This document will introduce a few of the basic commands and techniques for using Maple on RCS. Enter the commands into Maple as you read this document.
To bring up Maple, left click on Maple in the RCS Applications menu or type maple & in a Unix window.
> 1+1;
2
The > is the prompt from Maple to type a command. All Maple
commands end with a semicolon. This result looks like what we
might get from any desk calculator...
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> x+x;
2 x
...but Maple can manipulate variables too!
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> 17!/15^17
> ;
3903291392
----------------
1081219482421875
The factorial function is denoted ! and ^ raises to a power. If you
forget the semicolon, you can put it on the next line. Notice that
Maple always tries to do exact rational arithmetic...
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> evalf(");
-5
.3610082370*10
The evalf command evaluates numerical results as floating point.
You can refer to the previous result as ".
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> y:=x^3 - x**2 - x + 1;
3 2
y := x - x - x + 1
The assignment operator := causes y to be assigned the value of the
expression of the right-hand side. Exponentiation can be indicated
by either ^ or **.
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> subs(x=2,y);
3
The subs command tells what y would be if x were 2, without giving
x that value...
> y;
3 2
x - x - x + 1
...and without changing the definition of y.
> x:=2;
x := 2
If we instead assign x the value of 2...
> y;
3
...then y changes to its corresponding value.
> x:='x';
x := x
We can cause x to be regarded as a variable once again by
unassigning it...
> y;
3 2
x - x - x + 1
...and since x is now unknown, y returns to being an expression in
terms of x.
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> solve(y=0,x);
-1, 1, 1
The solve command finds roots analytically if it's possible.
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> s:=solve(a*x^2+b*x+c=0,x);
2 1/2 2 1/2
- b + (b - 4 a c) - b - (b - 4 a c)
s := 1/2 ---------------------, 1/2 ---------------------
a a
When solving the quadratic equation, Maple returns the two
solutions in a list that we have called s...
> s[1];
2 1/2
- b + (b - 4 a c)
1/2 ---------------------
a
...and we can refer to the individual entries of this list using
brackets.
> s[2];
2 1/2
- b - (b - 4 a c)
1/2 ---------------------
a
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> fsolve(x^5-3*x+2=0,x);
-1.446857248, .7412709106, 1.
The fsolve command returns all real roots for a polynomial.
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> sum(k^2,k=1..n);
3 2
1/3 (n + 1) - 1/2 (n + 1) + 1/6 n + 1/6
The sum command is quite useful.
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> simplify(");
3 2
1/3 n + 1/2 n + 1/6 n
Did you recall this formula for the sum of the squares of the first
n integers?
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> f:=x^(x^x);
x
(x )
f := x
Let's consider this harmless-looking function.
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> subs(x=4,f);
13407807929942597099574024998205846127479365820592393\
37772356144372176403007354697680187429816690342769003\
1858186486050853753882811946569946433649006084096
The backslash denotes continuation of the result (in this case, a
big integer).
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> fp:=diff(f,x);
x / x \
(x ) | x x |
fp := x |x (ln(x) + 1) ln(x) + ----|
\ x /
The diff command gives the derivative of f with respect to x.
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> int(fp,x);
x
(x )
x
The indefinite integral of fp is a function whose derivative is fp.
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> plot(cos(x),x=-Pi..Pi);
Maple is good at plotting. This plot will come up in a separate
window. To leave the plot, choose Exit under the File menu in the
plot window. The Appendix of this tutorial contains an example of
including a plot in your Maple session.
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Printing Your Maple Session:
To print your Maple session, you must first click on the word File
in the upper left-hand corner of your Maple window. This drops
down a menu. Click on Print.... A new Page Setup Dialog window will
appear with an Output to File: field above a Printer Command: field.
The Printer Command: field contains the default print command, lpr.
Left click on the diamond button to the left of Printer Command:.
Then left click on the OK button at the bottom of the Page Setup
Dialog window. Your Maple session will print on the nearest
PostScript printer.
The Appendix below includes information on how to save your Maple
session so that you can restart it at a later time.
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Quitting Out of Maple:
To quit Maple, enter the Maple command
> quit;
or click on the word Exit in the File menu.
