An Introduction to Maple V

Frontmatter

1. What MAPLE Can Do for You

Abstract

In this chapter, we have gathered a few typical examples of MAPLE’s use. Except for a few comments, no explanation is provided about the functions used. Further explanation is provided in the following chapters; these sections may be found through the table of contents or index.

Jack-Michel Cornil, Philippe Testud

2. Introduction

Abstract

When a session starts, MAPLE displays a prompt (in general the symbol >). The user can then request that something be executed by entering a MAPLE assertion — i.e. a mathematical expression, an assignment or other instructions.

Jack-Michel Cornil, Philippe Testud

3. Arithmetic

Abstract

The functions iquo (integer quotient) and irem (integer remainder) are used to compute the quotient and the remainder respectively of integer division. If a and b are two integers

• iquo (a,b) returns the quotient of the integer division of a by b,
• irem (a, b) returns the remainder of the integer division of a by b.

Jack-Michel Cornil, Philippe Testud

4. Real Numbers, Complex Numbers

Abstract

An integer value is automatically set in integer form and always displayed in base 10. Output may possibly span several lines on the screen. An integer value has the type integer.

Jack-Michel Cornil, Philippe Testud

5. Two-Dimensional Graphics

Abstract

This section explains how to use MAPLE to plot curves defined by an equation y = f (x). Such a curve may be defined either by an expression or by a function.

Jack-Michel Cornil, Philippe Testud

6. Equations and Inequations

Abstract

The function solve determines all the solutions of a polynomial equation in one unknown. If p is a polynomial expression in the free variable x, the evaluation of solve (p,x) returns all solutions in x of the equation p=0. One may simply write solve (p) if p contains no variable other than x. If p contains at least one element of type float, all the numerical values occuring in the expression of p’s roots are computed as floating point numbers and expressed with n digits, where n is given by the system variable Digits.

Jack-Michel Cornil, Philippe Testud

7. Limits and Derivatives

Abstract

The function limit computes limits of expressions in a single real variable. If p is an expression depending on the free variable x

• limit (p,x=a) returns the limit of p as x tends towards a
• limit (p,x=a, right) returns the limit of p as x tends towards a+
• limit (p,x=a, left) returns the limit of p as x tends towards a-
• limit (p,x=infinity) returns the limit of p as x tends towards +∞
• limit (p,x=-infinity) returns the limit of p as x tends towards —∞
• limit (p, x=inf inity, real) returns the limit of p as

Jack-Michel Cornil, Philippe Testud

8. Truncated Series Expansions

Abstract

The function series computes truncated series expansions of expressions depending on a single free variable.

Jack-Michel Cornil, Philippe Testud

9. Differential Equations

Abstract

The MAPLE function dsolve solves differential equations or systems of differential equations with or without initial conditions (or boundary conditions). It can be used to obtain an exact solution, a truncated series expansion of a solution, or a procedure that compute approximate numerical values of a solution.

Jack-Michel Cornil, Philippe Testud

10. Integration and Summation

Abstract

The function int gives the means to compute integrals or primitives.

Jack-Michel Cornil, Philippe Testud

11. Three-Dimensional Graphics

Abstract

In this part, we study surfaces defined by an equation z = f(x, y). In MAPLE, such a surface can be defined either by an expression depending on the variables x and y or by a function in two variables.

Jack-Michel Cornil, Philippe Testud

12. Polynomials with Rational Coefficients

Abstract

Polynomials in one or several variables have a specific type in MAPLE, the type polynom. However, introducing such objects doesn’t require any specific declaration. The user simply enters the polynomials with the help of the usual arithmetic operators +, -, * and ^.

Jack-Michel Cornil, Philippe Testud

13. Polynomials with Irrational Coefficients

Abstract

In the previous chapter, the study of the functions divide, factor, etc. was restricted to polynomials with rational coefficients. In this chapter, we generalize their use to polynomials whose coefficients may be algebraic, i.e. roots of a polynomial equation with integer coefficients.

Jack-Michel Cornil, Philippe Testud

14. Rational Functions

Abstract

Rational functions have a specific type ratpoly. However, as with polynomials, introducing such objects doesn’t require an explicit type declaration. Rational functions are simply entered by the user with the help of the basic operators +, -, *, ^ and /. Like the type polynom, the type ratpoly isn’t a basic type returned by whattype, but it can be tested for with the help of the function type.

