INTRODUCTION TO MAPLE
Teresa Dern Henson, Lecturer
Department of Mathematics
Naval Postgraduate School
September, 1995

SECTION I: ALGEBRA TOOLS

GOALS:

In this section you will learn how to
  1. perform arithmetic operations with Maple,
  2. define expressions and functions;
  3. evaluate expressions and functions;
  4. solve equations and systems of equations; and
  5. simplify expressions.
 REMEMBER: Every input line must end with either a semicolon (;) or a colon (:). Use a semicolon if you want to see the output.

BASIC ALGEBRA WITH MAPLE

1. Arithmetic. Arithmetic operations are performed using the symbols

+ plus; addition as in 3 plus 5: 3 + 5
- minus (hyphen); subtraction as in 3 minus 5: 3 - 5
* asterisk; multiplication as in 3 times 5: 3 * 5
/ forward slash; division as in 3 divided by 5: 3/5
^ caret(hat); raise a number to a power as in 3 to the fifth power: 3^5 Maple uses the standard order of operations:  Example: To execute the mathematical statement 4 ¸ 3 ´ 8 + 7 in Maple you would input The operations would be carried out in the following order: first divide 4 by 3, multiply that result by 8, then add 7.
To override the order of operations you may group parts of the expression using parentheses.

EXAMPLE: The expression

would be executed by first multiplying 3 and 8 together, then dividing 4 by that result and, lastly, adding 7.

2. Roots. In mathematics we express the notion of a root of a number either with a radical sign (Ö ) or by means of a rational power. Thus, both and 21/2 indicate the square root of 2; both  and 21/3 indicate the cube root of 2. In Maple we may use the Maple function sqrt or the rational power, ½ to indicate the square root of a number; for other roots we must use the surd function if we want Maple to look for the principal real root of the number.

EXAMPLE:  To enter in Maple, input

NOTE the use of parentheses in the second version of the input.

EXAMPLE:  To calculate 
you will need to use the surd command. The command can be used only after opening a library containing the surd function. This library need be opened only once in a session. To open the library, input

Then, enter to calculate the real cube root of -8. The output will appear as to get the (expected) answer of -2, you need to use this command in conjunction with the evalf command. See item 5 below.

NOTE If you input >(-8)^(1/3); you will get the principal cube root of -8 over the complex number field, which is a complex number.

NOTE If you wish to define a cube root function, you should use surd to ensure evaluation and plotting of the function over the real numbers. Maple can differentiate and integrate such functions.

3. Functions and Expressions. In Maple, as in formal mathematics, a distinction is made between functions and expressions. Maple input such as

assigns the expression on the right-hand side of the := symbol to the variable name of the left-hand side. The output would look like These expressions may be evaluated, plotted, differentiated, integrated, etc. That is, they act like functions, but in a formal, algebraic sense, they are not functions.

To define a function in Maple, you must define a "mapping". Mapping notation denotes a correspondence between an independent variable, typically called x (although any variable name is acceptable), and a functional expression. Here is an
EXAMPLE:  The function  would be formally defined as a mapping by writing .

The Maple version of this statement is

The symbol -> is made by typing a minus sign followed by a "greater than" symbol.

Function notation has the advantage that you may then use the familiar form of functional evaluation; that is, having defined the function f as above, to evaluate f at x= 4 you would input

On the other hand, once you have defined a function, you must use the f (x) notation consistently when referring to the function. For this reason, it is often much easier to use expressions instead of functions.

4. subs. The subs command is used to evaluate an expression at a chosen value of x. Multiple variables may be evaluated all at once. To evaluate expr at the point x = a, input

a may be a number (real or complex) or it may be a symbolic expression.

If expr has been assigned to a variable named f, then you may use

EXAMPLE:  To evaluate x^2 sin x at x = p /2, input NOTE: In Maple, Pi is the irrational number p , whereas pi is the Greek letter p .

EXAMPLE:  To evaluate at x = x – h, y = y - k, input

5. evalf. The input evaluates the expression expr using 10-digit floating-point arithmetic (this is the default); the result will be accurate to 10-digits and will be in decimal form. You can specify n-digit floating-point arithmetic where n is any positive integer. The output will be accurate to n digits. EXAMPLE:  To calculate p (ten-digit accuracy), input >evalf( Pi );
EXAMPLE:  To calculate  (5-digit accuracy), input >evalf( sqrt(200), 5 );
EXAMPLE:  To calculate , input >evalf( surd(-8,3) );

6. eval. The input

evaluates the expression expr using exact arithmetic. When eval is invoked, expressions like sin p/4 will be evaluated exactly; for instance, if you input the output will be We know that  should be . To produce this form of the answer, follow the line just input with This will cause Maple to evaluate the expression  exactly, producing the output .

In contrast, if you use evalf instead of eval, then the output will be a decimal approximation of  which is accurate to ten digits.

7. solve. Maple's solve function can be used to

 When using solve to solve an equation in one variable, Maple attempts to find the solutions algebraically. Therefore, solve works best with polynomial and rational equations. See fsolve for an alternative equation solver.

The basic syntax for the solve command is

where eqtn is the equation to be solved and var is the variable you wish to solve for. To solve a system of equations, the lists of equations to be solved and the list of variables are enclosed in braces.

EXAMPLE:  To solve the equation for b2 input

EXAMPLE:  To find the points where the parabola and the straight line y=x+1 intersect, input solve works well in situations where you wish to solve an equation for one variable in terms of another or for solving equations involving polynomials of small degree. It is of limited utility when solving equations involving non-polynomial expressions.

8. fsolve. Just as evalf calculates a quantity using floating-point arithmetic, the fsolve function solves equations or systems of equations using floating-point arithmetic. The basic syntax is similar to that of solve

fsolve is more flexible than solve in that it allows the user to specify an interval in which Maple should search for a solution by including an optional argument in the form of an interval.

EXAMPLE:  To solve the equation sin(x) = x, input

EXAMPLE:  To solve the equation you must take into account the fact that this equation may have as many as five real solutions. Since these solutions would be the x-intercepts of the function , plotting this function can give you some idea where to search for solutions. You can then specify the solution intervals in the fsolve function. In this example, the plot of the function reveals that it has three real roots: One ties between -3 and -1; one lies between -1 and 1; and the third lies between 1 and 3. (Observe that the function is odd.) To find the three roots you would input If the expression  has been assigned to the variable f, then you can use the somewhat simpler form  
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