$*&-%\"fG6#%\"nG\"\"\")F &,$F.!\"\"F//F.;\"\"!%)infinityG" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 497 7 "Example" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ztrans(2^n,n,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"zG\"\"\",&F$F%!\"#F%!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Maple calculates i nverse Z-transforms for rational polynomials and some special function s:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "invztrans((6*z^2-13* z)/(z^2-5*z+6),z,t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&)\"\"$%\"t G\"\"&)\"\"#F&\"\"\"" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 19 " III. Maple Examples" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 " 1 Laplac e analysis of a circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Maple can be used for complex circuit analysis. Th is circuit is described by the s-domain matrix mesh equation:\n" }} {PARA 0 "" 0 "" {MPLTEXT 0 21 42 " [Z(s)][I(s)]=[ Vs(s)]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(linalg,matrix,inverse):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "wi th(inttrans,laplace,invlaplace):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Z:=matrix(2,2,[20+80/s,-60/s,-60/s,3*s+3+60/s]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG-%'MATRIXG6#7$7$,&\"#?\"\"\"*$% \"sG!\"\"\"#!),$F-!#g7$F1,(F.\"\"$F5F,F-\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Vs:=matrix([[20*V],[0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VsG-%'MATRIXG6#7$7#,$%\"VG\"#?7#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "V:=s->laplace(10*cos(3*t),t,s):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "V(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"sG\"\"\",&*$F%\"\"#F&\"\"*F&!\"\"\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Vs(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7#,$*&%\"sG\"\"\",&*$F*\"\"#F+\"\"*F+!\" \"\"$+#7#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Ivector:= evalm(inverse(Z)&*Vs(s)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "I1:=s->Ivector[1,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " I1(s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(\",4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,(*$%\"sG\"\"#\"\"\"F'F)$\"#?\"\"! F)F)F'F(,**$F'\"\"$F)F&$\"\"&F,F'$\"#CF,F*F)!\"\",&F&F)$\"\"*F,F)F4$\" #5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "i1:=t->evalf(invlap lace(I1(s),s,t),4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "i1(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%$expG6#,$%\"tG$!\"\"\"\"!$\"% w6!\"$*&-F%6#,$F($!\"#F+\"\"\"-%$cosG6#,$F($\"\"%F+F5$\"%rBF.*&F0F5-%$ sinGF8F5$!%4(*F.-F76#,$F($\"\"$F+$\"%`kF.-F@FD$\"%&e\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "i1F:=t->6.453*cos(3.*t)+1.585*sin(3 .*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i1F(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%\"tG$\"\"$\"\"!$\"%`k!\"$-%$si nGF&$\"%&e\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(\{i1 (t),i1F(t)\},t=0..6);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7as7 $\"\"!$\"1++++++`k!#:7$$\"1++]ilyM;!#<$\"1inR!)Q%H_'F+7$$\"1+++DJdpKF/ $\"1W&RaV,sd'F+7$$\"1++](ofV!\\F/$\"1>m(p=Uch'F+7$$\"1+++]i9RlF/$\"1#) Q)Hpt\"QmF+7$$\"1++]7G$R<)F/$\"1-(o)p(3)*F/$\"1C-xA1 LNmF+7$$\"1++v$f]V9\"!#;$\"1BFa$)G'*4mF+7$$\"1+++]#HyI\"FN$\"1%pIjb*po lF+7$$\"1++voozw=FN$\"1SI4eA!HI'F+7$$\"1++]([kdW#FN$\"1!H\"\\JU\"R&eF+ 7$$\"1++D\"3Gc3$FN$\"1iv$G@gl9&F+7$$\"1+++v;\\DPFN$\"1vf$)ze9]UF+7$$\" 1++++nfpVFN$\"1/u+vp=!>$F+7$$\"1+++Dgyfy;F+7$$\"1++](G[W[(FN$!1:JIEuD#z#F+7$$\"1++]P@%)*4)FN$!1`en8yn ^QF+7$$\"1++]()fB:()FN$!150y&f!>!y%F+7$$\"1++](=x;N*FN$!1JPa3]BpbF+7$$ \"1++](Q=\"))**FN$!1)>J\"o!oe:'F+7$$\"1+++X=`I5F+$!1AbjPRsljF+7$$\"1++ D^=Di5F+$!1S[KJ*yz^'F+7$$\"1+]Pa=6y5F+$!1Wt(p$[0slF+7$$\"1++]d=(R4\"F+ $!1\"\\$*\\Nb7h'F+7$$\"1+]ig=$)46F+$!10btd<\\NmF+7$$\"1++vj=pD6F+$!1%e 'z(=4Zk'F+7$$\"1]iSEe+U6F+$!1ng,u()\\QmF+7$$\"1+D1*y>$e6F+$!1#\\_@G!R; 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Pole s exist in the right side of the s-plane." }}{PARA 0 "" 0 "" {TEXT -1 39 "2. Poles exist on the imaginary axis." }}{PARA 0 "" 0 "" {TEXT -1 36 "3. Multiple poles exist at s = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f:=(Pi*(exp(-s)+2))/s;# The \+ function f is defined." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(%#Pi G\"\"\",&-%$expG6#,$%\"sG!\"\"F'\"\"#F'F'F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Fa:=limit(s*f,s=0);# Final Value Formula" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FaG,$%#PiG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(\");# The floating point value of the above limit." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+izxC%*!\"*" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 " 3 Statistical analysis" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(stats): # The statical package" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "R eadings:=[9.2,9.3,8.9,9.1,9.0];" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ReadingsG7'$\"##*!\"\"$\"#$*F($\"#* )F($\"#\"*F($\"#!*F(" }}}{PARA 0 "" 0 "" {TEXT 443 4 "Mean" }{TEXT -1 30 " the average of the readings? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "describe[mean](Readings);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++\"*!\"*" }}}{PARA 0 "" 0 "" {TEXT 442 9 "Varian ce " }{TEXT -1 76 "is the average of the squares of the deviations (in dividual reading - mean)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "describe[variance](Readings): evalf (\",4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%+?!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 444 18 "Standard deviation " }{TEXT -1 40 " is just the squareroot of the variance." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "sqrt( \");#The squareroot of the variance is the standard deviation." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 38 "Get the standard dev iation right away:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "descri be[standarddeviation](Readings);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+iN@99!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The most likely va lue read is: " }{XPPEDIT 18 0 "mean" "I%meanG6\"" }{TEXT -1 7 " +/- \+ " }{XPPEDIT 18 0 "(standard deviation)/sqrt(N)" "*(%)standardG\"\"\"% *deviationGF$-%%sqrtG6#%\"NG!\"\"" }{TEXT -1 6 " \000" }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 26 " is the number of readings" }}{PARA 0 "" 0 "" {TEXT -1 16 "9.100000000 +/- " }{XPPEDIT 18 0 ".1414213562/s qrt(5)" "*&$\"+iN@99!#5\"\"\"-%%sqrtG6#\"\"&!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 28 "Linear Regression with Maple" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 445 31 "fit [leastsquare[[var,eq]](data)" }{TEXT -1 191 " from the stat package is used to find the best line through the data points to minimize the de viation of data point to line. 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The buttons on the graphic tool bar can change the axis, color/shading and method of display.\n" } {TEXT 652 101 "Note: For a black and white print out, try grid only an d designate \"color=black\" in the command line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Frequency response plots \+ of the magnitude and angle of " }{XPPEDIT 18 0 "H(j*omega)" "-%\"HG6#* &%\"jG\"\"\"%&omegaGF'" }{TEXT -1 28 " can be obtained as follows:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sigma:=0;# The real part of s is set to zero.\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&sigmaG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Now w e can plot the magnitude of " }{XPPEDIT 18 0 "H(j*omega)" "-%\"HG6#*&% \"jG\"\"\"%&omegaGF'" }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(abs(H),omega=0..50);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7`o7$\"\"!$\"++++?8!\"*7$$\"+P@Ki8!#5$\"+I`b?8 F+7$$\"+tUkCFF/$\"+3$=AK\"F+7$$\"+5k'p3%F/$\"+0)z\\K\"F+7$$\"+X&)G\\aF /$\"+2V#)G8F+7$$\"+=G$R<)F/$\"+S`mR8F+7$$\"+4x&)*3\"F+$\"+duYa8F+7$$\" +#R(*Rc\"F+$\"+6yI)Q\"F+7$$\"+uq8Q?F+$\"+Fm0H9F+7$$\"+OnNrDF+$\"+#*fZv 9F+7$$\"+(RwX5$F+$\"+ggU5:F+7$$\"+X\\wQKF+$\"+Cgp::F+7$$\"+#\\`HP$F+$ \"+\\#*4>:F+7$$\"+R?92NF+$\"+u7W?:F+7$$\"+&eI8k$F+$\"+:F+7$$\"+L\" >bx$F+$\"+7CK;:F+7$$\"+zwq4RF+$\"+O7l5:F+7$$\"+Ei*Q/%F+$\"+R\"3D]\"F+7 $$\"+sZ3yTF+$\"+jf!