Download Maple : A Brief Introduction

Getting Started with Maple V
Introduction
Maple V is a Symbolic Computation System or Computer Algebra System for mathematical
computation  symbolic, numeric and graphical. It does all the usual numerical operations to
arbitrary precision. More importantly, it can manipulate symbolic expressions: polynomials can
be factored, expanded and terms collected; functions can be integrated and differentiated;
differential equations can be solved; random numbers can be generated and manipulated and
problems of matrix algebra can be handled by Maple. Maple also has extensive facilities for
plotting functions of one and two variables.
This document describes some of the basic Maple operations and functions. It only serves as a
guide to get a beginner acquainted with the basic features of Maple. For details please refer to
references given at the end of this document. Like most packages, on line help is available.
You can start Maple by double-clicking on the Maple icon on your desktop. Maple
should start and present you with a new worksheet.
Your Maple window will look like the one shown below:
At the top of the window is the menu bar containing such menu items as File and Edit.
Immediately below the menu bar is the tool bar, which contains button-based shortcuts to
common operations such as opening, saving and printing. Below that is the context bar which
contains controls specific to the task you are currently performing. The next area is large and
displays your worksheet, the region in which you work. At the very bottom is the status bar,
which displays system information. The worksheet is an integrated environment in which you
interactively solve problems and document your work.
Basics
Maple is an interactive program. You type in an expression or command, and Maple evaluates or
executes it and returns the result. For example :
> 3*2;
6
The angle bracket > is the Maple prompt. As expressions can continue over more than one line,
Maple does not begin to process your input until you indicate you have finished by typing a semicolon ; . This is illustrated by the following (fairly silly) example :
> 2+2
> +2+2
> +2+2;
12
The use of a colon : suppresses the output of the answer completely.
> 2+2:
(No output will be displayed)
To exit Maple, use the File-Exit menu like any other Windows program or just type ‘quit’ at the
prompt. Note : there is no need to use a semi-colon for this command!
> quit
Numbers and constants
Maple understands all the usual numbers (rational, floating point and complex) and numerical
operations. In order to enter a number like 2.3456 10 6 , we would type: 2.3456*10^6. Try
the following:
> 2.3456*10^6 , 6.598*7.6986, 176/345+6768/1019;
.2345600000 10 7 , 50.7953628 ,
2514304
351555
Capital I is used to represent the square root of negative 1 when entering complex numbers. i.e.
I  1 .
> sqrt(3+4*I);
2 + I
The constant  is denoted by Pi in Maple. (Note that P in " Pi " must be in capitals)
> Pi;

