- Getting started
- Arithmetic
- Algebra
- User-defined Functions and Expressions
- Plots
- Differentiation and Integration
- Differential Equations
- Maple Worksheets
- Printing your Maple Worksheet
- Saving your Maple Worksheet
- Opening a Previously Saved Worksheet
- Ending your Maple Session
- Obtaining Copies of the Labs in Worksheet Form
- Useful Commands and Techniques
- Getting Help From Other Students

From an x-terminal in one of the public labs (e.g.,
in ARC) you should log in to ** eden**, then start Maple by typing the
command ` xmaple &` (note that ``xmaple'' is in lower case; typing the
& allows you to continue using your xterm window for other purposes while
Maple is running). A Maple window will open. At the top of the Maple window
is the * Menu Bar*, consisting of a row of * menu buttons* (**
File, Edit** etc.). Underneath the row of menu buttons is the * Tool
Bar*, most of which are shortcuts to menu commands. Directly below the
* Tool Bar* is the * Context Bar*, which consists of more
buttons which are shortcuts to menu commands. If you place the mouse pointer
on any of these buttons and hold down the left mouse button, a rough idea of
what the button does will appear at the bottom of your Maple window. Finally,
below the * Context Bar* is a window with the label * Untitled
(1)*. You can now give an instruction to Maple by placing the mouse
pointer in this window (just to the right of the prompt >), typing a Maple
command, and then typing a carriage return. Maple carries out the command and
prints a response. For example, if you type ` 1 + 1;`, Maple will
respond with ** 2**. Note that you must type a semicolon after each Maple
command, to indicate that you have finished entering the command.

Since you will not be given a Maple manual, you should learn to use Maple by
using the built in ** Help** facility. To get help on a particular topic,
such as the ` plot` command, type, after the prompt >, the command `
?plot;`. A window will open describing the basic structure of the
command and giving examples of its use. To close this window when you have
finished with it, click the left mouse button on the rectangle in the upper
left corner of the window and then drag the mouse pointer down, releasing the
button when ** Close** is highlighted. Another way to invoke ** Help**
is to place the mouse pointer on the word ** Help** at the far right of the
Menu Bar, click the left mouse button, and drag the mouse pointer down,
releasing the button when the desired item (such as ** Topic Search**) is
highlighted. We shall refer to this process as * choosing* ** Topic
Search** from the ** Help** menu. Fill in the ** Topic** box, click
the left mouse button on the item that you want, and then click on ** OK**.

To continue, you will need to know some basic commands and syntax of Maple.

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The operations addition, subtraction, multiplication, division, and
exponentiation are indicated by ` +, -, *, /, ^` respectively. All
grouping of expressions is done with the left and right parentheses ( and ).
The product xy must be written ` x*y`, not ` xy` or ` x
y`; if you type ` xy`, Maple assumes you are referring to a
variable called ``xy''. Thus, to enter the expression

(2 x +y ^{2})/(2 x + e ^{x}) +1
into Maple, you type:

` (2*x +y^2)/(2*x + exp(x)) + 1;`

Note that the exponential function is built into Maple and is
referred to as ` exp`. Similarly, Maple knows the functions `
log`, ` sin`, ` cos`, ` tan`, and many more standard
functions.

When operating on integers, Maple does exact arithmetic, rather than using
decimal approximations. To get a decimal approximation, use the Maple command
` evalf`. The Maple command ` evalf(4/7,20);` produces a 20
digit approximation to 4/7. Typing only ` evalf(4/7);` will produce
a 10 digit approximation--as will typing ` 4.0/7.0;`.

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To help you do algebraic manipulations, Maple has
the commands ` expand`, ` factor`, and ` simplify`, which
you can learn about by using the ** Help** facility. You can also solve
algebraic equations by using the commands ` solve` and ` fsolve`.

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In Maple, ` x^2 - 2*x +3` is an ** expression**. You can assign a
name to this expression for future use by typing ` g:= x^2 - 2*x +3;`.
Expressions can contain several variables, as in ` h:= y*t - sin(y)`.
To evaluate an expression at a particular value, use the Maple command `
subs`. For example, ` subs(x=2,g);` will produce the value
3. Maple also has a construct called a ** function**, with the syntax `
f:=x -> x^2 - 2*x +3;`. We will use these only occasionally.