>\\\"F+7$$\"+hyG7ZF+$\"+Q-VB$)F+$\"+1ctdzF/7$$\"+j()4_))F+$\"+0Q/LtF/7$$\"+Mbw!Q*F+$\"+ +wc)y'F/7$$\"+0j$o/\"!\")$\"+jU6Fft$\"+6&4U\"R(GF/7$$\"+#pW`(>Fft$\"++#*=7FF/7$$ \"+\"f#=$3#Fft$\"+W#**\\b#F/7$$\"+t(pe=#Fft$\"+(o&z@CF/7$$\"+uI,$H#Fft $\"+$\\CsH#F/7$$\"+rSS\"R#Fft$\"+Y\"))Q>#F/7$$\"+`?`(\\#Fft$\"+%GF/7$$\"+g4t.FFft$\"+_o%4#>F/7$$\"+!Hst!GFft$ \"+ci5X=F/7$$\"+ERW9HFft$\"+T8$Hx\"F/7$$\"+KE>>IFft$\"+Zlp27F/7$$\"+ye,tUFft$\"+fFY)=\"F/7$$\"+fO=yVFft$\"+.42f6F/7$$ \"+E>#[Z%Fft$\"+.7LL6F/7$$\"+(G!e&e%Fft$\"+#e<_5\"F/7$$\"+&)Qk%o%Fft$ \"+o8C\"3\"F/7$$\"+UjE!z%Fft$\"+_(4o0\"F/7$$\"+60O\"*[Fft$\"+V$[W.\"F/ 7$$\"#]F($\"+F(f9,\"F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$ %&omegaG%!G-%%VIEWG6$;F(Fc`l%(DEFAULTG" 2 369 262 262 2 0 1 0 2 9 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 0 130 176 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The angle of " }{XPPEDIT 18 0 "H(j*omega)" "-%\"HG6#*&%\"jG\"\"\"%&omegaGF'" }{TEXT -1 39 " in \+ degrees can be plotted as follows:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(180/Pi*argument(H),omega=0..50);\n" }}{PARA 13 " " 1 "" {INLPLOT "6%-%'CURVESG6$7X7$\"\"!$\"$!=F(7$$\"+4x&)*3\"!\"*$\"+ Lsn_:!\"(7$$\"+#R(*Rc\"F.$\"+h)y2W\"F17$$\"+uq8Q?F.$\"+w#=WK\"F17$$\"+ OnNrDF.$\"+[c;(=\"F17$$\"+(RwX5$F.$\"+(>(>V5F17$$\"+&eI8k$F.$\"+GBlO*) !\")7$$\"+sZ3yTF.$\"+l**>TuFK7$$\"+hyG7ZF.$\"+.B\"=,'FK7$$\"+]4\\Y_F.$ \"+b\"zIp%FK7$$\"+'flVB$)F.$!+eDA\\JF.7$$\"+Mbw!Q*F.$!+ytE48FK7 $$\"+0j$o/\"FK$!+0cQ<@FK7$$\"+_>jU6FK$!+91k)p#FK7$$\"+j^Z]7FK$!+A0IWKF K7$$\"+)=h(e8FK$!+*)>t.PFK7$$\"+Q[6j9FK$!+_?W\"3%FK7$$\"+\\z(yb\"FK$!+ )*pF!Q%FK7$$\"+b/cq;FK$!+<)*G\"p%FK7$$\"+FK$!+[eVa`FK7$$\"+\"f#=$3#FK$!+`4gUbFK7$$\"+t( pe=#FK$!+Bda/dFK7$$\"+uI,$H#FK$!+CQ3eeFK7$$\"+rSS\"R#FK$!+or)p)fFK7$$ \"+`?`(\\#FK$!+5um9hFK7$$\"++#pxg#FK$!+OVJOiFK7$$\"+g4t.FFK$!+tE:MjFK7 $$\"+!Hst!GFK$!+8pKKkFK7$$\"+ERW9HFK$!+liVElFK7$$\"+KE>>IFK$!+C417mFK7 $$\"+#RU07$FK$!+l\"\\%*o'FK7$$\"+?S2LKFK$!+q]qpnFK7$$\"+$p)=MLFK$!+H(3 s$oFK7$$\"+*=]@W$FK$!+^c\"\\!pFK7$$\"+]$z*RNFK$!+6'4F'pFK7$$\"+kC$pk$F K$!+]NNAqFK7$$\"+3qcZPFK$!+3=PvqFK7$$\"+/\"fF&QFK$!+zl$y7(FK7$$\"+0Ogb RFK$!+t\"Qk<(FK7$$\"+nAFjSFK$!+1&)oCsFK7$$\"+&)*pp;%FK$!+9l!)osFK7$$\" +ye,tUFK$!+OHr6tFK7$$\"+fO=yVFK$!+t[@_tFK7$$\"+E>#[Z%FK$!+cjv(Q(FK7$$ \"+(G!e&e%FK$!+k:lEuFK7$$\"+&)Qk%o%FK$!+3U))fuFK7$$\"+UjE!z%FK$!+,a!Q \\(FK7$$\"+60O\"*[FK$!+LA!\\_(FK7$$\"#]F($!+a-#pb(FK-%'COLOURG6&%$RGBG $\"#5!\"\"F(F(-%+AXESLABELSG6$%&omegaG%!G-%%VIEWG6$;F(F]\\l%(DEFAULTG " 2 369 260 260 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 163 179 0 0 0 0 0 0 }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 446 2 " " }{TEXT 637 12 "5 Bode Plots" }}{PARA 0 "" 0 "" {TEXT -1 117 "Semilog plots can be produced with Maple V Release 4. In this example, a transfer functio n for a filter is analyzed:" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "H(s)=4 9/400*s/(s+20)/(s+700)^2*(s+2000)^2" "/-%\"HG6#%\"sG*.\"#\\\"\"\"\"$+% !\"\"F&F),&F&F)\"#?F)F+*$,&F&F)\"$+(F)\"\"#F+,&F&F)\"%+?F)F1" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "The function i s first broken into a constant (k) and basic filter functions h1(s), h 2(s), h3(s). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "restart, with(plots,semilogplot):# Loads Semil og\n" }}}{PARA 0 "" 0 "" {TEXT -1 66 "The gain constant (k). Since (k ) is positive its angle is 0 deg. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k:=49/400:" }}}{PARA 0 "" 0 "" {TEXT -1 63 "h1(s) is \+ a highpass filter, with a cutoff frequency (W) of 20. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h1:=s/(s+20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a1:=abs(h1): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h 1G*&%\"sG\"\"\",&F&F'\"#?F'!\"\"" }}}{PARA 273 "" 1 "" {TEXT -1 70 "h2 (s) is two lowpass filters, each with a cutoff frequency (W) of 700." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h2:=(700^2/(s+700)^2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a2:=abs(h2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h2G,$*$,&%\"sG\"\"\"\"$+(F)!