Introduction to Maple: A Computer Algebra System.

General remarks.

• general syntax:
  • '>' denotes the Maple prompt. It automatically precedes each input statement.
  • Maple is case sensitive: 'xvalue', 'XVALUE' and 'Xvalue' refer to different registers.
  • Maple does not recognize implied multiplication: 'AB' refers to register AB , not to A*B.
  • Greek letters are spelled out: 'pi' , 'PI' denote lowercase and uppercase letter pi respectively
    whereas 'Pi' refers to the constant p = 3.141592 ..... Use exp(1) to refer to euler's number e=2.71828... 
  • To force a numeric rather than a symbolic value, and evaluate to floating point, use the 'evalf'
    command:               > phi:= Pi*2/3; phi;
                                  >evalf(phi);
    
    For the first 100 digits of p,   >evalf(Pi,100);
    
  • To save a MAPLE worksheet, click 'file' and select the 'save as' option, and type a name.
    Save to C:\windows\temp\ directory on your lab-PC or to your 3.5" diskette in drive A:\
                          
• Every Maple statement ends with ';' to echo the input, or ':', to have the echo suppressed.
   
• ':=' denotes the assignment operator:             > a:=5; a;
  To un-assign a variable name:                       > a:='a'; a;
   

• functions may be entered as expressions:         > f:=x^2;
  which must be evaluated by substitution:         > subs(x=george,f);
  Alternately, functions may be entered as such:   > f:=x->x^2;
  which may be evaluated directly:                    > f(george);

• Plotting points, e.g. {(1,2), (2,4), (3,3), (4,5), (5,8)} in the (x,y)-plane:
  we enter the data point as a list:              > points:=[[1,2],[2,4],[3,3],[4,5],[5,8]];
  and plot the points:                      > plot(points, style=POINT);

• Plotting graphs of functions of a single variable e.g, the linear function y=f(x)=mx+b,
  the function f is entered as an expression:    > m:=1; b:=1; f:=m*x+b;
  and plotted for x values from 0 to 5:         > plot(f, x=0..5);
  Note that the input is not designated when plotting expressions: then function is referred to as f.

      Alternately, f may be entered as a function:  > f:=x->m*x+b;   (-> emulates an arrow)
       and plotted for x from 0 to 5:             > plot(f(x),x=0..5);
      Note that functions do require an input designation: the function is referred to as f(x).
 

• Simultaneous plotting of graphs,
  make the 'plots' package available:         > with(plots);
  write the first plot to a file with name 'p1' :  > p1:=plot(points, style=POINT):  (Note: use ':' to suppress file output)
                                      > p2:=plot(f(x),x=0..5):

      To display this set of plots:               > display({p1,p2});
 

• Graphing implicit relations (expressions not explicitly written in the dependent variable),
  such as the unit circle x²+y² = 1:                      > implicitplot(x^2+y^2=1,x=-1..1,y=-1..1);
  Explore the difference in projection:, by right clicking on the plot and selecting constrained vs unconstrained.
  
• Graphing parametric relations (expressions for x and y in terms of a parameter, say, t,
  such as the unit circle x(t)=cos(t), y(t)=sin(t):                      > plot([cos(t),sin(t),t=0..2*Pi]);

Assignments:

      1. Plot the data points: {(12,3),(10,2.6),(8,2.1),(6,1.3),(4,.5),(2,-0.2),(0,-0.8)} simultaneously
      with the regression line y = mx + b. Describe your method of obtaining estimates for m and b.

      2. Graph the ellipse x2/2 + y2/3 = 1 and the line y = 2x - 1 simultaneously and estimate the coordinates
       of the points of intersection. Obtain exact values for the coordinates of these points by solving the system of equations.

                       Table 1 lists Wind-Chill values as a function of temperature, T (F) and wind speed, W (mph):
 

• Plotting data points in 3-space, we enter the wind chill data points as a 'list of lists':

            > points:=[[35,5,33],[30,5,27],[25,5,21],[20,5,16],[15,5,12],[10,5,7],[5,5,0],[0,5,-5],
                      [35,10,22],[30,10,16],[25,10,10],[20,10,3],[15,10,-3],[10,10,-9],[5,10,-15],[0,10,-22],
                      [35,15,16],[30,15,9],[25,15,2],[20,15,-5],[15,15,-11],[10,15,-18],[5,15,-25],[0,15,-31],
                      [35,20,12],[30,20,4],[25,20,-3],[20,20,-10],[15,20,-17],[10,20,-24],[5,20,-31],[0,20,-39],
                      [35,25,8],[30,25,1],[25,25,-7],[20,25,-15],[15,25,-22],[10,25,-29],[5,25,-36],[0,25,-44]];
          and make the plotting commands available:   > with (plots);

          To plot the data pointwise, with standard axes:

            > pointplot3d(points,axes=normal,title=`wind-chill data`); (title in back-quotes)

       To plot a surface which passes through the data, we use the surfdata command.
       For this, the data-rows need to be designated:

            >points:=[[[35,5,33],[30,5,27],[25,5,21],[20,5,16],[15,5,12],[10,5,7],[5,5,0],[0,5,-5]],
                     [[35,10,22],[30,10,16],[25,10,10],[20,10,3],[15,10,-3],[10,10,-9],[5,10,-15],[0,10,-22]],
                     [[35,15,16],[30,15,9],[25,15,2],[20,15,-5],[15,15,-11],[10,15,-18],[5,15,-25],[0,15,-31]],
                     [[35,20,12],[30,20,4],[25,20,-3],[20,20,-10],[15,20,-17],[10,20,-24],[5,20,-31],[0,20,-39]],
                     [[35,25,8],[30,25,1],[25,25,-7],[20,25,-15],[15,25,-22],[10,25,-29],[5,25,-36],[0,25,-44]]];

            > surfdata(points);

       Plotting graphs of a plane, e.g. A: -2x+y+z = 1 as a functions of a two variables, x and y,
       express the linear function explicitly for z: z = 2x - y + 1, and plot:
            > plot3d(2x-y+1,x=-3..3,y=-3..3, title=`the plane 2x-y-z=-1` );

       or plot the surface implicitly using the defining implicit relation -2x+y+z = 1:
            > implicitplot3d(2*x-y-z=-1,x=-3..3,y=-3..3,z=-3..3);
 

• Simultaneous plotting of graphs of planes:

            > p1:=implicitplot3d(x=0,x=-3..3,y=-3..3,z=-3..3): (Note: use ':' to suppress output)

            > p2:=implicitplot3d(y=0,x=-3..3,y=-3..3,z=-3..3):

            > p3:=implicitplot3d(z=0,x=-3..3,y=-3..3,z=-3..3):

           To display this set of plots: > display3d({p1,p2,p3});

• Graphing more complicated implicit relations in 3-space, such as the (unit) sphere: x²+y²+z² = 1,
         > implicitplot3d(x^2+y^2+z^2=1,x=-1..1,y=-1..1,z=-1..1);
  Check on the difference in projection: constrained vs unconstrained by right clicking on the plot,
  and changing the projection option.
  
  Other examples of implicit relations are (elliptic) cones, e.g.:

            > implicitplot3d(2*x^2+y^2-z=1,x=-3..3,y=-3..3,z=-3..3);

       and ellipsoids, e.g.:

            > implicitplot3d((x/2)^2+(y/3)^2+(z/4)^2=1,x=-5..5,y=-5..5,z=-5..5);

Extreme values:  Minima, Maxima and Saddles

We can use Maple's plotting facility to visualize extremal behavior of functions of two variables:

• the function z = x² + y² exhibits an absolute minimum: a point with a function (z) value less than any other:

            > f1:=x^2 + y^2;

            > plot3d(f1, x=-3..3, y=-3..3, title=`an absolute minimum` );
 

• the function z = 4 - x² - y² has an absolute maximum: a point with a function (z) value greater than any other:

            > f2:=4-x^2 - y^2;

            > plot3d(f2, x=-3..3, y=-3..3, title=`an absolute maximum` );

•  Some functions exhibit mixed extremal behavior. The function z = x² - y² for example has a maximum
  along the section x=0 whereas the section in the plane y=0 shows a minimum:

            > f3:=x^2 - y^2;

            > p1:=plot3d(f3,x=-2...2, y=-2..2, title=`a saddle point`);

            > p2:=implicitplot3d(x=0, x=-2..2, y=-2..2, z=-2..2):

            > p3:=implicitplot3d(y=0, x=-2..2, y=-2..2, z=-2..2):

            > display3d({p1,p2,p3});
 
           To view the sections separately, draw simple 2-d plots with the plane equation substituted in the surface expression:

            > plot(subs(x=0, f3), y=-2..2); to view the minimum along the y-axis, and
         > plot(subs(y=0, f3), x=-2..2); to see the maximum behavior along the x-axis.

        These points of mixed behavior are called 'saddles' or 'saddle points'.
 

• Sometimes a contour plot is useful in graphing a function of two variables.
  DEFINITION: A contour is a curve in 3-space, connecting points with equal function value:
               thus the contour C has representation C: {(x,y,z)|f(x,y)=c} for some constant c.

            > plot3d(x^2+y^2, x=-3..3, y=-3..3, style=contour, title=`contour lines` );

        DEFINITION: A level curve is a curve in the x,y-plane, connecting points with equal function value:
                     thus the level curve L has representation L: {(x,y)|f(x,y)=c} for some constant c.
       Note the missing z coordinate in the level curve - it is a plane curve.

            > contourplot(x^2+y^2, x=-3..3, y=-3..3, title=`level curves` );

          Thus, the level curves of a function z=f(x,y) are the projection of the contour lines of the function onto the x,y-plane.
        A contour plot is a collection of level curves.
 

Assignments:

1. Plot the circular cone z = 2(x² + y²)^1/2; simultaneously with the plane z - x= 2.
   What is the shape of the curve that constitutes this conic section?

2. Use Maple to sketch the plane that passes through the points: P(1,2,1),  Q(-2,2,2) and  R(-3,-2,-3).

3. Find the extreme value(s) of the function z = 2x² - xy + y² - 2x + y + 2
   using a 3d-plot and a contour plot by zooming in on the apparent extrema.

4. Plot the function z = xy/(x² + y²) and inspect the contour plot for this function.

• The levelcurves appear to meet. Where?
• Is it possible for the levelcurves of a function to intersect?
• Plot the sections y=ax for different values of a=-2,-1,0,1,2. Conclusion?