An Introduction to Maple V

1. What MAPLE Can Do for You.- 1.1 Arithmetic.- 1.2 Numerical Computations.- 1.3 Polynomials and Rational Functions.- 1.4 Trigonometry.- 1.5 Differentiation.- 1.6 Truncated Series Expansions.- 1.7 Differential Equations and Systems.- 1.8 Integration.- 1.9 Plot of Curves.- 1.10 Plot of Surfaces.- 1.11 Linear Algebra.- 2. Introduction.- 2.1 First Steps.- 2.1.1 Keyboarding an Expression.- 2.1.2 Operators, Functions and Constants.- 2.1.3 First Computations.- 2.2 Assignment and Evaluation.- 2.2.2 Identifiers.- 2.2.3 Assignment.- 2.2.4 Free Variables and Evaluation.- 2.2.5 Full Evaluation Rule.- 2.2.6 Use of Apostrophes: Partial Evaluation.- 2.3 Evaluation of Function Arguments.- 2.3.1 Fundamental Operations.- 2.3.2 The Function expand.- 2.3.3 The Function factor.- 2.3.4 The Function normal.- 2.3.5 The Function convert in Trigonometry.- 2.3.6 First Approach to the Function simplify.- 2.3.7 Simplification of Radicals: radnormal and rationalize.- 2.3.8 The Functions collect and sort.- 2.4 First Approach to Functions.- 2.4.1 Functions of One Variable.- 2.4.2 Functions of Several Variables.- 2.4.3 The Difference Between Functions and Expressions.- 2.4.4 Links Between Expressions and Functions.- 2.5 Simplification of Power Functions.- 2.5.1 The Functions exp, In and the Exponentiation Operator.- 2.5.2 The Function simplify.- 2.5.3 The Function combine.- 3. Arithmetic.- 3.1 Divisibility.- 3.1.1 Quotient and Remainder.- 3.1.2 G.c.d. and Euclid's Algorithm.- 3.1.3 Decomposition into Prime Factors.- 3.1.4 Congruences.- 3.2 Diophantian Equations.- 3.2.1 Chinese Remainder Theorem.- 3.2.2 Solution of Equations Modulo n.- 3.2.3 Classical Equations.- 4. Real Numbers, Complex Numbers.- 4.1 The Real Numbers.- 4.1.1 Display of Real Numbers.- 4.1.2 Approximate Decimal Value of Real Numbers.- 4.2 The Complex Numbers.- 4.2.1 The Different Types of Complex Numbers.- 4.2.2 Algebraic Form of the Complex Numbers.- 4.2.3 Trigonometric Form of the Complex Numbers.- 4.2.4 Computing with Expressions with Complex Coefficients.- 4.2.5 Approximate Decimal Value of the Complex Numbers.- 5.1 Curves Defined by an Equation y = f (x).- 5.1.1 Graphic Representation of an Expression.- 5.1.2 Graphic Representation of a Function.- 5.1.3 Simultaneous Plot of Several Curves.- 5.1.4 Plot of a Family of Curves.- 5.2 The Environment of plot.- 5.2.1 The plot Menu in Windows.- 5.2.2 The Options of plot.- 5.3 Parametrized Curves in Cartesian Coordinates.- 5.3.1 Plot of a Parametrized Curve.- 5.3.2 Simultaneous Plot of Several Parametrized Curves.- 5.3.3 Plot of a Family of Parametrized Curves.- 5.4 Curves in Polar Coordinates.- 5.4.1 Plot of a Curve in Polar Coordinates.- 5.4.2 Plot of a Family of Curves in Polar Coordinates.- 5.5 Curves Defined Implicitly.- 5.5.1 Plot of a Curve Defined Implicitly.- 5.5.2 Plot of a Family of Implicit Curves.- 5.5.3 Precision of the Plot of Implicit Curves.- 5.6 Polygonal Plots.- 5.7 Mixing Drawings.- 5.7.1 How Does plot Work.- 5.7.2 The Function display.- 5.8 Animation.- 5.9 Using Logarithmic Scales.- 6. Equations and Inequations.- 6.1 Symbolic Solution: solve.- 6.1.1 Univariate Polynomial Equations.- 6.1.2 Other Equations in One Variable.- 6.1.3 Systems of Equations.- 6.1.4 Inequations.- 6.2 Approximate Solution of Equations: fsolve.