Jack-Michel Cornil, Philippe Testud

15. Construction of Vectors and of Matrices

Abstract

Most functions¹ for linear algebra operations don’t belong to MAPLE’s kernel but to the linalg library. There are several ways to proceed when one wishes to use them.

Jack-Michel Cornil, Philippe Testud

16. Vector Analysis and Matrix Calculus

Abstract

In this chapter, we assume that the linalg library has been loaded with the command with (linalg).

Jack-Michel Cornil, Philippe Testud

17. Systems of Linear Equations

Abstract

In this chapter, we always assume that the linalg library has been loaded with the command with (linalg).

Jack-Michel Cornil, Philippe Testud

18. Normalization of Matrices

Abstract

In this chapter, we assume that the linalg library has been loaded with the command with (linalg).

Jack-Michel Cornil, Philippe Testud

19. Orthogonality

Abstract

In this chapter, we use functions from the linalg library, which is assumed to have been loaded at the beginning of the session using the command with (linalg).

Jack-Michel Cornil, Philippe Testud

20. Vector Analysis

Abstract

The functions that are described in this chapter belong to the linalg library. We always assume that the linalg library has been been loaded using the command with (linalg) for the remainder of this chapter.

Jack-Michel Cornil, Philippe Testud

21. The MAPLE Objects

Abstract

MAPLE classifies the basic expressions, involving only rational constants, names of variables and the operators +, -, *, / and ^, according to three types: the type +, the type * and the type ^.

• Expressions like x+y and x+y+z are of type +, as are x-y+z, which MAPLE stores as x+(-l)*y+z, and (x*y)+(z*t), which is the sum of the two terms x*y and z*t. For MAPLE, the operator + isn’t a binary operator that is generalized by recurrence but an n-ary operator whose operands all play a symmetric role.
• Expressions like x*y and x*y*z are of type *, as are x*y/z, which MAPLE stores as x*y*z^(- l), and (x+y)*(z+t) which, written like this, is the product of the two terms (x+y) and (z+t). Like the operator +, the operator * is an n-ary operator.
• Expressions like x^y, 1/x, (x+y) ^(-3) as well as x^ (1/2) or (x-y+z) ^ (1/3) are of type ^. For MAPLE, the operator ^ is a binary operator and MAPLE doesn’t accept x^y^z, one must thus specify x^(y^z) or (x^y)^z.

Jack-Michel Cornil, Philippe Testud

22. Working More Cleverly with the Subexpressions

Abstract

The function subs replaces elements of a MAPLE expression by the given objects. This substitution is syntactic, not algebraic: the function subs only modifies subexpressions of the initial expression, i.e. terms which can be obtained using the function op.

Jack-Michel Cornil, Philippe Testud

23. Programming: Loops and Branches

Abstract

Although MAPLE is an interactive language, it is programmable. Despite the language’s simplicity, one can quickly and effectively code most problems encountered in Mathematics, in Physics or in Chemistry and even more generally in symbolic computation. This chapter is devoted to the study of loops and branches, while functions and procedures will be studied in Chapter 24.

Jack-Michel Cornil, Philippe Testud

24. Programming: Functions and Procedures

Abstract

In PASCAL, a function returns a result while a procedure carries out an action: a modification of variables may be the outcome of such an action, but a procedure doesn’t return a value, and is not directly usable within an expression. Such a distinction doesn’t exist with MAPLE whose functions and procedures both return a result which is usable within an expression: the difference mainly lies in the complexity of the definition.

Jack-Michel Cornil, Philippe Testud

25. The Mathematical Functions

Abstract

The ordinary mathematical functions are either internal functions or functions of the standard library. The internal functions, written in C, belong to the kernel and are automatically loaded when MAPLE is started. As for the functions of the standard library, some are automatically loaded when they are used for the first time, while other must be explicitly loaded with the command readlib.

Jack-Michel Cornil, Philippe Testud

26. MAPLE Environment in Windows

Abstract

Once MAPLE is started, the user is presented with a typical Windows window that looks like Within this window, you see from top to bottom

• the header bar, with MAPLE V Release 4 — [Untitled(l)],
• the menu bar containing the options: File Edit View Insert Format …,
• the tool bar allowing direct access to some functions through icons.
• the context bar, which changes according to the current area: command area, text area or graphical area.
• the worksheet itself (MAPLE Worksheet) where the user works.
• the status line, at the bottom, where certain information is displayed.

Jack-Michel Cornil, Philippe Testud

Backmatter