\"#\"'++\\" }}}{PARA 274 "" 1 "" {TEXT -1 81 "h3(s) is two inverted lowpass filters, each w ith a cutoff frequency (W) of 2000 ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "h3:=((s+2000)^2/2000^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a3:=abs(h3):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h3 G,$*$,&%\"sG\"\"\"\"%+?F)\"\"##F)\"(+++%" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Frequency response of the transfer function is the characteristic of interest here. Thus, any Real component (" }{XPPEDIT 18 0 "sigma " "I&sigmaG6\"" }{TEXT -1 41 "), of a complex input is irrelevant, and " }{XPPEDIT 18 0 "s=j* omega" "/%\"sG*&%\"jG\"\"\"%&omegaGF&" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s:=I*omega:" }}} {PARA 0 "" 0 "" {TEXT -1 81 "The decibel (dB) is the logarithmic measu re of amplitude ratio. Decibel gain is " }{XPPEDIT 18 0 "g(omega)= 20 *log(a(omega))" "/-%\"gG6#%&omegaG*&\"#?\"\"\"-%$logG6#-%\"aG6#F&F)" } {TEXT -1 120 ". The sum of the individual filter gains g1, g2, g3 and k, plotted here individually, is the transfer function gain, g." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "semilogplot([20*log10(k),20* log10(a1),20*log10(a2), 20*log10(a3)],omega=1..8000,dB=-20..20);" }} {PARA 13 "" 1 "" {INLPLOT "6)-%'CURVESG6$7S7$\"\"!$!+BysB=!\")7$$\"+M# >RC#!\"*F)7$$\"+%)*GY^#F/F)7$$\"+x:%pp#F/F)7$$\"+(>#pDGF/F)7$$\"+S>b'QF/F)7$$\"+ >-![(QF/F)7$$\"+??[%)QF/F)7$$\"+h3b$*QF/F)7$$\"+()**3.RF/F)-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-F$6$7in7$F($!+sV9.EF+7$$\"+e$fY4)!#5$!+&)>3E5F+7 $$\"+,`Wv5F/$!+3#Hu#eF/7$$\"+*)[>R7F/$!+'oBDn$F/7$$\"+G(GyN\"F/$!+S!\\ $zCF/7$$\"+2A_F:F/$!+o')*4J\"F/7$$\"+fjA\\;F/$!+wb=hzFgu7$$\"+j(e?#=F/ $!+'e>Lx$Fgu7$$\"+b)e`%>F/$!+q;ly@Fgu7$$\"+1FGT?F/$!+6$oJT\"Fgu7$$\"+k _!)>@F/$!+56(=*)*!#67$$\"+]4G'=#F/$!+)4w]I(Fgx7$F-$!+Ij,8cFgx7$$\"+*)R .+CF/$!+wF7WFFgx7$F1$!+%=s4i\"Fgx7$F4$!+'pm#3q!#77$F7$!+k;&\\(QF[z7$F: $!+#RP$fCF[z7$F=$!+N()*4u\"F[z7$F@$!+#z:WG\"F[z7$FC$!+#)*R:y*!#87$FF$! +o[G-xF[[l7$FI$!+yt$f='F[[l7$FL$!+?)zF>&F[[l7$FO$!+GQF[[l7$F^o$!+!H&QQ9F[[l7$Fgo$!+VH?!H\"F[[l7$Fjo$!+M([i=\"F[[l7$F]p$!+%z,w3\"F[[l7$F`p$! 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In this example, k is pos itive and its angle is zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "theta:=((argument(h1)+argument(h2)+argument(h3)) *180/Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG,$*&,(-%)argumentG6#*(%\"IG\" \"\"%&omegaGF-,&*&F,F-F.F-F-\"#?F-!\"\"F--F)6#*$,&F0F-\"$+(F-!\"#F--F) 6#*$,&F0F-\"%+?F-\"\"#F-F-%#PiGF2\"$!=" }}}{PARA 0 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "semilogplot(theta,omega =1..8000,angle=-90..90);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7 \\o7$\"\"!$\"+T)=Jq)!\")7$$\"+e$fY4)!#5$\"+gdHWrF+7$$\"+,`Wv5!\"*$\"+u '[()z&F+7$$\"+*)[>R7F5$\"+NS+AZF+7$$\"+G(GyN\"F5$\"+]R(R)QF+7$$\"+2A_F :F5$\"+\"\\B8r#F+7$$\"+fjA\\;F5$\"+]YJU>F+7$$\"+j(e?#=F5$\"+#HMRt*F57$ $\"+b)e`%>F5$\"+jC**oMF57$$\"+k_!)>@F5$!+Zw`^^F57$$\"+M#>RC#F5$!+NW!*f 6F+7$$\"+*)R.+CF5$!+XrFh?F+7$$\"+%)*GY^#F5$!+OzE,GF+7$$\"+#[#G:EF5$!+, Xr#\\$F+7$$\"+x:%pp#F5$!+^qUdSF+7$$\"+(>#pDGF5$!+iY!=([F+7$$\"+3K(y(GF 5$!+\"3fX9&F+7$$\"+SsXc%F+7$$\"+f.4FMF5$!+Oy^#Q%F+7$$\"+SV@^MF5$!+KxFMUF+7$$\"+,woxMF5$!+ *zr%pSF+7$$\"+*)f%)*\\$F5$!+QspIRF+7$$\"+CL#H_$F5$!+,'Rhy$F+7$$\"+^T\" Qa$F5$!+)4Gel$F+7$$\"+'f!fkNF5$!+DzAFNF+7$$\"+g@$Ge$F5$!+SkY:MF+7$$\"+ Wao,OF5$!+3TO,LF+7$$\"+J*Q/i$F5$!+_\\_*=$F+7$$\"+g&Hhj$F5$!+'fyt4$F+7$ $\"+*QiCl$F5$!+3w'H+$F+7$$\"+KXroOF5$!+4Vl5HF+7$$\"+?k/%o$F5$!+9H9DGF+ 7$$\"+kFQ)p$F5$!+$HFmu#F+7$$\"+)HlPr$F5$!+4p)Rm#F+7$$\"+ys8FPF5$!+$*Q` $f#F+7$$\"+nV(4u$F5$!+pi,ADF+7$$\"++J9`PF5$!+yXIgCF+7$$\"+Wy1mPF5$!+lc )fR#F+7$$\"+Jv)yx$F5$!+5:GQBF+7$$\"+.!3**y$F5$!+Zlp!G#F+7$$\"+!3Z8!QF5 $!+y)HpA#F+7$$\"+@%3I\"QF5$!+Dr:t@F+7$$\"+27&R#QF5$!+R7lB@F+7$$\"+DP'[ $QF5$!+E+?v?F+7$$\"+o;UXQF5$!+/G>H?F+7$$\"+6@!\\&QF5$!+Sog))>F+7$$\"+! 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If it looks like a " }{TEXT 509 16 "Lissajous fig ure" }{TEXT -1 18 ", and acts like a " }{TEXT 510 20 "Lissajous figure ..