- 6.2.1 Algebraic Equations in One Variable.- 6.2.2 Other Equations in One Variable.- 6.2.3 Systems of Equations.- 6.3 Solution of Recurrences: rsolve.- 6.3.1 Linear Recurrences.- 6.3.2 Homographic Recurrences.- 6.3.3 Other Recurrence Relations.- 7. Limits and Derivatives.- 7.1 Limits.- 7.1.1 Limit of Expressions.- 7.1.2 Limit of Expressions Depending on Parameters.- 7.1.3 Limit of Functions.- 7.2 Derivatives.- 7.2.1 Derivatives of Expressions in a Single Variable.- 7.2.2 Partial Derivatives of Expressions in Several Variables.- 7.2.3 Derivatives of Functions in One Variable.- 7.2.4 Partial Derivatives of Functions in Several Variables.- 8. Truncated Series Expansions.- 8.1 The Function series.- 8.1.1 Obtaining Truncated Series Expansions.- 8.1.2 Generalized Series Expansions.- 8.1.3 Regular Part of a Series Expansion.- 8.1.4 Obtaining an Equivalent.- 8.1.5 Limits of the Function series.- 8.2 Operations on Truncated Series Expansions.- 8.2.1 Sums, Quotients, Products of Truncated Series Expansions.- 8.2.2 Compositions and Inverses of Truncated Series Expansions.- 8.2.3 Integration of a Truncated Series Expansion.- 8.3 Series Expansion of an Implicit Function.- 9. Differential Equations.- 9.1 Methods for Solving Exactly.- 9.1.1 Differential Equations of Order 1.- 9.1.2 Differential Equations of Higher Order.- 9.1.3 Classical Equations.- 9.1.4 Systems of Differential Equations.- 9.2 Methods for Approximate Solutions.- 9.2.1 Numerical Solution of an Equation of Order 1.- 9.2.2 Numerical Solution of an Equation of Higher Order.- 9.2.3 Computing a Truncated Series Expansion of the Solution.- 9.3 Methods to Solve Graphically.- 9.3.1 Differential Equation of Order 1.- 9.3.2 The Options of DEplot for a Differential Equation.- 9.3.3 Differential Equation of Order n.- 9.3.4 Necessity of the Option stepsize.- 9.3.5 Differential System of Order 1.- 9.3.6 Study of an Example.- 10. Integration and Summation.- 10.1 Integration.- 10.1.1 Exact Computation of Definite and Indefinite Integrals.- 10.1.2 Generalized Integrals.- 10.1.3 Inert Form Int.- 10.1.4 Numerical Evaluation of Integrals.- 10.2 Operations on Unevaluated Integrals.- 10.2.1 Integration by Parts.- 10.2.2 Variable Substitution in an Integral.- 10.2.3 Differentiation Under the Integral Sign.- 10.2.4 Truncated Series Expansion of an Indefinite Integral.- 10.3 Discrete Summation.- 10.3.1 Indefinite Sums.- 10.3.2 Finite Sums.- 11. Three-Dimensional Graphics.- 11.1 Surfaces Defined by an Equation z = f (x, y).- 11.1.1 Plot of a Surface Defined by an Expression.- 11.1.2 Plot of a Surface Defined by a Function.- 11.1.3 Simultaneous Plot of Several Surfaces.- 11.2 The Environment of plot3d.- 11.2.1 The Menu of plot3d in Windows.- 11.2.2 The Options of plot3d.- 11.3 Surface Patches Parametrized in Cartesian Coordinates.- 11.4 Surfaces Patches Parametrized in Cylindrical Coordinates.- 11.5 Surface Patches Parametrized in Spherical Coordinates.- 11.6 Parametrized Space Curves.- 11.6.1 Plot of a Parametrized Curve.- 11.6.2 Simultaneous Plot of Several Parametrized Curves.- 11.7 Surfaces Defined Implicitly.- 11.8 Mixing Plots from Different Origins.- 12. Polynomials with Rational Coefficients.- 12.1 Writing Polynomials.- 12.1.1 Reminders: collect, sort, expand.