A Day with Maple

or

How I learned to love the computer.

Before you start a few notes on notation and the use of Maple.

  1.  Don’t feel shy about using the Help menu. When you click on Help, a menu will come up. For our purposes, you should then click on "Topic Search". You should then type "graphics" in the Topics box. What will then come up is a page with help on graphics commands. These will probably be incomprehensible unless you have some experience with computers. However, what you will probably find most helpful is to go down to the bottom of the document for examples (in red) of how to write commands. You can even cut and paste these into a notebook. The commands written on this sheet are in boldface, but you do not need to give them to the machine that way.
  2. For some of the plots you will need to write with(plots): or with(plottools): before you put in your command.
  3. If you want to tell the computer you want to square x , you should write x^2
  4. Multiplication is denoted by using * so that what we normally write as 2x should be written as 2*x.
  5. Be sure to end every command with a colon or semi-colon. We'll talk about when to use which later.

WHAT TO DO WITH THE COMPUTER

Text Box:Click on the Maple icon

  1. After the Maple screen comes up, and clears, type in the following exactly:
    plot(sin(x), x=0..2*Pi);
    and then press the Return or Enter key to the right of the keyboard.
    You should see a picture similar to the picture at right. To analyze what you typed in:
    "plot" tells the machine to plot a graph in two dimensions. Inside the parentheses the function to be plotted is next (in this case sin(x) ), followed by the range of x values over which it should be plotted. Note that pi must be spelled with a capital P for the machine to know that it is pi. Try changing the graph by increasing the range of x to 4*Pi and by changing the function to x*sin(x).
  2. To plot in 3 dimensions, use the command "plot3d". To plot a graph in 3 dimensions type plot3d((x^2 + y^2), x=-3..3,y=-3..3);
    and then press the Enter key. You should see a the graph of a paraboloid. If you click on the graph you can use the buttons across the top of the window to alter how the picture looks. In particular when you hold down the left mouse button and move the mouse when the cursor is on the graph, you can change the view of the surface.

                             






  If you want to display two graphs simultaneously you type in the following:

with(plots):
F:=plot3d(9-x^2, x=-3..3, y=-3..3):
G:=plot3d(x^2 + y^2, x=-3..3, y=-3..3):
display3d({G,F});

Observe several things about this list of commands:

  1. All except the last ends with a colon
  2. The second and the third commands use "F" and "G", respectively, to name the graphs that are generated
  3. The fourth command simply shows the surfaces that were generated in the two previous commands.
  4. There are many options that you can use. Check the Help section under the topic "plot3d"to see what some of them are. It is possible to plot in cylindrical and spherical coordinates as well.

SOME USEFUL NOTATIONS AND COMMANDS

  1. * denotes multiplication, e. g. 2*x means 2 multiplies x.
  2. ^ denotes exponentiation, e. g x^2 means x squared.
  3. / denotes division
  4. exp(x) denotes the exponential function ex.
  5. diff(f(x),x); computes the derivative of f(x).
  6. int(f(x),x); computes the antiderivative or an indefinite integral of f(x) i. .

    int(f(x),x=1..5); computes the definite integral of f(x) from 1 to 5, i.e .
    Thus if we want to evaluate the integral , we would type in

    int(x^2, x=2..5);

    Try this. You should get an answer of 39.
  7. This also works for a multiple integral. If you want to evaluate the double integral
    , the Maple command would be:

    int(int(x+y,y=x^2..x),x=0..1)
    Try this. You should get 3/20. Note that if you are integrating first with respect to y , it is the integration that is the furthest inside the nested int commands, and its range of integration is noted as y=x^2..x.
  8. The triple integral is similar: is evaluated by

    int(int(int(x+y+z, z=x^2-4..8-y^2),y=0..(4-x^2)^(1/2)), x=0..2);

    Try it. You should get .
  9. plot3d(9-x^2-y^2,x=0..3,y=0..(9-x^2)^.5);
     This command is used to plot the 3-dimensional surface:
  10. with(plots):
    spacecurve([cos(t),sin(t),1+sin(5*t)],t=0..5*Pi,color=blue,thickness=2,numpoints=500);
     This command plots the spacecurve: . Note that the function is enclosed in [……]. The option, "numpoints=500" tells the program how many points to compute. The higher the number, the smoother the curve will be usually.
  11. plot([t^2,t^3-2*t,t=0..3],color=blue);
    This command will do the same as the above in 2 dimensions.
  12. f:=[t^2,sin(t)-t*cos(t),cos(t)+t*sin(t)]
    g:=[t^2,t^3,t^4]:
    These two statements define two vector-valued functions, f and g. To evaluate these functions at the point , write the command eval(f, t=3) or eval(g, t=3). You can define any kind of function this way.
  13. diff(f,t)
    This command finds the derivative of the function f with respect to the variable t.
  14. plot3d([3*sin(u)*cos(v),3*sin(u)*sin(v),3*cos(u)],v=0..2*Pi,u=0..Pi);
    This command plots the parametric surface:
    .
    Again, note the [……] around the vector-valued function.
  1. with(plots):contourplot(9-x^2-y^2,x=-4..4,y=-4..4,contours=[9,7,5,3,1],numpoints=3000,color=blue);
    This command shows the function with contours at .
  2. f := transform((x,y) -> [x,y,0]):
    This command allows you to show a 2-dimensionsl graphic in a 3-dimensional graphic. In this case the third coordinate on the right being 0 means that the 2-d graphic will appear on the plane. If there were a 3 there, it would appear on the plane. You will need to write with(plottools): to use this function.
  1. with(plots):
    fieldplot( [x,-y],x=-2..2,y=-2..2,color=blue, thickness=2);

     This plots a blue vector field in two dimensions. Remember that the thickness numbers run from 1 to ten, with 10 being the thickest. Note the [….] around the vector function
  2. with(plots):
    fieldplot3d([xy, yz, zx], x=-3..3, y=-3..3, z=-3..3, color=red, thickness = 2);

     This plots a red vector field in three dimensions.
  3. with(plots):
    gradplot( x^2+y^2,x=-2..2,y=-2..2,thickness=3);

     This plots the gradient of the function.
  4. with(plots):
    gradplot3d( (x^2+y^2+z^2+1)^(1/2),x=-2..2,y=-2..2,z=-2..2,color=black);

     This plots the gradient of the function in 3 dimensions. "

 

Some Spherical Coordinates Material

 

 with(plots):
 plot3d(12+sin(8*theta)*sin(8*phi),theta=0..2*Pi,
 phi=0..Pi,numpoints=5000,coords=spherical);
 
 int(int(int(rho^2*sin(phi),rho=0..12+sin(8*theta)*sin(8*phi)),
phi=0..Pi),theta=0..2*Pi,coords=spherical);