THE MAPLE BOOK
CONTENTS 1 GETTING STARTED STARTING A MAPLE SESSION 1 2 DI?ERENT VERSIONS OF MAPLE BASIC SYNTAX EDITING MISTAKES HELP A SAMPLE SESSION AND CONTEXT MENUS PALETTES SPREADSHEETS QUITTING MAPLE 2 MAPLE AS A CALCULATOR EXACT ARITHMETIC AND BASIC FUNCTIONS FLOATING-POINT ARITHMETIC 3 HIGH SCHOOL ALGEBRA POLYNOMIALS AND RATIONAL FUNCTIONS FACTORING A POLYNOMIAL EXPANDING AN EXPRESSION COLLECTING LIKE TERMS SIMPLIFYING AN EXPRESSION SIMPLIFYING RADICALS WORKING IN THE REAL DOMAIN SIMPLIFYING RATIONAL FUNCTIONS DEGREE AND COE?CIENTS OF A POLYNOMIAL SUBSTITUTING INTO AN EXPRESSION RESTORING VARIABLE STATUS EQUATIONS LEFT- AND RIGHT-HAND SIDES FINDING EXACT SOLUTIONS FINDING APPROXIMATE SOLUTIONS ASSIGNING SOLUTIONS FUN WITH INTEGERS COMPLETE INTEGER FACTORIZATION QUOTIENT AND REMAINDER GCD AND LCM PRIMES INTEGER SOLUTIONS REDUCTION MOD P UNIT CONVERSION TRIGONOMETRY DEGREES AND RADIANS TRIGONOMETRIC FUNCTIONS SIMPLIFYING TRIGONOMETRIC FUNCTIONS 4 DATA TYPES SEQUENCES SETS LISTS TABLES ARRAYS DATA CONVERSIONS OTHER DATA TYPES 5 CALCULUS 5 1 DE?NING FUNCTIONS COMPOSITION OF FUNCTIONS SUMMATION AND PRODUCT LIMITS 5 5 DI?ERENTIATION EXTREMA INTEGRATION TECHNIQUES OF INTEGRATION SUBSTITUTION INTEGRATION BY PARTS PARTIAL FRACTIONS TAYLOR AND SERIES EXPANSIONS THE STUDENT PACKAGE 6 GRAPHICS TWO-DIMENSIONAL PLOTTING RESTRICTING DOMAIN AND RANGE PARAMETRIC PLOTS MULTIPLE PLOTS POLAR PLOTS PLOTTING IMPLICIT FUNCTIONS PLOTTING POINTS TITLE AND TEXT IN A PLOT PLOTTING OPTIONS SAVING AND PRINTING A PLOT OTHER PLOT FUNCTIONS THREE-DIMENSIONAL PLOTTING SPACE CURVES CONTOUR PLOTS PLOTTING SURFACES DE?NED IMPLICITLY THREE-DIMENSIONAL PLOTTING OPTIONS OTHER THREE-DIMENSIONAL PLOT FUNCTIONS ANIMATION 7 MAPLE PROGRAMMING THE MAPLE PROCEDURE LOCAL AND GLOBAL VARIABLES CONDITIONAL STATEMENTS BOOLEAN EXPRESSIONS THE “FOR” LOOP TYPE DECLARATION THE “WHILE” LOOP RECURSIVE PROCEDURES EXPLICIT RETURN ERROR STATEMENT ARGS AND NARGS INPUT AND OUTPUT FORMATTED OUTPUT INTERACTIVE INPUT READING COMMANDS FROM A ?LE READING DATA FROM A ?LE WRITING DATA TO A ?LE WRITING AND SAVING TO A ?LE GENERATING C AND FORTRAN CODE VIEWING BUILT-IN MAPLE CODE THE MAPLE INTERACTIVE DEBUGGER WRITING YOUR OWN PACKAGES PACKAGES AS TABLES MODULES FOR PACKAGES ANSWERS TO PROGRAMMING EXERCISES 8 DI?ERENTIAL EQUATIONS 8 1 SOLVING ORDINARY DI?ERENTIAL EQUATIONS IMPLICIT SOLUTIONS INITIAL CONDITIONS SYSTEMS OF DI?ERENTIAL EQUATIONS 8 2 FIRST-ORDER DI?ERENTIAL EQUATIONS ODEADVISOR INTEGRATING FACTORS DIRECTION ?ELDS NUMERICAL SOLUTIONS SECOND- AND HIGHER ORDER LINEAR DES CONSTANT COE?CIENTS VARIATION OF PARAMETERS REDUCTION OF ORDER SERIES SOLUTIONS THE METHOD OF FROBENIUS THE LAPLACE TRANSFORM THE HEAVISIDE FUNCTION THE DIRAC DELTA FUNCTION THE DETOOLS PACKAGE DE PLOTTING FUNCTIONS DYNAMICAL SYSTEMS DE MANIPULATION LIE SYMMETRY METHODS DI?ERENTIAL OPERATORS CLOSED FORM SOLUTIONS SIMPLIFYING DES AND RIFSIMP 9 LINEAR ALGEBRA VECTORS, ARRAYS, AND MATRICES MATRIX AND VECTOR ENTRY ASSIGNMENT THE MATRIX AND VECTOR PALETTES MATRIX OPERATIONS MATRIX AND VECTOR CONSTRUCTION SHORTCUTS VIEWING LARGE MATRICES AND VECTORS MATRIX CONTEXT MENU THE EXPORT AS SUBMENU THE NORM SUBMENU THE SELECT ELEMENT SUBMENU THE SOLVERS SUBMENU THE CONVERSIONS SUBMENU THE IN-PLACE OPTIONS SUBMENU ELEMENTARY ROW AND COLUMN OPERATIONS GAUSSIAN ELIMINATION INVERSES, DETERMINANTS, MINORS, AND THE ADJOINT SPECIAL MATRICES AND VECTORS BAND MATRIX CONSTANT MATRICES AND VECTORS DIAGONAL MATRICES GIVENS ROTATION MATRICES HANKEL MATRICES HILBERT MATRICES HOUSEHOLDER MATRICES IDENTITY MATRIX JORDAN BLOCK MATRICES RANDOM MATRICES AND VECTORS TOEPLITZ MATRICES VANDERMONDE MATRICES ZERO MATRICES AND VECTORS SYSTEMS OF LINEAR EQUATIONS ROW SPACE, COLUMN SPACE, AND NULLSPACE EIGENVECTORS AND DIAGONALIZATION JORDAN FORM INNER PRODUCTS, AND VECTOR AND MATRIX NORMS THE DOT PRODUCT AND BILINEAR FORMS VECTOR NORMS MATRIX NORMS LEAST SQUARES PROBLEMS QR-FACTORIZATION AND THE GRAM-SCHMIDT PROCESS LU-FACTORIZATION OTHER LINEARALGEBRA FUNCTIONS THE LINALG PACKAGE MATRICES AND VECTORS CONVERSION BETWEEN LINALG AND LINEARALGEBRA MATRIX OPERATIONS IN LINALG THE FUNCTIONS IN THE LINALG PACKAGE 10 MULTIVARIABLE AND VECTOR CALCULUS VECTORS VECTOR OPERATIONS LENGTH, DOT PRODUCT, AND CROSS PRODUCT PLOTTING VECTORS LINES AND PLANES LINES PLANES VECTOR-VALUED FUNCTIONS DI?ERENTIATION AND INTEGRATION OF VECTOR FUNCTIONS TANGENTS AND NORMALS TO CURVES CURVATURE THE GRADIENT AND DIRECTIONAL DERIVATIVES LOCAL EXTREMA AND SADDLE POINTS LAGRANGE MULTIPLIERS MULTIPLE INTEGRALS DOUBLE INTEGRALS TRIPLE INTEGRALS THE JACOBIAN 10 7 VECTOR ?ELD PLOTTING A VECTOR ?ELD DIVERGENCE AND CURL POTENTIAL FUNCTIONS LINE INTEGRALS GREEN’S THEOREM SURFACE INTEGRALS FLUX OF A VECTOR ?ELD STOKE’S THEOREM THE DIVERGENCE THEOREM 11 COMPLEX ANALYSIS ARITHMETIC OF COMPLEX NUMBERS POLAR FORM NTH ROOTS THE CAUCHY-RIEMANN EQUATIONS AND HARMONIC FUNCTIONS ELEMENTARY FUNCTIONS CONFORMAL MAPPING TAYLOR SERIES AND LAURENT SERIES CONTOUR INTEGRALS RESIDUES AND POLES 12 OPENING, SAVING, AND EXPORTING WORKSHEETS OPENING AN EXISTING WORKSHEET SAVING A WORKSHEET 12 3 OPENING A MAPLE TEXT ?LE EXPORTING WORKSHEETS AND LATEX 13 DOCUMENT PREPARATION ADDING TEXT INSERTING MATH INTO TEXT ADDING TITLES AND HEADINGS CREATING A SUBSECTION CUTTING AND PASTING BOOKMARKS AND HYPERTEXT 14 MORE GRAPHICS THE PLOTTOOLS PACKAGE TWO-DIMENSIONAL PLOT OBJECTS THREE-DIMENSIONAL PLOT OBJECTS TRANSFORMATION OF PLOTS THE GEOMETRY PACKAGE THE GEOM3D PACKAGE REGULAR POLYHEDRA QUASI-REGULAR POLYHEDRA THE ARCHIMEDEAN SOLIDS 15 SPECIAL FUNCTIONS OVERVIEW OF MATHEMATICAL FUNCTIONS BESSEL FUNCTIONS THE GAMMA FUNCTION HYPERGEOMETRIC FUNCTIONS ELLIPTIC INTEGRALS THE AGM JACOBI’S THETA FUNCTIONS ELLIPTIC FUNCTIONS THE RIEMANN ZETA-FUNCTION ORTHOGONAL POLYNOMIALS INTEGRAL TRANSFORMS FOURIER TRANSFORMS HILBERT TRANSFORM MELLIN TRANSFORM FAST FOURIER TRANSFORM ASYMPTOTIC EXPANSION 16 STATISTICS INTRODUCTION DATA SETS NUMERICAL METHODS FOR DESCRIBING DATA DESCRIBING THE CENTER OF A DATA SET DESCRIBING THE DISPERSION OF A DATA SET DESCRIBING CHARACTERISTICS OF A DATA SET TRANSFORMING DATA GRAPHICAL METHODS FOR DESCRIBING DATA HISTOGRAM BOX PLOT SCATTER PLOT LINEAR REGRESSION ANOVA DISTRIBUTIONS EVALUATING DISTRIBUTIONS GENERATING RANDOM DISTRIBUTIONS 17 OVERVIEW OF OTHER PACKAGES 17 1 FINITE ?ELDS POLYNOMIALS GROUP THEORY COMBINATORICS THE COMBINAT PACKAGE THE NETWORKS PACKAGE THE COMBSTRUCT PACKAGE NUMBER THEORY THE NUMTHEORY PACKAGE THE GAUSSINT PACKAGE P-ADIC NUMBERS NUMERICAL APPROXIMATION MISCELLANEOUS PACKAGES THE ALGCURVES PACKAGE THE CODEGEN PACKAGE THE DI?ALG PACKAGE THE DI?ORMS PACKAGE THE DOMAINS PACKAGE THE ?NANCE PACKAGE THE GENFUNC PACKAGE THE GROEBNER PACKAGE THE LIESYMM PACKAGE THE LRETOOLS PACKAGE THE ORE ALGEBRA PACKAGE THE PDETOOLS PACKAGE THE POWSERIES PACKAGE THE PROCESS PACKAGE THE SIMPLEX PACKAGE THE SLODE PACKAGE THE SPREAD PACKAGE THE SUMTOOLS PACKAGE THE TENSOR PACKAGE NEW PACKAGES THE CURVEFITTING PACKAGE THE EXTERNALCALLING PACKAGE THE LINEARFUNCTIONALSYSTEMS PACKAGE THE LINEAROPERATORS PACKAGE THE LISTTOOLS PACKAGE THE MATHML PACKAGE THE ORTHOGONALSERIES PACKAGE THE RANDOMTOOLS PACKAGE THE RATIONALNORMALFORMS PACKAGE THE REALDOMAIN PACKAGE THE SOCKETS PACKAGE THE SOLVETOOLS PACKAGE THE STRINGTOOLS PACKAGE THE UNITS PACKAGE THE XMLTOOLS PACKAGE STARTING A MAPLE SESSION ON MOST SYSTEMS A MAPLE SESSION IS STARTED BY DOUBLE CLICKING ON THE MAPLE ICON IN THE UNIX X WINDOWS VERSION, MAPLE IS STARTED BY ENTERING THE COMMAND XMAPLE IN THE COMMAND-LINE (TTY) VERSION, THE MAPLE LOGO APPEARS FOLLOWED BY THE > PROMPT IN MOST VERSIONS A WINDOW WITH MENUS WILL APPEAR SEE FIGURE 1 1 BELOW AT THE TOP ARE THE MENUS FILE, EDIT, VIEW, INSERT, FORMAT, SPREADSHEET, OPTIONS, WINDOW AND HELP BENEATH ARE TWO ROWS OF BUTTONS THE ?RST ROW OF BUTTONS IS CALLED THE TOOL BAR AND CONTAINS 24 BUTTONS: CREATE A NEW WORKSHEET OPEN AN EXISTING WORKSHEET OPEN A SPECI?ED URL SAVE THE ACTIVE WORKSHEET PRINT THE ACTIVE WORKSHEET CUT THE SELECTION TO THE CLIPBOARD COPY THE SELECTION TO THE CLIPBOARD PASTE THE CLIPBOARD CONTENTS INTO THE CURRENT WORK- SHEET UNDO THE LAST OPERATION UNDO THE LAST “UNDO ”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’T WORRY ABOUT THE BUTTONS TOO MUCH AT THIS STAGE GETTING STARTED DIFFERENT VERSIONS OF MAPLE THE CURRENT VERSION OF MAPLE IS MAPLE THE PREVIOUS VERSION WAS MAPLE BEFORE THAT, THERE WAS MAPLE V RELEASE BASIC SYNTAX IN MAPLE THE PROMPT IS THE SYMBOL > MAPLE COMMANDS ARE ENTERED TO THE RIGHT OF THE PROMPT EACH COMMAND ENDS WITH EITHER “:” OR “;” IF THE COLON IS USED, THE COMMAND IS EXECUTED BUT THE OUTPUT IS NOT PRINTED WHEN THE SEMICOLON IS USED, THE OUTPUT IS PRINTED TRY TYPING 105/25: FOLLOWED BY A RETURN (OR ENTER) > 105/25: OBSERVE THAT THE OUTPUT WAS NOT PRINTED NOW TYPE 105/25; > 105/25; BELOW IN FIGURE 1 2 IS A RENDERING OF HOW THIS LOOKS IN THE WORKSHEET WINDOW THE MAPLE BOOK FIGURE 1 2 MAPLE COMMANDS WITH OUTPUT TRY THESE > 105/25-1/5; > %+1/5; SEE SECTION 1 3 FOR A METHOD FOR EDITING MISTAKES RESULTS CAN BE ASSIGNED TO VARIABLES USING THE COLON-EQUALS “:=” GETTING STARTED > F+G; 21/5 + G > # OBSERVE THAT MAPLE IS CASE SENSITIVE > F+G;