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Appendix:
Other Maple Features
> evalf(Pi);
3.141592654
Maple knows several special constants, whose names you should not
use as variables. This one is Pi, not pi.
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> evalf(E);
2.718281828
E is the base of the natural logarithms, usually denoted in
mathematics by e.
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> I*I;
-1
Maple uses I for the square root of -1.
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> evalf(""");
3.141592654
Here the result to be evalf'd is the third one back. That's as far
back as you can refer using the ".
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> limit(sin(x)/x,x=0);
1
Maple is good at taking limits. Notice that Maple uses x=0 to
denote x approaches 0.
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> y:=(1+1/x)^x;
x
y := (1 + 1/x)
> limit(y,x=infinity);
e
So y approaches e, the base of the natural logarithms, as x
approaches infinity.
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> f:=abs(cos(x));
f := abs(cos(x))
Maple uses the abs command for absolute value.
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> solve(x^3-3*x+9=0,x);
1/3 1
- - -----,
1/3
1/3 1 1/2 / 1/3 1 \
1/2 + ------- + 1/2 I 3 |- + -----|,
1/3 | 1/3|
2 \ /
1/3 1 1/2 / 1/3 1 \
1/2 + ------- - 1/2 I 3 |- + -----|
1/3 | 1/3|
2 \ /
1/2
:= 9/2 + 1/2 77
Maple sometimes uses substitutions with names like to make
complicated expressions look simpler. Notice that the above list
gives one real root followed by two complex roots (separated by
commas).
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> f:=sin(x^2)/(1+x^2);
2
sin(x )
f := -------
2
1 + x
> int(f,x);
/ 2
| sin(x )
| ------- dx
| 2
/ 1 + x
Maple cannot find this indefinite integral...
> int(f,x=0..1);
1
/ 2
| sin(x )
| ------- dx
| 2
/ 1 + x
0
...so it cannot find a closed form expression for this definite
integral.
> evalf(");
.2014627982
But it can evaluate the above definite integral numerically.
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> solve(cos(x)=x,x);
Maple cannot solve this equation analytically, so a null response
is returned...
> fsolve(cos(x)=x,x);
.7390851332
...but Maple can solve the equation numerically using the fsolve
command.
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> plot({cos(x),x^2},x=-3..Pi);
Maple can plot several functions on the same axes. To embed this
plot into your Maple session, use the Edit menu in the plot window
to Copy... and then use the Edit menu in the Maple window to Paste.
You may want to resize the plot (by resizing the plot window)
first.
To insert a prompt after the plot, place your cursor in the region
of the embedded plot and left click. This gives a "big" cursor to
the left of the plot. Then left-click on the Insert Prompt entry
in the Edit menu. You can also insert text or a separator line
using the same technique.
We can restrict the range of the y-axis to get a better plot...
> plot({cos(x),x^2},x=-3..Pi,-2..2);
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Saving Your Maple Session:
To save your Maple session to a new file so that you can read it in
to Maple at a later date, click on the entry Save As... in the File
menu. A Save Session window will appear. Click on the Selection
field, and type in a file name. (You can specify a directory at
the same time by naming it directory_name/file_name.) This will
create a Maple Script file in your home directory (or in the
directory you specified). If you will want all your variables to
be set correctly when you read this session back in, which will
usually be the case, then "turn on" the button next to the words
Save kernel state
if necessary, by clicking on the button. (The button is "off" when
light grey and "on" when a darker grey.) Then click on Save.
To save your Maple session to an existing file, click on Save
rather than Save As... in the Maple File menu. If you loaded the
kernel state when you read the file in, then the kernel state will
also be saved automatically.
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Retrieving a Previous Maple Session:
To retrieve a previous Maple Session, click on Open...in Maple's
File menu. A Load Session window will appear. If you see the file
name you want in the Files sub-window, click on it; otherwise, type
the file name in the Selection field. If you want your variables
to be set correctly, "turn on" the button next to the words
Load kernel state, if possible
by clicking on it, if necessary. Then click on Load.
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Click on an item listed below to access additional information on a variety of Maple-related topics.
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