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The basic plotting command in Maple is ` plot`. This command has many
forms--for example, several functions can be plotted at once--so you should
look carefully at the examples given at the end of its ** Help** page to
get some idea of its flexibility. There are many other plotting commands in
Maple; in Math 251 we used ` plot3d` and ` implicitplot` and in
this course we will use ` dfieldplot`, ` odeplot`, and `
DEplot` (see the section on ** Differential Equations ** for a
description of these commands). Before using the ` plot3d` and `
implicitplot` commands, as well as some other plotting commands, you must
first issue the command ` with(plots):` To use the commands for solving
and plotting solutions to differential equations, you must first issue the
command ` with(DEtools):` It is frequently useful to enlarge the size
of a plot. To do this, place the mouse pointer in the region occupied by the
plot and click the left mouse button. This places a box around the plot and
changes the Menu Bar and Control Bar so that new options are offered to
manipulate the plot. The plot is resized by placing the mouse pointer at one
of the small dots along the edge of the box and dragging the pointer. To
enter additional commands, move the mouse down to the next prompt > and click
the left mouse button. When viewing three dimensional plots, it is useful to
view the plot from different viewpoints. First, place a box around the plot
by moving the mouse pointer in the region occupied by the plot and clicking
the left mouse button. Now, place the mouse pointer inside the box and while
holding down the left mouse button, move the mouse pointer to different
positions. A cube will appear which rotates as the mouse pointer moves. When
you are at the desired viewpoint, hold down the right mouse button and choose
** Redraw**. Explore the effects of using the ** Axes** and other
commands on the ** Menu** Bar. (Note that each time a change is made, you
must also use ** Redraw**.)

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In addition to performing
basic calculations and evaluating standard functions, Maple can also
differentiate and integrate functions. The ` diff` command
differentiates expressions. For example, ` diff(x*sin(x), x);`
differentiates x sin(x) with respect to x, and either ` diff(x*sin(x),
x, x);` or ` diff(x*sin(x),x$2);` gives the second derivative. If
you have typed ` g:=x^2*y;`, then ` diff(g,x);` is the first
partial derivative of x^{2} y with respect to x and
` diff(g,x,y);` is
the mixed second partial derivative.

The command for both definite and indefinite integration is ` int`. If
Maple cannot evaluate a definite integral exactly, numerical integration may
be used. Type ` ?int` and ` ?int[numerical]` for details on
integration in Maple.

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Maple has a special set of commands to deal with differential equations. To
access these, you must first type ` with(DEtools):` Maple can obtain
exact solutions of some differential equations and can obtain numerical
solutions in cases in which it can not find an exact solution. The basic
command for this is ` dsolve`. There are four types of solutions this
command can be used to find: ** exact, series, lapace** and ** numerical
**. To plot the direction field of a single first order differential
equation or a system of two autonomous first order equations, the commands
` dfieldplot` or ` DEplot` may be used. The standard `
plot` command is used to graph exact solutions of differential equations
found by the ` dsolve` command. To plot the numerical solutions found
by ` dsolve`, the command ` odeplot` may be used. Another
alternative which directly plots numerical solutions to differential equations
is the ` DEplot` command. However, this command only gives a graph of
the solution. It cannot be used to give the value of a numerical solution at a
particular point.

Consider the differential equation d^{2}y/dx^{2}-y=x^{3}. To describe this equation in
Maple, we need to communicate that the unknown solution y is a function of x,
and to describe the relevant derivatives of y with respect to x. To
communicate to Maple that y is the dependent variable and x is the independent
variable, we write y(x) instead of y when referring to this variable. Since
the first derivative is ` diff(y(x),x)}`, the second `
diff(y(x),x$2)`, etc., the differential equation d^{2}y/dx^{2}-y=x^{3} should be
expressed to Maple as ` diff(y(x),x$2) -y(x)=x^3` . Notice that Maple
echoes it back in symbolic form. To use this equation later we give it a name
by typing ` de:=diff(y(x),x$2) -y(x)=x^3;` so that the symbol `
de` will refer to this equation. We can use the commands ` rhs`
and ` lhs` to refer to the right or left side of the differential
equation. To place other conditions on y(x), we group additional equations
together with ` de` inside braces. For example, the initial value
problem d^{2}y/dx^{2}-y=x^{3}, y(0)=1, y'(0)=2 can be described to Maple via `
ivp:={de,y(0)=1,D(y)(0)=2};` . Note the use of D as another notation for
differentiation. Any conditions on higher derivatives at a point are
described via the D notation as well, for example ` D(D(y))(0)=3`.

Once a differential equation has been described to Maple, Maple can attempt to
find its general solution, particular solutions, or to plot solutions.
We briefly describe the relevant functions here. Find more about a function
by typing ` ?functionname`. For example ` ?dsolve` will tell
you all about the Maple function ` dsolve`.

** Brief summary of relevant Maple commands for Differential Equations**

` dfieldplot` produces a plot of the direction field of a single first
order differential equation or a system of two first order autonomous
differential equations.