2
Digits and evalf
The evalf function returns a floating point approximation of the expression given to it
(evalf is read "e-val-F" and is short for "EVALuate as Floating point"). This enables us to
look at the values in more familiar guise :
> evalf(Pi), evalf(2/3)
3.141592654 , .6666666667
An optional argument to evalf allows us to work with any number of digits of precision. We
could display  with 20 digits by entering :
> evalf(Pi,20);
3.1415926535897932385
Alternatively, we could set the number of digits in all our operations using the Digits
command. For example :
> Digits:=15: evalf(Pi);
3.14159265358979
Variables and Functions
One of the keys to Maple’s usefulness is the ability to use arbitrary variables in expressions as in
the following examples :
> 2*x+3, x^2+2*x-1;
2 x  3 , x2  2 x - 1
Note the explicit multiplication operator * in the previous example. Unfortunately, it is
compulsory as implicit multiplication is not recognised by Maple :
> 2x+3;
Syntax error, missing operator or `;`
The assignment operator := is used to assign names to expressions.
> y := 4*x+5;
y := 4 x + 5
Then we can assign values to the variable x and see the value of y change.
> x:=10; y;
x := 10
45
3
We may repeat the command, setting a different value for x. You may do this either by
retyping the line, or moving your cursor to the previous command and changing the "10" to "-3"
and hitting the return key.
> x:=-3; y;
x := -3
-7
To recover the general form of y we use the following trick to remove the value assigned to x :
> x:= ’x’; y;
x := x
4 x + 5
Note that the above is not the same as saying " y is a function of x ". In order to do that, we use
the functional operator ->
> y := x -> 4*x+5;
y := x -> 4 x + 5
We can read this as “y is the function that takes x and gives 4x+5”.
Besides these user-defined functions, there are also a number of pre-defined functions in Maple.
For instance, the exponential function, e x , is defined as exp .
We can use y and exp like an ordinary function, evaluating them at specific points or at different
arbitrary points. For example :
> y(10), y(-3), y(x), y(z+20/z^2);
45, -7, 4 x + 5, 4 z +
80
+ 5
z2
> exp(1), evalf(exp(1)), exp(2), y(exp(y(x)));
e, 2.718281828, e 2 , 4 e (4x + 5) + 5
Besides the exponential function, Maple has definitions for almost all of the so-called elementary
functions : all the trigonometric functions and their inverses (sine, arcsine, and so on), natural
logarithm and many others. Some examples are given below :
> sin(Pi/4), cos(3*Pi/2), ln(1), 12! ;
1
2 , 0, 0, 479001600
2
4
A sample list of functions found in Maple is given in the table below :
Function
Maple name
ex (exponential)
exp(x)
ln x (natural logarithm)
ln(x) or log(x)
log10 x (logarithm to the base 10)
log10(x)
x
sqrt(x)
| x| (absolute value)
abs(x)
sine, cosine, tangent (in radians)
sin(x), cos(x), tan(x)
secant, cosecant, cotangent
sec(x), csc(x), cot(x)
inverse trigonometric functions
arcsin(x), arccos(x), arctan(x),
arcsec(x), arccsc(x), arccot(x)
Hyperbolic functions
sinh(x), cosh(x), tanh(x),
sech(x), csch(x), coth(x)
Inverse hyperbolic functions
arcsinh(x), arccosh(x), arctanh(x),
arcsech(x), arccsch(x), arccoth(x)
Factorial of n
n!
Close and Start new worksheets
At this point, it may be useful to close your present worksheet and start a new worksheet.
To close a worksheet, click FILE from menu bar, and select CLOSE.
To start a new worksheet, click FILE from the menu bar and select NEW.
Plotting 2-D Graphs
When plotting an explicitly given function, y  f (x) , Maple needs to know the function and the
domain. For example,
> plot(sin(x), x = -2*Pi..2*Pi);
5
Maple can also graph user-defined functions.
> f:=x -> x^3-3*x^2-5*x+2:
> plot(f(x),x=-3..5);
To graph more than one function in the same plot, you need to specify a list of functions. In
addition, you may also specify the range for y, as well as the colours for the individual graphs.
> plot([sin(x),1/x],x=-2*Pi..2*Pi,y=-3..3,color=[blue,black]);
Help!!
On-line help is available on the Maple system. Simply type a ? followed immediately by the
topic on which you need help. Alternatively, you may click on the Help item on the menu bar
and a help window will pop up.
References
1. B.W. Char, et al., “First Leaves : A Tutorial Introduction to Maple V”, Springer-Verlag, New
York, 1992.
2. B.W. Char, et al., “Maple V Language Reference Manual”, Springer-Verlag, New York,
1991. [QA 155.7 E4 Map]
3. R.M. Corless, “Essential Maple : An Introduction for Scientific Programmers”, SpringerVerlag, New York, 1995. [QA 76.95 Cor]
4. A. Heck, Springer-Verlag, New York, 1993. [QA 155.7 E4 Hec]
5. Heal, K.M., et al, “Maple V Learning Guide”, Springer-Verlag, New York, 1996.
NTU/NIE/99/AKC
6
Solving Algebraic Equations
Maple can be used to solve algebraic equations. For example, to solve for a single unknown in
the equation, 3x+4=5x :
> solve(3*x+4=5*x);
2
If you give an expression rather then an equation, Maple automatically assumes that the
expression is equal to zero.
> solve(x^3-13*x+12,x);
1, 3, -4
To solve an equation with more than one solution :
> solve(ln(x^2-1)=a,x);
e a  1,
 ea  1
Maple may also be used to solve simultaneous equations. For example, to solve the
simultaneous equations, 3x + 4y = 8
and 2x - 3y = 2 :
> eqs := {3*x + 4*y = 8, 2*x - 3*y = 2}:
> solve(eqs);
{y 
10
32
,x }
17
17
Other Maple Features
Try the following commands :

solve(a*x^2+b*x+c=0,x);

diff(cos(x),x);

int(x^3*cos(x),x);

expand((x-2)^4);

factor(x^3-3*x^2+3*x-1);

sum(i,i=1..100);

factor(sum(i,i=1..n));

plot(sin(x),x=-Pi..Pi);
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