DISCOVERING MATHEMATICS WITH MAPLEAn Interactive Exploration for Mathematicians, Engineers, and EconometriciansChapter 1. A Tour of Maple VWorksheet 1a. NumbersIt is a well-known saying that the practical side of a subject can not be fully mastered by merely reading about it or listening to experts, one needs to be actively involved. It is certainly true that in order to get some proficiency with the Maple system you need to sit down in front of the computer, switch on and start learning by doing.Information about this Computer Algebra CourseBut before you start, you should know a thing or two about the worksheets that are part of this Computer Algebra course. The main subdirectory Book\134R5 itself has six subdirectories, each corresponding to a chapter. These subdirectories contain one session, two worksheets and two assignment worksheets.[Update Note:
The Maple 12 Standard .mw worksheets are now on-line. Maple Release 5 is ancient!] The Maple sessions are aimed at demonstrating the many possibilities Maple has to offer. Apart from the first session - this is the one we have just gone through in the preceding section - each of the remaining five Maple sessions focuses on a specific field of application. All sessions with their input groups, output, and text regions are reproduced entirely in printed form.The worksheets can be recognized by the chapter's name: TourW1a.mw is the first worksheet associated with the session TourS1.mw and Chapter 1: A tour of Maple V. The printed form of these twelve worksheets does not include the full output, and a few input lines are left out as well.You are expected to conscientiously work through the worksheets, with close attention to the remarks, suggestions and instructions.Likewise, there are twelve assignment worksheets with similar names, for instance: TourA1b.mw is the second assignments worksheet associated with the first chapter. It goes without saying that the assignments are text only.In the present worksheet (TourW1a.mw), the first of twelve, we shall get acquainted with Maple on a very basic level. We shall see how instructions and commands should be given (input), how input errors can be discovered and removed, and learn about the way in which Maple deals with integers, rational numbers, and reals. In the worksheet TourW1b.mw we shall pay attention to computations with variables. In Chapter 2 we will use Maple to examine functions and sequences. Chapter 3 is about matrices and vectors. In Chapter 4 attention will be paid to summation and to Maple's random generator. In Chapter 5 we will look at derivatives and integrals. Finally in Chapter 6 we will consider linear transformations, eigenvalues and eigenvectors. Giving Instructions to MapleUsually, it will be clearly indicated what should be typed in, but occasionally you will have to decide for yourself. The different regions or groups, composed of lines of input, output, or text are clearly distinguishable by the differences in colour and type of font. In the printed form these differences are less pronounced. Let us start by making the active worksheet as large as possible. Next, place the cursor on the first input line, immediately after the prompt. This can be achieved by clicking the left mouse button on this position, or by using the arrow keys. Now type the following Maple instruction, and make sure to include spaces and the semicolon. Then hit the <Enter> key. 5*(17 + 21) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Observe that Maple immediately produces the correct answer to this most simple of exercises (5 times the sum of 17 and 21) in a newly created output region. Also note that the cursor jumps to the next input line where it patiently waits for your next instruction. In order to prevent the cursor from jumping too far ahead, which might make it difficult for you to read the suddenly disappearing text block in between these successive input regions, we have inserted a dummy input line following each regular input group. In print this dummy line is not reproduced. As soon as you have finished reading the text, you can move the cursor with the mouse or with the arrow keys (use the scroll bar) to the next input line following the text block. A mistake in the input can be easily corrected by moving the cursor to the relevant position in the input line after which you can remove the error with either <Del> or <BackSpace> keys. The output of a Maple instruction cannot be edited, only copied or removed, if so required. Just place the cursor somewhere inside the output region and press the <Del> key to remove the entire output. Selecting a (large) part of the worksheet can be done by moving the cursor to the beginning of the selection and clicking the left mouse button right at the end of the selected part while keeping the <Shift> key pressed down.Now move the cursor to the previous input line, replace the plus sign by a minus sign, and press the <Enter> key. Maple adjusts the answer by replacing old output by new.Maple does not act on an instruction until the <Enter> key is pressed. Even then, if the instruction is not closed off by a semicolon or colon, still nothing happens. Or so it seems. In fact Maple is waiting until the input instruction is properly terminated by semicolon or colon, indicating that the input instruction is complete. Let us try and see.Type the next instruction without closing symbol (; or :) and hit <Enter>.2*(175 - 16)The cursor jumps to the next input line and a warning message is issued which speaks for itself. Completing the instruction or closing it off with a semicolon (in the present case) is sufficient to force Maple into action.[Update Note.
This is no longer necessary in a Standard Worksheet in 2d Math input. But a colon at the end supresses the output in cases where we do not need to see an ugly or large expression. If we wish to put more than one Maple input in the same execution group, then they must be separated either by a colon or semicolon. Except for the above two inputs regions and one below which are in 1d Math input (red boldface unslanted characters), the default input type is 2d Math input (black slanted italic characters) for all of these worksheets. One can convert from one to the other by selecting the input and right clicking on it, then choosing "Convert to", 1d Math input or 2d Math input.]Maple ignores spaces, a feature we can use to enhance the readability of the instructions, not for the benefit of Maple, but for ourselves. Only within numbers or names of commands no spaces must occur.Next we shall find out how Maple treats numbers and in what way Maple works with the ordinary arithmetic operations of addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^). In reverse order this is the usual order of operation: multiplication before addition and exponentiation before multiplication. In case of equal priority, the execution order is from left to right. There is only one exception to this rule. Two successive exponential operations, like in 2^3^4, is not allowed. If in doubt, use parentheses or some other kind of bracketing, because operations of bracketed expressions are always executed first.Carefully type in the following nine expressions, exactly as they appear below. Observe that they are separated by commas. In this way a sequence of instructions is created, which will produce a similar sequence of output results.2-3+4, 2-(3 + 4), 2/(3/4), 2/3/4, (2 + 5/3)*3^2, 2 + 5/3*3^2, 2 + 5/(3*3)^2, 2^(3^2), (2^3)^2;[Update Note:
This becomes the following in 2d Math input:]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 the resulting output and make sure you agree with the outcome of each instruction. Observe with care the placing of the parentheses. Now give the instruction:2^3^4;and consider the error message that Maple returns.You probably noticed that fractions appearing in Maple output are not replaced by approximate numerical values. But Maple does write fractions in lowest terms. Let us consider a large integer: 2 to the power one thousand will do nicely. Give the Maple code for this number, hit the <Enter> key and see what happens.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The backslash (\134) marks the place where the output is broken off, to continue on the next line without interruption. Maple uses the backslash in output that is too long for one line.Next let Maple calculate 1000!, which is, as you know, the product of all positive integers up to and including 1000. Again, give the Maple code on the next line. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Before we continue our tour of Maple, it makes sense to remove from the screen the very large output caused by these numbers. This can be done in several ways. First, select the relevant output region and then press the <Del> key. Alternatively, replace the closing semicolon of the corresponding Maple instruction by a colon, and hit <Enter>. Output is now suppressed, because new output replaces old.At this point you can try and complete assignment 1 of worksheet TourA1a.mw. In order to find out how far we can go, let us try a huge integer next. What about 2 to the power ten million?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY3NiRRKTEwMDAwMDAwRidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOkYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGOkYrRkNGOg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Apparently, even for Maple this is too large an object. Dropping the exponent to one million leaves a power of 2 that is not too large for Maple' taste; even so, one should not instruct Maple to expand this huge number, because it would take Maple a very long time indeed. Moreover, once the calculation of this large number has started, it can not be interrupted. In other words, Maple does not react to clicking the <stop> button on the icon bar. The only way out is the brute force way of halting the Maple program by pressing simultaneously the <Ctrl>, <Alt>, and <Del> keys (followed by End Task under Windows95), which will cause the loss of unsaved information. Generally, a computational process consists of a sequence of so-called primary calculations. A primary calculation can not be interrupted, but has to run its full course. The computational process can only be interrupted or stopped (by clicking the <stop> button) when one primary calculation is completed and before the next one is started.Maple is able to keep track of the time needed to complete a calculation. This can be done by means of the Maple procedure time( ), a procedure that calculates the computing time (in seconds) elapsed since the start of the Maple session. The total amount of time Maple has used for computations can also be found in the Time window on the status bar.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY3NiRRJzEwMDAwMEYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjpGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjotSSNtb0dGJDYtUSI7RidGOi8lJmZlbmNlR0ZFLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRkNGOg==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Assignment 2 of worksheet TourA1a.mw asks for the determination of the computing time of a similar computation.Numerical ComputationsWe have established that Maple performs exact computation on integers and rational numbers, provided the integers in question are not outrageously large. These number types have in common that they can be expressed uniquely with only finitely many digits. Let us call any finite expression, exclusively built from finitely many rational numbers and the four standard arithmetic symbols (+,-,*,/), a finite representation. Irrational numbers do not have such finite representations. For instance, the number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== (the ratio of the circumference of a circle and its diameter), the number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (the base of the natural logarithm), and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== have no finite representation in this sense. In spite of this, we often like to know the approximate size of these numbers too, that is to say, how large they are compared with the unit of measurement, the number 1. We know that all real numbers can be written as infinite decimal fractions in essentially one way only. For instance, in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== = 3.141592653... the first ten digits of the number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== are given. But how does this decimal expansion continue? Nobody knows for sure what the billionth decimal digit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is. Nevertheless, it is not a matter of choice, there is only one such digit. Not knowing it is of no great importance, but is does mean that in order to include LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== in our calculations we are forced to make do with only finitely many digits and thus with an inexact, approximate value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== . On request, Maple will transform exact values into approximate values, and what is more, we may even choose the number of digits precision Maple should use in its calculations. Now give the following instruction: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This gives the floating point representation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== rounded to 48 digits. Because of rounding, only the final digit may be wrong, all other digits are correct. Check this, by comparing evalf(Pi,48) with evalf(Pi,50). With the Maple command Digits we can change the number of digits precision for internal computations with floats. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Setting Digits to 50 forces Maple to carry out all subsequent floating point calculations in 50 digits arithmetic with rounding.Now reset Digits to 10, its default value.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=As we mentioned before, exact or symbolic expressions like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSNsbkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiM0YnRjdGN0Y3RjdGK0Y3 and so on, can also be shown as floats. Naturally, this does not change their actual exact values; asking Maple to show 48 digits of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== does not alter the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== for future calculations! Further, floats are 'contagious' in the following sense. If an expression contains floats next to (exact) rational numbers then it automatically evaluates to a float.Try this out by giving the following instructions: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=In this way it is easy to calculate (approximate) values of well-known functions such as ln and sin without having to use Maple's evalf command.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=In the third input line we used the command evalb. Recall that with evalb Boolean expressions are evaluated with two possible outcomes: true or false. In the next chapter we shall consider in detail the mathematical functions known to Maple, and the use of functions in general. Here let it suffice to ask Maple for information on the logarithmic function.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Carefully read the information on this function; don't skip the examples, much can be learned from them.Now turn to assignment 3 of worksheet TourA1a.mw, in which the logarithmic function should be used to determine the number of digits of the integer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjU2JFEpMTAwMDAwMDBGJ0Y4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y4RitGOA== .This is where we conclude the first worksheet.