Level Curves and Surfaces

The graph of a function of two variables is a surface in space. Pieces of graphs can be plotted with Maple using the command plot3d. For example, to plot the portion of the graph of the function f(x,y)=x²+y^{2
corresponding to x between -2 and 2 and y between -2 and 2, type}

^{> with (plots);}
^{> plot3d(x^2+y^2,x=-2..2,y=-2..2);}

^{To draw the coordinate exaes,
use either "Normal" from the Axes menu of the 3D display window, or enter
axes=normal as
an option in plot3d.
To obtain more accurate graphs of certain functions, it is helpful to use
the option numpoints
, for example,}

^{> plot3d(x^2+y^2,x=-2..2,y=-2..2,axes=normal,numpoints=1000);}

^{To see the details of the graph
around a particular point, it is useful to zoom in. To do this, for example,
around the point (1,1), make a new plot with a different ranges of x and
y:}

^{> plot3d(x^2+y^2,x=-0.9..1.1,y=-0.9..1.1,axes=normal);}

^{Level curves}

^{The graph of a function
f(x,y) can be studied by drawing level curves f(x,y)=c corresponding
to various values of the constant c. There is a Maple command
that will plot several level curves in the same plot for a given function
with evenly spaced values of c. For example, to plot level
curves for the above function f(x,y), use the command}

^{> contourplot(x^2+y^2,x=-2..2,y=-2..2,scaling=constrained,}
^{axes=normal);}

^{The option scaling=constrained
is added here to make for a more accurate
plot - without it the level curves (which are circles) would look like
ellipses. To increase or decrease the number of level curves drawn
you may want to use option contours
. For example, contours=15 specifies
15 contours in the plot.}

^{In general, the result of
contourplot looks
like a topographical map of a landscape where the value of f
at a point represents the elevation. Such a map shows the general features
of the landscape by drawing curves through points which have the same elevation.
Notice that in the example above the level curves are not evenly spaced
- their higher concentration further away from the origin indicates that
the graph is getting steeper. Also notice that there is a minimum
in the center of the picture. Try the same plot with a larger number of
contours to see a family of smaller and smaller circles shrinking to the
origin.}

^{The command contourplot
does not allow to plot a level curve
f(x,y)=c for a specific value of c. There is however another
Maple command that will do just that. For example,}

^{> implicitplot(x^2+y^2=1,x=-2..2,y=-2..2,scaling=constrained);}

^{will plot a circle of radius
1.}

^{Functions of three variables}

^{For a function of three variables
f(x,y,z) the notion corresponding to the level curve of a two-variable
function is a level surface, f(x,y,z)=c. This is generally a surface,
which can be plotted with the help of Maple. For example,}

^{> implicitplot3d(9*x^2 + 4*y^2 + z^2 = 1,x=-1..1,y=-1..1,z=-1..1,}
^{scaling=constrained);}

plots the ellipsoid 9x²+4y²+z²=1 which is a level surface of the function f(x,y,z)=9x²+4y²+z².