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Maple Basics

Here are some basic Maple examples.    They illustrate the syntax
of some useful commands for calculus.   You can make
the commands ``happen'' by hitting  RETURN at the end of any input
line.   Input lines are in  red  on a color monitor, and they start
with a    >  symbol.



Define an expression to work with in various ways:

> f :=  x^2*sin(x);

                                  2
                            f := x  sin(x)



Note that every command ends with a semicolon.  Having
defined  f  , we can use it in various ways.   Maple will remember
what's meant by the symbol f.


Now let's try doing some calculus-style things to  f, such as differentiating
f with respect to x.

> diff(f,x);

                                      2
                        2 x sin(x) + x  cos(x)

Look what happens if we use another variable than  x :

> diff(f,y);

                                  0

Do you agree with the answer?   (You should.) 

Let's do something else.

> g := int(f,x);

                      2
               g := -x  cos(x) + 2 cos(x) + 2 x sin(x)

The previous command   defined   g  as a new expression ---
 an antiderivative of  f.   

> diff(g,x);

                               2
                              x  sin(x)

That's reasssuring ...      
Let's try plotting something.    Maple can plot all kinds of things.

> plot( f, x=-3..3 );
As you see, a new window pops up.   You can do various things with the buttons
in the menu at the top of the plot window, such as changing the style of axes.
When you're done with the window you can use the File  menu to close
it or kill it.

There are many variations on the plot command.   To find out more, use
the built-in help system.   To do so, give a command like this:

> ?plot
A help window pops up, with lots more information than you want.   The most
useful stuff is often the examples at the bottom of the window.   Note that you 
can try any  example   by highlighting it with the mouse, then moving to an
input position in the Maple window, and pressing the middle mouse button.  (This
copies the highlighted stuff from one window into the input position.)

Here are a few more plot examples, to illustrate the possibilities:
 
> plot( sin(x), x=-Pi .. Pi, -2 .. 2 );

> plot( sin(x), x=-Pi .. Pi, -5 .. 5 );
> plot( [sin(t),cos(t), t=0..Pi] );
The plot above is a parametric plot---it produces a semi-circle.
If the plot doesn't look right to you,   try something in  the Projection
menu in the plot window to see what it does.

Note carefully how the square brackets are used.   In particular, the
t-range is included   INSIDE  the square brackets, for some reason.

In the next example, the last two bits of information control the ``window''
in which the plot is drawn.

> plot( [sin(t),cos(t), t=0..Pi] , -5..5, -5..5);
> plot( [sin(t),cos(t), t=0..2*Pi]);

Try playing with any of the commands above, by using the mouse and
arrow keys to change whatever you want.   

Plotting in 3d


Maple is especially useful for plotting surfaces and other objects in three dimensions.

The basic 3-d plotting command has the  following form:


> plot3d( x^2+y^2, x=-3..3, y=-3..3 );

The result is a surface in xyz-space, as you'd expect.    Note that you might
have to fool with some of the menu items at the top to get axes, different
color schemes, etc.    After you've made new choices for such things, either  click 
again on the picture (use the MIDDLE mouse button) or type  ``p''  to get
the new plot.

Try clicking on the picture and dragging the bounding
box around to see the surface from different angles.      


Try changing the function or the domain region to see what happens.



Other 3d plotting tools

 Maple has many other 3d plotting tools.   To get access to most of them, use
this command:

> with( plots );

This loads a lot of new plotting functions into Maple.    For example, you can type


> contourplot( x^2+y^2, x=-3..3, y=-3..3 );

The result is a set of level curves for the function.   (You may want to experiment
with some of the menu items at the top of the plot window to get axes, etc.)

For more information about any function, such as   contourplot , you can
always use the Maple help system.   Type, e.g., 


> ?contourplot

You'll get a window full of information.   The bottom of the window often contains
the most useful information and examples.




Defining functions 


Above we showed how to define   f   and   g   as EXPRESSIONS.   Doing
so can save a lot of typing and retyping,  but it does NOT define   f and g
as FUNCTIONS   in the  usual  mathematical sense.    For example,
we might like to find   f(3),   but   Maple  won't do this properly (yet):


> f(3);
We got some nonsense, but not what we wanted.

Here's how to define   f  as a  FUNCTION, not an expression.  Notice
the use of the  ``arrow''---it's actually just a hyphen and a  greater than  sign.


> f :=  x -> x^2*sin(x);
Now we've defined   f  successfully as a function in the usual sense.  


> f(3);
To get a decimal form of the answer above,  type something like this:

> evalf(%);
The    evalf    command  tries to evaluate anything to a decimal (or ``floating 
point'')  number.   The  percent  sign represents the previous output.

If you feed   f   a decimal number to start with, it will give a decimal output.

> f(3.0);
>