- 12.1.2 Indeterminates of a Polynomial.- 12.1.3 Value of a Polynomial at a Point.- 12.2 Coefficients of a Polynomial.- 12.2.1 Degree and Low Degree.- 12.2.2 Obtaining the Coefficients.- 12.3 Divisibility.- 12.3.1 The Function divide.- 12.3.2 Euclidean Division.- 12.3.3 Resultant and Discriminant.- 12.4 Computation of the g.c.d. and the I.c.m.- 12.4.1 The Functions gcd and lem.- 12.4.2 Content and Primitive Part.- 12.4.3 Extended Euclid's Algorithm: The Function gcdex.- 12.5 Factorization.- 12.5.1 Decomposition into Irreducible Factors.- 12.5.2 Square-Free Factorization.- 12.5.3 Irreducibility Test.- 13. Polynomials with Irrational Coefficients.- 13.1 Algebraic Extensions of ?.- 13.1.1 Irreducibility Test.- 13.1.2 Roots of a Polynomial.- 13.1.3 The Function RootOf.- 13.1.4 Numerical Values of Expressions Containing RootOf's.- 13.1.5 Conversion of RootOf Into Radicals.- 13.2 Computation Over an Algebraic Extension.- 13.2.1 Factorization Over a Given Extension.- 13.2.2 Incompatibility Between Radicals and RootOf.- 13.2.3 Irreducibility, Roots Over a Given Extension.- 13.2.4 Factorization of a Polynomial Over Its Splitting Field.- 13.2.5 Divisibility of Polynomials with Algebraic Coefficients.- 13.2.6 G.c.d. of Polynomials with Algebraic Coefficients.- 13.3 Polynomials with Coefficients in ?/p?.- 13.3.1 Basic Polynomial Computations in ?/p?.- 13.3.2 Divisibility of Polynomials in ?/p?.- 13.3.3 Computation of the G.c.d. of Polynomials in ?/p?.- 13.3.4 Euclidean Division, Extended Euclid's Algorithm.- 13.3.5 Factorization of the Polynomials in ?/p?.- 14. Rational Functions.- 14.1 Writing of the Rational Functions.- 14.1.1 Irreducible Form.- 14.1.2 Numerator and Denominator.- 14.2 Factorization of the Rational Functions.- 14.2.1 Rational Functions with Rational Coefficients.- 14.2.2 Rational Functions with Any Coefficients.- 14.2.3 Factorization Over an Algebraic Extension.- 14.3 Partial Fraction Decomposition.- 14.3.1 Decomposition of a Rational Function Over ?(x).- 14.3.2 Decomposition Over ?(x) or Over ?(x).- 14.3.3 Decomposition of a Rational Function with Parameters.- 14.4 Continued Fraction Series Expansions.- 15. Construction of Vectors and of Matrices.- 15.1 The linalg Library.- 15.2 Vectors.- 15.2.1 Definition of the Vectors.- 15.2.2 Dimension and Components of a Vector.- 15.3 Matrices.- 15.3.1 Definition of Matrices.- 15.3.2 Dimensions and Coefficients of a Matrix.- 15.4 Problems of Evaluation.- 15.4.1 Evaluation of Vectors.- 15.4.2 Evaluation of the Matrices.- 15.4.3 Example of Use of Matrices of Variable Size.- 15.5 Special Matrices.- 15.5.1 Diagonal Matrix and Identity Matrix.- 15.5.2 Tri-Diagonal or Multi-Diagonal Matrix.- 15.5.3 Vandermonde Matrix.- 15.5.4 Hilbert Matrix.- 15.5.5 Sylvester Matrix and Bézout Matrix.- 15.5.6 Matrix of a System of Equations.- 15.6 Random Vectors and Matrices.- 15.6.1 Random Vectors.- 15.6.2 Random Matrices.- 15.7 Functions to Extract Matrices.- 15.7.1 Submatrices.- 15.7.2 Column Vector and Row Vector.- 15.8 Constructors of Matrices.- 15.8.1 Block-Diagonal Matrices.- 15.8.2 Blockmatrices.- 15.8.3 Juxtaposition and Stack of Matrices.- 15.8.4 Copying a Matrix Into Another.- 16. Vector Analysis and Matrix Calculus.- 16.1 Operations upon Vectors and Matrices.- 16.1.1 Linear Combinations of Vectors.