An Introduction to Maple V

ISBN-13: 9783540664420 / Angielski / Miękka / 2001 / 470 str.

Jack-Michel Cornil;Cornil; P. Testud

Opis

MAPLE is a computer algebra system which, thanks to an extensive library of sophisticated functions, enables both numerical and formal computations to be performed. Until recently, such systems were only available to professional users with access to mainframe computers, but the rapid improvement in the performance of personal computers (speed, memory) now makes them accessible to the majority of users. The latest versions of MAPLE belong to this new generation of systems, allowing a growing audience of users to become familiar with computer algebra. This work does not set out to describe all the possibilities of MAPLE in an exhaustive manner; there is already a great deal of such documentation, including extensive online help. However, these technical manuals provide a mass of information which is not always of great help to a beginner in computer algebra who is looking for a quick solution to a problem in his own speciality: mathematics, physics, chemistry, etc. This book has been designed so that a scientist who wishes to use MAPLE can find the information he requires quickly. It is divided into chapters which are largely independent, each one being devoted to a separate subject (graphics, differential equations, integration, polynomials, linear algebra, ... ), enabling each user to concentrate on the functions he really needs. In each chapter, deliberately simple examples have been given in order to fully illustrate the syntax used.

Recenzje

"I do not like manuals. ... I prefer introductions like this one: After some unavoidable preliminaries, numerous examples (arranged by topic, not by command name) show how Maple can be used to solve problems, and (very important!) also what one can do wrong. Normally, these samples can easily be modified, and will (after some trial and error, maybe) soon lead to a solution of one's problem. - Those who think like me will find this book very helpful."

P.Schmitt, Monatshefte für Mathematik, Vol. 141, Issue 1, 2004

"... Thus the book is designed for students, teachers, engineers and researchers aiming to master the Maple computer algebra system at an introductory level. The book achieves this aim. The coverage of material is quite extensive. ... I worked through selected parts of the book attempting to view the instructions from a beginner's perspective. The instructions are clear and the examples are relevant for most of the mathematics covered at undergraduate level by engineers and scientists. There are adequate warnings along the way as to where things might go wrong. This feature is a particularly useful aid to the beginner. I believe I would have found this book very helpful when I first started using Maple. ... The structure of the book allows the beginner to locate easily practical helpful information to deal with the problem at hand. ... ...I would recommend that some copies be placed in your computer laboratory to be on hand when students are working alone, as I am sure they will find it most helpful."

Australian Mathematical Gazette, Volume 29, Number 1

 "[...] There are several ways to make yourself familiar with a computer algebra system. This book provides a useful and easy way to get acquainted with MAPLE V. This is an excellent book written about MAPLE V. We can recommend this book to mathematicians, physicists, chemists, engineers, and to all who aren't loath to use computer in computations."

Acta Math.Scient. 68, p. 494, 2002

Spis treści

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.