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 638 15 " 7 Convolution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "The convolution of two functions x(t) and h(t)is defined as:" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{XPPEDIT 18 0 "x(t)*h(t)=int(x(lambda)*h(t-lambda),lambda=-infinity ..infinity)" "/*&-%\"xG6#%\"tG\"\"\"-%\"hG6#F'F(-%$intG6$*&-F%6#%'lamb daGF(-F*6#,&F'F(F2!\"\"F(/F2;,$%)infinityGF6F:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 186 "There is no built in function in Maple f or this operation, but the integral itself is easily evaluated after d efining x(t) and h(t). The following example illustrates the evaluatio n of " }{XPPEDIT 18 0 "y(t)=x(t)*h(t)" "/-%\"yG6#%\"tG*&-%\"xG6#F&\" \"\"-%\"hG6#F&F+" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "h:=t-> Heaviside(t)-Heaviside(t-10);# rectangle of width 10" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG:6#%\"tG6\"6$%)operatorG%&arrowGF(,&-% *HeavisideG6#9$\"\"\"-F.6#,&F0F1!#5F1!\"\"F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "x:=t-> exp(-t)*Heaviside(t);# causal exponenti al" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG:6#%\"tG6\"6$%)operatorG%& arrowGF(*&-%$expG6#,$9$!\"\"\"\"\"-%*HeavisideG6#F1F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "y:=t-> int(x(lambda)*h(t-lambda), l ambda=-infinity..infinity);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"y G:6#%\"tG6\"6$%)operatorG%&arrowGF(-%$intG6$*&-%\"xG6#%'lambdaG\"\"\"- %\"hG6#,&9$F4F3!\"\"F4/F3;,$%)infinityGF:F>F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 72 "Note that Maple has truncated the lower limit and omitted the Heaviside(" }{XPPEDIT 18 0 "lambda" "I'lambdaG6\"" }{TEXT -1 7 ") for x(" }{XPPEDIT 18 0 "lambda" "I'lambdaG6\"" }{TEXT -1 3 ").\n" }} {PARA 0 "" 0 "" {TEXT -1 35 "To see the waveform of y(t), use,:\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(y(t),t=-1..20);\n" }} {PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7co7$$!\"\"\"\"!$\"1D9jMmUhE! #F7$$!1,++DwfAa!#;$!1%*[41!oS!=!#E7$$!1+]Pf4@JMF1$\"1?(Gwq=pH)F-7$$!1* **\\PHC)R9F1$!1ak!G,_z$=F47$$!1%\\(=#\\O%*z)!#<$\"1w'e!zG&Hd\"F47$$!1% *\\(o/I1?$FB$\"1prOvoZB7F47$$\"11vV)Rw\")R#FB$\"1!Q$*Gl['pBFB7$$\"11+v VG)p*zFB$\"1*[HL8!e&o(FB7$$\"1,vVt&f%>>F1$\"1U3A&G&[YEN?\"!#:$\"1A)GA) )f')*pF17$$\"1+]P\\:a69Fdp$\"13!o$eyKivF17$$\"1+]i+pb>;Fdp$\"1;A(p#f8? !)F17$$\"1+]7[Z%\\$=Fdp$\"1+lW!zwPS)F17$$\"1+]i&fK.0#Fdp$\"1!H*ef!zIr) F17$$\"1+]iN9%e\\#Fdp$\"1bmk[$Hd<*F17$$\"1+]7B:#*RHFdp$\"1kQhB7Ir%*F17 $$\"1++]xCr'R$Fdp$\"11a!*z$Fdp$\"1/(4$)f!3w(*F17$ $\"1++]#o&*>D%Fdp$\"1OZ'e/Uw&)*F17$$\"1++]()pz1ZFdp$\"1U!=wLj'4**F17$$ \"1++](***F17 $$\"1,+D1\\lI')Fdp$\"1>7DBX@)***F17$$\"1+]7$4(*Q/*Fdp$\"1'=fM!*=))***F 17$$\"1***\\(=Yj*[*Fdp$\"1sMn#oV#****F17$$\"1*\\(=FE8@(*Fdp$\"1P&y*z** R****F17$$\"1**\\iN1j_**Fdp$\"1(y]v(R_****F17$$\"1]iS[!R`+\"!#9$\"1s') >zrmz%*F17$$\"1++DL]T:5F]x$\"1@'HhSU5d)F17$$\"1]P4=5\\D5F]x$\"1JoX!>4& \\xF17$$\"1+v$H+nb.\"F]x$\"1\\me=)>n+(F17$$\"1v$4EM\\k/\"F]x$\"15#G\"4 &[UG'F17$$\"1]7G#oJt0\"F]x$\"1L'3b3sij&F17$$\"1DJ&>-9#o5F]x$\"1CIV!H4^ 0&F17$$\"1+]ihj4z5F]x$\"1elG!*3(Q`%F17$$\"1+D109e,6F]x$\"1J%oH'4\"4i$F 17$$\"1++][k1C6F]x$\"1_>W5')y\"*GF17$$\"1+D\"onjg9\"F]x$\"1\"okM/z2K#F 17$$\"1+]7041o6F]x$\"1V$G3V?D'=F17$$\"1]PfaVM*=\"F]x$\"1T))GdkX0:F17$$ \"1+D1/yi57F]x$\"1D!fHDXo@\"F17$$\"1+](y3\"*yD\"F]x$\"1H;\\\"z@:$FB7$$\"1 +DcE8z'Q\"F]x$\"1QN,!=*4!4#FB7$$\"1++DM;rJ9F]x$\"1IRXk&oPL\"FB7$$\"1+D 1V\"yRZ\"F]x$\"1v4,u&f,u)!#=7$$\"1+DJB)e\"=:F]x$\"1.^5<)>)=cF_^l7$$\"1 ++v8NNh:F]x$\"1?z-)>%)zk$F_^l7$$\"1+Dc^Vd1;F]x$\"1n$[[FE4K#F_^l7$$\"1+ ++$RF,l\"F]x$\"1tn*p'pX,:F_^l7$$\"1+]Pomm%p\"F]x$\"1N3l@D#zh*!#>7$$\"1 +D1Or$)QP%f#=F]x$\"1d(o3cGze#Fi_l7$$\"1+++J/bn=F]x$\"1yS=W342\">F]x$\"1)f_C;ia4\"Fi_l7$$\"1+v$RTrV&>F]x$\"1I]H(4)pkr!#?7$$\"# ?F*$\"1g4*3'oyRXFhal-%'COLOURG6&%$RGBG$\"#5F)F*F*-%+AXESLABELSG6$%\"tG %!G-%%VIEWG6$;F(Fjal%(DEFAULTG" 2 426 214 214 2 0 1 0 2 9 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 0 180 246 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 89 "There are three distinct regions of y(t) as (t) changes f rom (-)infinity to (+)infinity:\n" }}{PARA 0 "" 0 "" {TEXT -1 76 " y(t )=\{ 0 \+ " }{XPPEDIT 18 0 "t<=0" "1%\"tG\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }{XPPEDIT 18 0 "int(e^(-lambda),lambda= 0..t)" "-%$intG6$)%\"eG,$%'lambdaG!