- 16.1.2 Linear Combination of Matrices.- 16.1.3 Transposition of Matrices and of Vectors.- 16.1.4 Product of a Matrix by a Vector.- 16.1.5 Product of Matrices.- 16.1.6 Inverse of a Matrix.- 16.1.7 Powers of Square Matrices.- 16.2 Basis of a Vector Subspace.- 16.2.1 Subspace Defined by Generators.- 16.2.2 Kernel of a Matrix.- 16.2.3 Subspace Generated by the Lines of a Matrix.- 16.2.4 Subspace Defined by Equations.- 16.2.5 Intersection and Sum of Vector Subspaces.- 16.2.6 Rank of a Matrix.- 16.2.7 Evaluation Problem.- 16.2.8 An Exercise About the Commuting Matrices.- 17. Systems of Linear Equations.- 17.1 Solution of a Linear System.- 17.1.1 Linear System Given in Matrix Form.- 17.1.2 Linear System Specified by Equations.- 17.2 The Pivot's Method.- 17.2.1 Operations on the Rows and the Columns of a Matrix.- 17.2.2 The Function pivot.- 17.2.3 Gaussian Elimination: The Function gausselim.- 17.2.4 Gaussian Elimination Without Denominator: ffgausselim.- 17.2.5 Optional Parameters of gausselim and ffgausselim.- 17.2.6 Gauss-Jordan Elimination.- 18. Normalization of Matrices.- 18.1 Determinant, Characteristic Polynomial.- 18.1.1 Determinant of a Matrix.- 18.1.2 Characteristic Matrix and Characteristic Polynomial.- 18.1.3 Minimal Polynomial of a Matrix.- 18.2 Eigenvalues and Eigenvectors of a Matrix.- 18.2.1 Eigenvalues.- 18.2.2 Eigenvectors, Diagonalization.- 18.2.3 Testing if a Matrix Can Be Diagonalized.- 18.2.4 Matrices That Have an Element of Type float.- 18.2.5 The Inert Function Eigenvals.- 18.2.6 Normalization to the Jordan Form.- 19. Orthogonality.- 19.1 Euclidean and Hermitean Vector Spaces.- 19.1.1 Scalar Product, Hermitean Scalar Product.- 19.1.2 Norm.- 19.1.3 Cross Product.- 19.1.4 Gram-Schmidt Orthogonalization.- 19.1.5 Positive Definite and Positive Semidefinite Real Symmetric Matrices.- 19.1.6 Hermitian Transpose of a Matrix.- 19.1.7 Orthogonal Matrix.- 19.1.8 Normalization of Real Symmetric Matrices.- 19.2 Orthogonal Polynomials.- 19.2.1 Chebyshev Polynomials of the First Kind.- 19.2.2 Chebyshev Polynomials of the Second Kind.- 19.2.3 Hermite Polynomials.- 19.2.4 Laguerre Polynomials.- 19.2.5 Legendre and Jacobi Polynomials.- 19.2.6 Gegenbauer Polynomials.- 20. Vector Analysis.- 20.1 Jacobian Matrix, Divergence.- 20.1.1 Jacobian Matrix.- 20.1.2 Divergence of a Vector Field.- 20.2 Gradient, Laplacian, Curl.- 20.2.1 Gradient.- 20.2.2 Laplacian.- 20.2.3 Hessian Matrix.- 20.2.4 Curl of a Vector Field of ?3.- 20.3 Scalar Potential, Vector Potential.- 20.3.1 Scalar Potential of a Vector Field.- 20.3.2 Vector Potential of a Vector Field.- 21. The MAPLE Objects.- 21.1 Basic Expressions.- 21.1.1 The Types +, * and ^.- 21.1.2 The Functions whattype, op and nops.- 21.1.3 The Type function.- 21.1.4 Structure of Basic Mathematical Expressions.- 21.2 Real and Complex Numerical Values.- 21.2.1 The Values of Type numeric.- 21.2.2 The Values of Type realcons.- 21.2.3 The Complex Values.- 21.3 Expression Sequences.- 21.3.1 The Function seq.- 21.3.2 The Operator $.- 21.3.3 Sequence of Results.- 21.3.4 Sequence of Components of an Expression.- 21.3.5 Sequence of Parameters of a Procedure.- 21.4 Ranges.- 21.5 Sets and Lists.- 21.5.1 The Operators { } and [ ].- 21.5.2 Operations Upon the Sets.- 21.5.3 Operations on Lists.- 21.5.4 Extraction.- 21.5.5 Back to the Function seq.- 21.6 Unevaluated Integrals.- 21.7 Polynomials.- 21.8 Truncated Series Expansions.- 21.8.1 Taylor Series Expansions.- 21.8.2 Other Series Expansions.- 21.9 Boolean Relations.- 21.9.1 The Type relation.- 21.9.2 The Type boolean.- 21.10 Tables and Arrays.- 21.10.1 Tables.- 21.10.2 Tables, Indexed Variables.- 22. Working More Cleverly with the Subexpressions.- 22.1 The Substitution Functions.- 22.1.1 The Function subs.- 22.1.2 The Function subsop.- 22.2 The Function map.- 22.2.1 Using map with a Function Which Has a Single Argument.- 22.2.2 Using map with a Function of Several Arguments.- 22.2.3 Using map Upon a Sequence.- 22.2.4 Avoiding the Use of map.- 23. Programming: Loops and Branches.- 23.1 Loops.- 23.1.1 for Loop with a Numerical Counter.- 23.1.2 for Loop Upon Operands.- 23.1.3 How to Write a Loop That Spans Several Lines.- 23.1.4 Echo of the Instructions of a Loop.- 23.1.5 Nested Loops and Echo on the Screen: printlevel.- 23.1.6 Avoiding for Loops.- 23.1.7 while Loop.- 23.2 Branches.- 23.2.1 The Conditional Branch: if ... then ... elif ... else.- 23.2.2 next and break.- 23.2.3 MAPLE's Three-State-Logic.- 24. Programming: Functions and Procedures.- 24.1 Functions.- 24.1.1 Definition of a Simple Function.- 24.1.2 Use of a Function.- 24.1.3 Function Using Tests.- 24.1.4 Evaluation Problem for a Function.- 24.1.5 Number of Arguments Passed on to a Function.- 24.1.6 Other Ways to Write a Function.- 24.1.7 Particular Values: remember Table.- 24.2 Procedures.- 24.2.1 Definition of a Procedure.- 24.2.2 Local Variables and Global Variables.- 24.2.3 Recursive Procedures.- 24.2.4 remember Table Versus Recursion.- 24.2.5 Structure of an Object Function or Procedure.- 24.3 About Passing Parameters.- 24.3.1 Automatic Verification of the Types of Arguments.- 24.3.2 Testing the Number and the Kind of Arguments Which Are Passed.- 24.3.3 How to Test a Type.- 24.3.4 Procedure Modifying the Value of Some Parameters.- 24.3.5 Procedure with a Variable Number of Arguments.- 24.3.6 Procedure with an Unspecified Number of Arguments.- 24.4 Follow-up of the Execution of a Procedure.- 24.4.1 The Variable printlevel.- 24.4.2 The Functions userinfo and infolevel.- 24.5 Save and Reread a Procedure.- 25. The Mathematical Functions.- 25.1 Catalogue of Mathematical Functions.- 25.1.1 Arithmetical Functions.- 25.1.2 Counting Functions and ? Function.- 25.1.3 Exponentials, Logarithms and Hypergeometric Function.- 25.1.4 Circular and Hyperbolic Trigonometric Functions.- 25.1.5 Inverse Trigonometric Functions.- 25.1.6 Integral Exponential and Related Functions.- 25.1.7 Bessel Functions.- 25.1.8 Elliptic Functions.- 25.2 How Does a MAPLE Function Work?.- 25.2.1 Numerical Return Values.- 25.2.2 An Example: The Function arcsin.- 25.2.3 Case of the Functions builtin.- 25.2.4 remember Table.- 26. Maple Environment in Windows.- 26.1 The MAPLE Worksheet.- 26.1.1 Text and Maple-input Modes.- 26.1.2 Groups and Sections.- 26.1.3 The Menu Bar.- 26.2 The File Menu.- 26.3 The Edit Menu.- 26.4 The View Menu.- 26.5 The Insert Menu.- 26.6 The Format Menu.- 26.7 The Options Menu.- 26.8 The Window Menu.- 26.9 On-line Help.- 26.9.1 The Help Menu.- 26.9.2 Accessing the On-line Help Directly.- 26.9.3 Structure of a Help Page.