An Introduction to Maple V

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.

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.- 2.2 Assignment and Evaluation.- 2.3.1 Fundamental Operations.- 2.4 First Approach to Functions.- 2.5 Simplification of Power Functions.- 3. Arithmetic.- 3.1 Divisibility.- 3.2 Diophantian Equations.- 4. Real Numbers, Complex Numbers.- 4.1 The Real Numbers.- 4.2 The Complex Numbers.- 5.1 Curves Defined by an Equation y = f (x).- 5.2 The Environment of plot.- 5.3 Parametrized Curves in Cartesian Coordinates.- 5.4 Curves in Polar Coordinates.- 5.5 Curves Defined Implicitly.- 5.6 Polygonal Plots.- 5.7 Mixing Drawings.- 5.8 Animation.- 5.9 Using Logarithmic Scales.- 6. Equations and Inequations.- 6.1 Symbolic Solution: solve.- 6.2 Approximate Solution of Equations: fsolve.- 6.3 Solution of Recurrences: rsolve.- 7. Limits and Derivatives.- 7.1 Limits.- 7.2 Derivatives.- 8. Truncated Series Expansions.- 8.1 The Function series.- 8.2 Operations on Truncated Series Expansions.- 8.3 Series Expansion of an Implicit Function.- 9. Differential Equations.- 9.1 Methods for Solving Exactly.- 9.2 Methods for Approximate Solutions.- 9.3 Methods to Solve Graphically.- 10. Integration and Summation.- 10.1 Integration.- 10.2 Operations on Unevaluated Integrals.- 10.3 Discrete Summation.- 11. Three-Dimensional Graphics.- 11.1 Surfaces Defined by an Equation z = f (x, y).- 11.2 The Environment 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.- 11.7 Surfaces Defined Implicitly.- 11.8 Mixing Plots from Different Origins.- 12. Polynomials with Rational Coefficients.- 12.1 Writing Polynomials.- 12.2 Coefficients of a Polynomial.- 12.3 Divisibility.- 12.4 Computation of the g.c.d. and the I.c.m.- 12.5 Factorization.- 13. Polynomials with Irrational Coefficients.- 13.1 Algebraic Extensions of ?.- 13.2 Computation Over an Algebraic Extension.- 13.3 Polynomials with Coefficients in ?/p?.- 14. Rational Functions.- 14.1 Writing of the Rational Functions.- 14.2 Factorization of the Rational Functions.- 14.3 Partial Fraction Decomposition.- 14.4 Continued Fraction Series Expansions.- 15. Construction of Vectors and of Matrices.- 15.1 The linalg Library.- 15.2 Vectors.- 15.3 Matrices.- 15.4 Problems of Evaluation.- 15.5 Special Matrices.- 15.6 Random Vectors and Matrices.- 15.7 Functions to Extract Matrices.- 15.8 Constructors of Matrices.- 16. Vector Analysis and Matrix Calculus.- 16.1 Operations upon Vectors and Matrices.- 16.2 Basis of a Vector Subspace.- 17. Systems of Linear Equations.- 17.1 Solution of a Linear System.- 17.2 The Pivot’s Method.- 18. Normalization of Matrices.- 18.1 Determinant, Characteristic Polynomial.- 18.2 Eigenvalues and Eigenvectors of a Matrix.- 19. Orthogonality.- 19.1 Euclidean and Hermitean Vector Spaces.- 19.2 Orthogonal Polynomials.- 20. Vector Analysis.- 20.1 Jacobian Matrix, Divergence.- 20.2 Gradient, Laplacian, Curl.- 20.3 Scalar Potential, Vector Potential.- 21. The MAPLE Objects.- 21.1 Basic Expressions.- 21.2 Real and Complex Numerical Values.- 21.3 Expression Sequences.- 21.4 Ranges.- 21.5 Sets and Lists.- 21.6 Unevaluated Integrals.- 21.7 Polynomials.- 21.8 Truncated Series Expansions.- 21.9Boolean Relations.- 21.10 Tables and Arrays.- 22. Working More Cleverly with the Subexpressions.- 22.1 The Substitution Functions.- 22.2 The Function map.- 23. Programming: Loops and Branches.- 23.1 Loops.- 23.2 Branches.- 24. Programming: Functions and Procedures.- 24.1 Functions.- 24.2 Procedures.- 24.3 About Passing Parameters.- 24.4 Follow-up of the Execution of a Procedure.- 24.5 Save and Reread a Procedure.- 25. The Mathematical Functions.- 25.1 Catalogue of Mathematical Functions.- 25.2 How Does a MAPLE Function Work?.- 26. Maple Environment in Windows.- 26.1 The MAPLE Worksheet.- 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.