\"\"/F(;\"\"!%\"tG" }{TEXT -1 35 " \+ " }{XPPEDIT 18 0 "010" "2\"#5%\"tG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 98 " This result may also be obtained by ev aluating the convolution integral in the three regions of " }{XPPEDIT 18 0 "y(t)" "-%\"yG6#%\"tG" }{TEXT -1 18 " as defined above." }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 8 State Variable Analysis" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "In this \+ example, Maple is used to analyze a linear circuit or system described by the following matrix:\n" }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }{XPPEDIT 18 0 "dq/dt=Aq+Bx" "/*&%#dqG\"\"\"% #dtG!\"\",&%#AqGF%%#BxGF%" }{TEXT -1 40 "\n \+ " }{XPPEDIT 18 0 "y=Cq+Dx" "/%\"yG,&%#CqG\"\"\"%#DxGF&" } {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 74 " where matrices A and C include a parameter (R) and are functions of (R).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "restart: with(linalg): with(inttrans):\n " }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }} {PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition for hilbert" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Defining \+ matrix parameters\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "A:=( R)->matrix(3,3,[-R/3,-1/3,0,60,-3,3,0,1,-1]):\nB:=matrix(3,2,[1/3,0,0, 60,0,-20]):\nC:=(R)->matrix(3,3,[-R,0,0,0,1,0,0,1,-1]):\nE:=matrix(3,2 ,[1,0,0,0,0,0]):\ni:=array(identity,1..3,1..3):# identity matrix" } {TEXT -1 1 "\n" }{TEXT 606 5 "Note:" }{TEXT -1 76 " \"D\" is Maple's n ame for the differential operator and can not be assigned. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "To find the cha racteristic polynomial P(s) in terms of R:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "P:=(s,R)->det(evalm((s*i-A(R)))):\nP(s,R);# to see \+ P(s)" }{TEXT -1 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$%\"sG\"\" $\"\"\"*$F%\"\"#\"\"%*&%\"RGF'F%F)#F'F&*&F,F'F%F'#F*F&F%\"#?F0F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "To find t he roots of P(s) for R=3:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:=solve(P(s,3)=0,s):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Now the " }{XPPEDIT 18 0 "vector (p)" "-%'vectorG6#% \"pG" }{TEXT -1 34 " contains three solutions of P(s)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "p;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\",&!\"#\"\"\"%\"IG\"\"%,&F%F&F'!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "To find the resolvent mat rix " }{XPPEDIT 18 0 "phi(s)=inverse(sI-A)" "/-%$phiG6#%\"sG-%(inverse G6#,&%#sIG\"\"\"%\"AG!\"\"" }{TEXT -1 2 ":\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "phi:=(s,R)->inverse((s*i-A(R))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "phi(s,3);# R=3, as an example:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%*(%\"sG\"\"\",&F)F*\"\"%F*F*,**$F)\" \"$F**$F)\"\"#\"\"&F)\"#C\"#?F*!\"\",$*$,(F0F*F)F,F4F*F5#F5F/,$*$F-F5F 57%,$F7\"#g*&,&F)F*F*F*F*F8F5,$F7F/7%,$F;F>F7*&,(F0F*F)F,\"#BF*F*F-F5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Give n q(0-), the zero input response can be found." }}{PARA 0 "" 0 "" {TEXT -1 28 " Q(s)=" }{XPPEDIT 18 0 "phi" "I$phi G6\"" }{TEXT -1 24 "(s)q(0-) and Y(s)=CQ(s)\n" }}{PARA 0 "" 0 "" {TEXT -1 33 "Define the initial value matrix:\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "q0:=matrix(3,1,[0,1,0]):\n" }}}{PARA 0 "" 0 " " {TEXT -1 36 "Evaluate q and phi in terms of (R).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q:=(s,R)->evalm(phi(s,R)&*q0):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Y:=(s,R)->evalm(C(R)&*Q(s,R) ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Ve rify with R=3\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "map(simpl ify,Y(s,3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7#*$,( *$%\"sG\"\"#\"\"\"F+\"\"%\"#?F-!