To access the Level Curves applet (based
on work of Tom Banchoff) follow the link:

http://www.slu.edu/classes/maymk/banchoff/LevelCurve.html

Spread the three graphs out
where you can see them all, as well as the control panel.

We want to start with a simple function whose
graph we understand and use it to see how level curves can be used to
build the surface which is the graph of a function in two
variables. The applet opens with the graph of the function f(x,y)
= x^2-y^2, a simple saddle where level curves are hyperbolas opening either
left and right or up and down, depending on the value of z0.

1) Use the opening example, f(x,y) = x^2-y^2, to
see how a graph can be
constructed.

Use
the controls on z0 to see how the intersections of various planes with
the surface give contours that help define the surface.

Using the "down the z axis" option from the view menu of the f(x,y)
window, notice that the contour map in the domain window can be
obtained form the f(x,y) window.

Uncheck the Surfaces checkbox and see how good a surface one can obtain
from the level curves. You can increase the number of curves by
increasing the number of steps for z.

Now unclick the Contour Sets checkbox so that you only see one contour at a time.

Scroll z0 through its range and describe the shape of the graph z=f(x,y).

Try visualizing the surface from the contours.

2) Repeat the process above with z=x*y.

3) Repeat the process with the linear function z=x/2+y/3.

Now move to less familiar curves and try to use contours to describe a surface.

4) Turn the Contour Sets and Surfaces boxes off, and set the view of f(x,y) to "down the z-axis.

Set f(x,y) equal to x^2+y^2 - x^2*y^2.

Scroll z0 through its range and describe the shape of the graph z=f(x,y).

Describe a "typical contour" and explain how the contours change in shape and location as z0 changes.

Note any value of z0 that have particularly interesting contours.

Note places where the pattern of contours indicates interesting points like minimums or saddle points.

Turn on the Contour Sets, and rotate f(x,y) by dragging the image to see how well it fits your prediction.

Turn on the Surfaces to see how good your prediction is.

5) Repeat the process with f(x,y) = sin(y-x^2)

Return to the Level Curves Applet page.

Last updated By Mike May, S.J.,
January 23, 2006.