Directions for Functions of Two Variables:

Wait for the applet to load and for the graph of the function to appear
in the gray area above.
(If there is no gray area, check your browser settings to make sure that
Java is enabled, or try with another browser)

The applet combines several tools for viewing functions of two variables.
Use the Show menu to switch from one mode to another.

The applet initially starts in the Input mode, which lets you choose a
function to plot (you can either enter it manually, or select one from the
dropdown list; click on the Plot button to create the new plot).

In Level curves mode (select it in the
Show menu) the left half of the display shows a contour plot
corresponding to the 3D plot in the right half. The slider control
in the lowerleft corner moves a level curve highlighted in yellow on both
plots.

In
Partial derivatives mode (select it in the Show menu)
the right half of the display still shows the graph of the function. The
lowerleft corner shows a small contour plot, with a pink dot representing
the point where we measure the partial derivatives. To move the
point, simply click somewhere in the contour plot.
There are two small graphs above the contour plot: these represent
slices of the graph of the function through the given point
(intersecting the 3D graph with planes parallel to the xz and
yzplanes: compare the small plots with the highlighted curves on the 3D plot). The partial
derivatives are, by
definition, the slopes of these graphs. Look at the bottomright corner of
the display for the values of f_{x} and f_{y}
at the selected point.

In
Directional derivatives mode (select it in the Show menu)
the right half of the display still shows the graph of the function. The
lowerleft corner shows a small contour plot, with a pink dot representing
the point where we measure the directional derivative. To move the
point, simply click somewhere in the contour plot.
The small graph above the contour plot shows a slice
of the graph of the function through the given point by a vertical plane
that makes the angle theta with the axis (theta is controlled
by the slider). This same slice is shown on the 3D graph.
By
definition, the directional derivative is the slope of this graph.
Look at the bottomright corner of
the display for the values of grad f and the directional derivative.