\"\"7#*&,&F+F-F-F-F-F)F07#*&F+F-F)F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Find t he Laplace inverse of Y(s) with R=10 and plot for R=10:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "y1:=(t)->evalf(invlaplace(Y(s,10)[ 1,1],s,t),4):\ny2:=(t)->evalf(invlaplace(Y(s,10)[2,1],s,t),4):\ny3:=(t )->evalf(invlaplace(Y(s,10)[3,1],s,t),4):" }{TEXT -1 1 "\n" }}}{PARA 0 "" 0 "" {TEXT -1 80 "Note: evalf(argument,4) converts the argument i nto a four digit floating point.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(y1(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, (*&-%$expG6#,$%\"tG$!++++=L!\"*\"\"\"-%$cosG6#,$F)$\"++++2UF,F-$!++++; T!#6*&F%F--%$sinGF0F-$\"++++nw!#5-F&6#,$F)$!++v>mpF<$\"++++;TF6" }}} {PARA 12 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(y2(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&-%$expG6 #,$%\"tG$!++++=L!\"*\"\"\"-%$cosG6#,$F)$\"++++2UF,F-$\"++++u'*!#5*&F%F --%$sinGF0F-$\"++++Rb!#6-F&6#,$F)$!++v>mpF6$\"++++bKF<" }}}{PARA 12 " " 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simpl ify(y3(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&-%$expG6#,$%\"tG$!+ +++=L!\"*\"\"\"-%$cosG6#,$F)$\"++++2UF,F-$\"++++v5F,*&F%F--%$sinGF0F-$ !++++a6!#5-F&6#,$F)$!++v>mpF;$!++++vu!#6" }}}{PARA 0 "" 0 "" {TEXT -1 51 "Note (y.(1..3)(t)) is same as (y1(t),y2(t),y3(t)).\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot([y1(t),y2(t),y3(t)],t=0..2);" }{TEXT -1 1 "\n" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7eo7$\"\"!F(7$$\" +X&)G\\a!#7$\"+b[C&y\"!#67$$\"+4x&)*3\"F/$\"+d$H&3NF/7$$\"+klyM;F/$\"+ 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,sin(q2),cos(q2),0,a2*sin(q2),0,0,1,0,0,0,0,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "T3:=(q3,a3)->matrix(4,4,[cos(q3),-sin(q3 ),0,a3*cos(q3) ,sin(q3),cos(q3),0,a3*sin(q3),0,0,1,0,0,0,0,1]):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "To see th e matrices;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "T1(q1,a1); T2 (q2,a2); T3(q3,a3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7 &7&-%$cosG6#%#q1G\"\"!,$-%$sinGF*!\"\"F,7&F.F,F(F,7&F,F0F,%#a1G7&F,F,F ,\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&-%$cosG6#%# q2G,$-%$sinGF*!\"\"\"\"!*&%#a2G\"\"\"F(F37&F-F(F0*&F2F3F-F37&F0F0F3F07 &F0F0F0F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&-%$cosG6# %#q3G,$-%$sinGF*!\"\"\"\"!*&%#a3G\"\"\"F(F37&F-F(F0*&F2F3F-F37&F0F0F3F 07&F0F0F0F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now compute the product T = T1 T2 T3;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "T:=(q1,q2,q3,a1,a2,a3)->T1(q1,a1)&*T2(q2,a2) \+ &*T3(q3,a3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The specific case [q1,q2,q3]=[0,-Pi/2,Pi/2] yields:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(T(0,-Pi/2,Pi/2,a1,a2,a 3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!F )%#a3G7&F)F)F(F)7&F)!\"\"F),&%#a2GF(%#a1GF(7&F)F)F)F(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "10 Electromagnetics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Maple is a useful tool f or evaluating the complicated integrals and differential equations tha t occur in electromagnetics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 447 16 "Flux of a Vect or" }}{PARA 0 "" 0 "" {TEXT -1 27 "A density of current J = r " } {XPPEDIT 18 0 "r *cos(theta)+(theta)*r^3+(phi)*r^2" ",(*&%\"rG\"\"\"-% $cosG6#%&thetaGF%F%*&F)F%*$F$\"\"$F%F%*&%$phiGF%*$F$\"\"#F%F%" }{TEXT -1 5 " (r, " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 74 " are vectors), emerging out of a surface bounded by spherical coordinates:" }}{PARA 0 "" 0 " " {XPPEDIT 18 0 "0<=r" "1\"\"!%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r<=2" "1%\"rG\"\"#" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "Pi/3<=(theta )" "1*&%#PiG\"\"\"\"\"$!\"\"%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta<=Pi" "1%&thetaG%#PiG" }{TEXT -1 5 ", " }{XPPEDIT 18 0 " 0<=(phi)" "1\"\"!