` dsolve(de,y(x));` or ` dsolve(ivp,y(x));` In these basic
forms, `dsolve` will attempt to solve the differential equation or the
initial value problem for y(x) described in ` ivp`, returning a
formula for y(x) if possible. In the first form, arbitrary constants `
_C1`, etc. will be used to express a general solution.

` soln:= dsolve(ivp,y(x),numeric);` In this form ` dsolve` finds
an approximate solution to the initial value problem described in `
ivp`. The output ` soln` is a Maple procedure. To get the value
of this approximate solution at a particular point, e.g., x=2.5, type `
soln(2.5);`. Because the solution is returned as a Maple procedure which
produces output of the form ` [x= 2.5, y(x) = 6.437]`, the `
plot` command cannot be used directly to graph the approximate solution.
Instead, use the ` odeplot` command. Sometimes it is useful to convert
the approximate solution from the form of a Maple procedure into a standard
function that just returns the y-value. This can be done by typing `
yapprox:= u-> subs(soln(u), y(x));`. A graph of the approximate solution
over the interval 0 < x < 5 can then be obtained by typing `
plot(yapprox, x=0..5);`.

` odeplot` is used to plot numerical solutions of differential
equation obtained from using the ` numeric` option of ` dsolve`.

` DEplot` may be used to plot numerical solutions of differential
equations with a set of initial conditions. In the case of a single first
order equation or a system of two first order autonomous equations, it can
also be used to plot the direction field of the equation, with or without
solution curves. In fact, the default option is to also produce the direction
field; to suppress the direction field, one must add the option ` arrows =
NONE`. For computing the numerical solution at a particular point, the
` numeric` option of ` dsolve` should be used. The default
numerical method used by the ` DEplot` command is the classical fourth
order Runge-Kutta method with a fixed step size. For some problems, the
default step size may be too large to produce an accurate solution. This may
be corrected by using the ` stepsize` option. In general, the default
method used by ` DEplot` is a poor choice of method, so it is better to
use the command with the option ` method=rkf45`, which is the default
method for the ` numeric` option of the ` dsolve` command.

** Examples of use of Maple commands for Differential Equations **

`
with(plots): with(DEtools):
de1:= diff(y(x),x) = - y(x) + 1/(1 + exp(x));
s1:=dsolve(de1,y(x));
t1:=rhs(dsolve({de1,y(0)=-2},y(x)));
plot(t1,x=-1..5);
dfieldplot(de1,y(x), x=-1..5,y= -6..4);
initval:={[y(0)=-2],[y(0)=1]};
DEplot(de1, y(x), x=-1..5, initval,y=-6..4);
t2:=dsolve({de1,y(0)=-2},y(x),numeric);
odeplot(t2,[x,y(x)],-1..5);
`

** Linear Algebra:** To use Maple's linear algebra commands, you first
enter ` with(linalg):` To see a list of commands in this package, type
` with(linalg);`, i.e., use a semicolon instead of a colon. In this
course, we will only use a few of Maple's linear algebra commands: `
matadd` to add two matrices, ` multiply` to multiply a matrix times
a vector, ` linsolve` to solve the linear system of equations A x =
b, ` det` to find the determinant of a matrix, and `
eigenvects` to find the eigenvalues and eigenvectors of a matrix.

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The window which appears when you start
Maple is called a ** Maple worksheet.** To complete each computer assignment
in Mathematics 251 you are asked to turn in an edited printout of your work;
such a printout can be obtained by editing, and then printing, your
worksheet. The printout should include only the numerical, symbolic, and
graphical output of Maple which is appropriate for the solution of the
problems assigned, plus text material in which labels are provided for
graphical output and explanations added. Maple includes various editing
capabilities which should enable you to produce neat and coherent output,
and which we now describe.

To remove an unwanted portion of your Maple worksheet (e.g., a region
containing commands that you typed incorrectly or that were not directly
relevant to the solution of the exercises), select the region to be deleted by
clicking the left mouse button at the beginning, then dragging the mouse
across to the end of the portion of the worksheet you wish to delete. The
region should now be highlighted. Then choose ** Cut** from the ** Edit**
menu. To copy a region to a new location, select the region as above, but now
choose ** Copy** from the ** Edit** menu. Then click the left mouse
button in the position in which you wish to insert your selected region and
choose ** Paste** from the ** Edit** menu.

To insert text, such as a label for a plot, into your worksheet, click the
mouse at the beginning or end of the plot and choose ** Text Input** from
the ** Insert** menu. Now type your label. To continue using your
worksheet, move the mouse pointer down to the next prompt > and click the left
mouse button. If there is no prompt, insert one by clicking on the prompt
symbol > in the Tool Bar Menu.

Sometimes it is useful to be able to place a comment after a Maple command, rather than insert text elsewhere in the worksheet. To do this, enter the sharp symbol #. Everything typed on a line following this symbol will be considered by Maple to be a comment, and therefore not executed.

To make your worksheet less cluttered, it is a good idea to have Maple
suppress the output of various commands, e.g., the command `
with(plots)` or a command given to assign a name to a plot. To do this,
end the command with a colon (:), instead of a semicolon (;).

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To print your Maple worksheet, choose ** Print** from the ** File** menu
or click on the print button in the ** Tool Bar** (this is the fourth
button from the left). A panel will appear with the options ``Output to
File'' and ``Print Command'' at the top. Clicking on ``Print Command''
directs your output to the printer, while clicking on ``Output to File''
places your output in a file, whose name is given in the adjacent box (the
default is * untitled.ps*). If you wish to change this name, click in
the box and type in a new name. Since the file will be a postscript file, the
file name should have the form * something.ps*. When you have finished
making your choices, click on the ``Print'' button.

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If your work is interrupted, you can save your work so that you can later
resume where you left off. To save your Maple worksheet, choose ** Save
As** from the ** File** menu. Then type the name of the file in which
you wish to save your worksheet in the ** Selection** box where the mouse
pointer is. (First delete the asterisk.) The file name should have the form
* something.mws*. After you have typed the filename, click on ``OK'' to
save the file. Once you have named the worksheet and saved it in a file, you
can save further changes by choosing ** Save** from the ** File** menu.
Maple automatically keeps track of the filename.

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To open a previously saved worksheet, choose ** Open** from the ** File**
Menu. Click on the name of the file you wish to open and then click on
** OK**. If you wish to close one of your open worksheets, make it active
by clicking in it, and then select ** Close** from the ** File** menu.
Maple will prompt you to save the worksheet if you have changed it since
the last save.

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To end your Maple session, choose ** Exit** from the ** File** menu.
A box will open, reminding you that all unsaved work will be discarded.
If you have saved what you need or wish to exit anyway, click on ``Exit.''

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In some of the labs, you will be asked to simply execute a string of Maple
commands to learn what they do. To avoid retyping these commands, you can
first obtain a modified copy of the lab in worksheet form. This modified
copy will omit instructions and problems and contain only strings of
Maple commands you are asked to execute. Once you obtain this file, you
can access it by following the instructions in the section
** Opening a Previously Saved Worksheet:** To obtain the file
* lab0.mws*, for example, login to eden, and before starting up Maple,
type * cp /eden/u8/falk/math251/lab0.mws lab0.mws* If you look
at the files in your home directory (by typing * ls*), you should
see the file * lab0.mws*. Now start up Maple and open the file
* lab0.mws* .

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If Maple gets hung up in a calculation or is taking too long, click the mouse
on the ** stop** button in the ** Tool Bar** menu (fifth button from the
right).

Any Maple command previously entered in your worksheet can be re-executed
without retyping it in a new location. Simply move the mouse to the position
of the command you wish to execute and hit the ** Return** key.

It is often useful to be able to refer later to the result of a
computation--the output of some command--in a simple way. To make this
possible, simply assign the output of the command to a variable. For
example, if you enter ` a:= evalf(2*Pi);` then you can later square the
result of ` evalf(2*Pi);` by typing ` a*a;`.

You can assign a name to a plot just as described above for assigning a name
to an expression. Several previously named plots can then be displayed on the
same graph by using the command ` display`. Type `
?plots[display]` for details.

If you assign the name ` a` as above and then continue your Maple
session, you may want to reassign the name ` a` to another expression.
To do so, first unassign ` a` by typing ` a:='a';`. To clear
all the assigned variables in a Maple session, type ` restart;`. One
common problem is to try to use such a variable without unassigning it,
forgetting that is has already been assigned a value. A simple way to avoid
this is to issue the ` restart ` command before beginning a new
problem. This will not change anything on your screen.

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The purpose of the Maple assignments is partly to learn about Maple, a very useful program for symbolic, numerical, and graphical computations, and partly to help you understand the material in the course. Just as with other homework assignments, it is permissible and helpful to discuss the Maple labs with other students. However, the Maple labs you are turning in are being graded and will be part of your final course grade. It is therefore expected that the work you hand in is your own in the sense that you have personally input everything that appears on your Maple worksheet, using notation that you have determined, in a style and layout that is your own, and with text comments that reflect the way you finally understand what is being asked. In light of the above, it seems highly unlikely that any two students would turn in nearly identical assignments, and if such an event occurs, the students involved would be considered in violation of the policy on academic integrity and subject to university sanctions.

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Last modified January 6, 1999 by
*Richard S. Falk, falk@math.rutgers.edu*