Running Maple

To start MAPLE 11 CLASSIC WORKSHEET, click the Start button on the bottom left hand corner of your screen, then select MAPLE11 from the popup menu, then select MAPLE11 CLASSIC WORKSHEET with the yellow icon.

When you want to save your work, select the disk button on the toolbar.  You will be prompted for a file name.

To exit Maple, simply select the "exit" option under the "file" submenu in the top left corner of the Maple window.

Introduction

You are about to undertake a tutorial that will guide you through portions of the symbolic manipulation package called Maple. Maple is a powerful program that can be used interactively to do a wide variety of things ranging from tedious algebra to plotting three-dimensional graphics. This tutorial is intended to provide you with an overview of some of Maple's capabilities, as well as warn you about a few of the common pitfalls. In the material that follows you will repeatedly encounter text in the form shown below:

> 1 + 1;  

2

When text appears like this, the entry in bold following the > sign indicates items that should be entered by you at the Maple cursor. The text that follows is the output produced by Maple in response to your input. In the operation above, for instance, Maple responded to 1 + 1 by performing the addition correctly. Good !

Every Maple command must end with a semicolon as shown in the example above (or a colon if you don't want output displayed). For example, for the sum 1 + 1, enter the following:

> 1 + 1;

to which Maple responds with:

2

The use of a semicolon is crucial in Classic Maple, but not in the regular Maple. Until you enter a semicolon, Classic Maple assumes that you are still in the process of entering a single command. For example, the command above is equivalent to,

> 1
> + 1;

2

Alternately,

> 1
> +
> 1
> ;

2

In either case, you can recognize that the semicolon behaves as a cue for Maple, indicating that the expression is ready for evaluation. This is a useful feature when dealing with long expressions that require multiple lines of input. You can also combine two commands one one input line, as long as you use the semicolon.

> 1+1;2+2;

2
4

Arithmetic

We have encountered addition within Maple above. Analogously, subtraction follows simply as:

> 5 - 2;

with output,

3

Multiplication requires an asterisk *,

> 5*5;

25

As you might expect, division is signified by a backslash /,

> 25/5;

5

To raise something to a power, proceed as follows:

> 8^2;

64

Note that if you forget a semicolon, you'll see a warning message,

> 10^20
warning, premature end of input

If you ignore the warning...

> 10^20
> 10^20
Error, unexpected number

evalf explicitly evaluates expressions as decimal numbers. Let's drive a little harder. How many decimal places do you know Pi to ? Let's request Maple for Pi to a 1000 decimal places. How do you suppose this number is calculated ? How do we know it is right ? explicitly evaluates expressions as decimal numbers. Let's drive a little harder. How many decimal places do you know Pi to ? Let's request Maple for Pi to a 1000 decimal places. How do you suppose this number is calculated ? How do we know it is right ? 

> evalf(Pi, 1000);

3.14159265358979323846264338327950288419716939937510582097494459230781640628620 8998628034825342117067982148086513282306647093844609550582231725359408128481117 4502841027019385211055596446229489549303819644288109756659334461284756482337867 8316527120190914564856692346034861045432664821339360726024914127372458700660631 5588174881520920962829254091715364367892590360011330530548820466521384146951941 5116094330572703657595919530921861173819326117931051185480744623799627495673518 8575272489122793818301194912983367336244065664308602139494639522473719070217986 0943702770539217176293176752384674818467669405132000568127145263560827785771342 7577896091736371787214684409012249534301465495853710507922796892589235420199561 1212902196086403441815981362977477130996051870721134999999837297804995105973173 2816096318595024459455346908302642522308253344685035261931188171010003137838752 8865875332083814206171776691473035982534904287554687311595628638823537875937519
57781857780532171226806613001927876611195909216420199

Make sure you understand the above instruction by determining E to a 100 decimal places. To compute E, you need to use the exp() function, as exp(1);

evalf works in other situations as well. In the following, we will use it to obtain approximate values for the fraction:

> 2 + 1/3;

7/3

To evaluate the fraction numerically, we will use evalf but this time with the symbol % which Maple recognizes as shorthand representing the expression on the previous line.

> evalf(%);

2.333333333

Maple permits the definition of variables. To define a variable called emu and assign to it a value of 12 we proceed as follows.

> emu := 12;

emu := 12

Here := signifies assigned to. We can now query Maple for the value of emu,

> emu;

12

or, we can multiply emu by 10,

> emu * 10;

120

Maple has an excellent Help capability.  To access it, you need to click on the Help button on the top right hand side of the Maple window.  You can then search the Help database by topic, by text, or complete MAPLE's own tutorial.

 

Algebra and Plotting

Almost everything we have done so far we could have done just as fast without ever using Maple. To really begin to take advantage of Maple we move on to some more advanced topics. Lets look at some symbolic calculations using Maple, where we tinker with variables without defining their numerical
values,
 

> (x + 2*y)^2;

Alternately, using shorthand % to refer to the last result, we can expand the expression,
 

> expand(%);

	   2          2
	x + 4xy + 4y 
                                             

Another example:
> (x*y - y^2) / (x - y);

            2
x y - y
--------
	                    x - y                                     

Maple does not simplify expressions automatically. In the previous example, for instance, as the numerator equals y*(x - y), the expression reduces to just y. For Maple to perform such a simplification, it must be explicitly requested,
 

> simplify(%);

y

where the " again refers to the previous result. Having warmed up with some simple algebra, we are in shape to attempt solving algebraic equations. First, consider our old friend the quadratic equation. To make the steps somewhat
clearer, we use the assigned to := operation to define a variable that will be our abbreviation for a quadratic equation,
 

> eland := x^2 + 8*x + 12;

       2
	                                 eland := x + 8 x + 12                                     

To solve the equation, eland = 0, enter the following:
 

> solve(eland = 0, x);

-2,-6

where x within the solve command specifies the variable to be solved for. Are these in fact solutions to the quadratic equation ? We can use Maple to carry out the check very easily. For example, substituting x = -6 into eland,
 

> subs(x = -6, eland);

0

Therefore, -6 is a solution to eland = 0. Try other choices for x to see if they are solutions.   

Maple can also solve sets of algebraic equations. In the following we'll use this capability to determine the intersection between two lines. As before, to keep things tidy, we will define variables as abbreviation,

> eq1 := y = a*x + c;

eq1 := y = ax + c

> eq2 := y = b*x + d;

eq2 := y = bx + d

These commands should strike you as a little odd. So far we have used  the assignment operation := to introduce variables as abbreviations for expressions. Here, on the other hand, variables eq1 and eq2 are abbreviations for entire equations. To determine intersections between the two, we need to solve the equations simultaneously. Maple does this really well:
 

> solve({eq1, eq2}, {x, y});

	          c - d            ad - cb
	{x = - ------- ,    y = ---------}
	         a - b             a - b
	                                    

Easy ! Observe that we have exploited Maple's ability to work with sets to use solve on multiple equations involving multiple unknowns. Within Maple a set of objects is defined by enclosing the objects in a pair of curly brackets { }. With this in mind, the solve command can be seen to instruct Maple to solve the set of equations {eq1, eq2} for the set of unknowns {x, y}.  

Solutions to simultaneous equations are always returned as a list {unknown 1, unknown 2 ...}.  We often want to extract the solution to just one unknown, so we can do further algebraic manipulations. For this purpose, the following trick is helpful

>soln := solve({eq1,eq2},{x,y}):

>soln[1];

               c - d      
x = -  ------- 
              a - b       

>soln[2];

        ad - cb
  y = ---------
                    a - b            

This lets you see the solutions one at a time.  MAPLE seems to give the solutions in a random order - when you run this tutorial you might find solution[1] is the equation for y, and solution[2] is the equation for  x.  I guess they do this to make life more interesting.

We can substitute these solutions into new equations

> eq3 := z = x^2 + y^2:

>subs(soln[1],eq3);

         2 
             (b-d)            2
  z = ------     + y
             2
             ( a - b)            

You can substitute more than one variable into an equation using the syntax subs({soln[1],soln[2]},eq3);

You can also use subs() to plug numbers into an equation.  For example

> subs({a=1,b=2,c=3,d=4},soln[1]);

x=1

Maple's equation solving is fantastic and we will make a great deal of use of it in EN3.  It can be a frustrating function to use, however, because if you make a mistake while typing in one of the equations, Maple will sometimes just sit there and won't tell you what's going on.  For example, try the following

>eq1 := a*x + b*y + c*z = p:
>eq2 := a*x + b*y + c*z = q:
>eq3 := g*x + h*y + i*z = r:
> solve({eq1,eq2,eq3},{x,y,z}); 

We get nothing...  That's because the system of equations actually has no solution.  In eq1 and eq2 the left hand sides are identical, but the right hand sides are different.  So Maple is stuck.  When you solve a complicated set of equations with lots of variables, a small typo can cause this to happen.  

However, if we try

>eq1 := a*x + b*y + c*z = p:
>eq2 := d*x + e*y + f*z = q:
>eq3 := g*x + h*y + i*z = r:
> solve({eq1,eq2,eq3},{x,y,z}); 

then Maple has no trouble at all...

Sometimes the equation of interest may not have a closed-form solution. In such cases, Maple can be used to find numerical solutions. For instance, what if we were interested in finding values of x at which cosh(x) = 2*x. To solve this equation numerically:

> fsolve(cosh(x)=2*x, x);