%$phiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi<=2Pi " "1%$phiG*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The integrals to evaluate are given by:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "int(int(r^3*cos(th eta)*sin(theta),theta=Pi/3..Pi)\\ ,phi=0..2*Pi);" "-%$intG6$-F#6$*(%\" rG\"\"$-%$cosG6#%&thetaG\"\"\"-%$sinG6#F-F./F-;*&%#PiGF.F)!\"\"F5/%$ph iG;\"\"!*&\"\"#F.F5F." }{TEXT -1 5 " - " }{XPPEDIT 18 0 "int(int(-r^ 4*sin(theta),phi=0..2*Pi),r=0..2);" "-%$intG6$-F#6$,$*&%\"rG\"\"%-%$si nG6#%&thetaG\"\"\"!\"\"/%$phiG;\"\"!*&\"\"#F/%#PiGF//F);F4F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The above integration is easily carried out in two steps:\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "restart; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "aflux:=int(int(r^3*cos(theta)*sin(theta), theta= Pi/3..Pi),phi=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&afluxG,$ *&%#PiG\"\"\"%\"rG\"\"$#!\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "bflux:=int(int(-r^4*sin(theta),phi=0..2*Pi) ,r=0..2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&bfluxG,$*&%#PiG\"\"\"-%$sinG6#% &thetaGF(#!#k\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r:=2: theta:=Pi/3:" }{TEXT -1 1 "\n" }}}{PARA 0 "" 0 "" {TEXT -1 85 "The in tegral may still be evaluated without specifying the values of (r) or \+ (theta).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "totalflux:=afl ux+bflux;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*totalfluxG,&%#PiG!\" '*&F&\"\"\"\"\"$#F)\"\"##!#K\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 448 38 "Radiation Resistance o f a Wire Antenna" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 177 "By \+ observing the input current to a transmitting antenna at its feed and \+ the average power radiated from the antenna, a relationship can be fou nd, called Radiation Resistance R(" }{XPPEDIT 18 0 "Omega" "I&OmegaG6 \"" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "R(Omega )=2*Pa ve/(Io)^2" "/-%\"RG6#%&OmegaG*(\"\"#\"\"\"%%PaveGF)*$%#IoGF(!\"\"" } {TEXT -1 9 " , where " }{TEXT 608 1 "P" }{TEXT 659 3 "ave" }{TEXT -1 35 " is the average radiated power and " }{XPPEDIT 18 0 "Io" "I#IoG6\" " }{TEXT -1 25 " is the rms feed current." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "The Pave is obtained by calculati ng the flux of the real part of the Poynting vector through an arbitra ry surface which encloses the antenna.\nFor a spherical surface and a \+ half-wave antenna (length " }{XPPEDIT 18 0 "(lambda)/2" "*&%'lambdaG \"\"\"\"\"#!\"\"" }{TEXT -1 61 " ), Pave is given by:\n \+ " }{XPPEDIT 18 0 "Pave=(eta /Pi)" "/%%PaveG*&% $etaG\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " int((cos(Pi*co s(theta)/2))^2/sin(theta),theta=0..Pi" "-%$intG6$*&-%$cosG6#*(%#PiG\" \"\"-F'6#%&thetaGF+\"\"#!\"\"F/-%$sinG6#F.F0/F.;\"\"!F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "eta" "I$etaG6\"" }{TEXT -1 110 " i s the intrinsic impedance of free space. It is the square root of the ratio of the electrical permutivity (" }{XPPEDIT 18 0 "epsilon" "I(ep silonG6\"" }{TEXT 660 1 "o" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "8.864*10 ^(-12)" "*&$\"%k))!\"$\"\"\")\"#5,$\"#7!\"\"F&" }{TEXT -1 32 ") to the magnetic permeability (" }{XPPEDIT 18 0 "mu" "I#muG6\"" }{TEXT 661 2 "o " }{TEXT -1 2 "= " }{XPPEDIT 18 0 "4*Pi*10^(-7)" "*(\"\"%\"\"\"%#Pi GF$)\"#5,$\"\"(!\"\"F$" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 43 "The integral is evaluated in two steps, ( " }{XPPEDIT 18 0 "eta" "I$etaG6\"" }{TEXT -1 14 " = squareroot(" }{XPPEDIT 18 0 "mu" "I#muG6 \"" }{TEXT 662 1 "o" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "epsilon" "I(epsil onG6\"" }{TEXT 663 1 "o" }{TEXT -1 25 ") has been substituted):\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "evalf(int(((cos(Pi/2*cos(theta)))^2)/sin(theta),the ta=0..Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$[LT4'!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalf(sqrt(4*Pi*10^(-7)/8.86 4/10^(-12))/Pi)*\";\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(3bQI(!\" )" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "47 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }