If you click and drag on the small 2D grid to the left of the 3D plot you,
a trace point moves around on the surface corresponding to the point
(x, y) on the 2D plot. The gradient and tangent vectors
at this point on the surface are also shown, with the gradient vector shown
in the xy-plane at the input point in the 3D plot.
At any time, you can use the scrollbar to move the trace point and
tangent vector along the current path of steepest ascent. (As
described above, you can move this path using the right mouse button.)
This helps you to consider the way the gradient vectors consistently point
in the compass direction you should move in to ascend the surface most
steeply.
There are also several useful menu options. In addition to the
Select 3D View menu (described below), there is an Options
menu and a Help menu. Using the Options menu, you can
choose the scale factor to use to shorten the appearance of the gradient vector and the
tangent vector, dividing by a value from 1 to 10. The default is 3 for
the first function in this applet, since the vectors lengths are otherwise quite long. (It is 1 for the
other function.) This means that the vectors are displayed at 1/3
their normal length to make them easier to view. To see them actual
size, simply set the scale factor to 1.
At any time, you can rotate the 3D graph to get a better perspective by using
the mouse to click on the 3D plot and then using the left and right arrow keys
to rotate the 3D graph about the z-axis. You can also click and drag anywhere
on the 3D plot to rotate it about the origin in the direction you drag the mouse.
There is a button to adjust the viewing region for the first function,
f (x, y) = 4 - x^{2}
- 2y^{2}. This allows you to adjust how much of the
surface is drawn so that you can more easily see the gradient vector
underneath the surface. (This button is only enabled for this
surface.) There is also a button for making the surfaces
semi-transparent. This gives another useful way to view the gradient
vector in the 3D plot, although it is slower to rotate, and it makes the 3D
effect less clear when used with 3D glasses (described in the next
paragraph).
A feature that you may find helpful is the Select 3D View menu.
On this menu, you can select from a list of 3D viewing options. Most
require 3D glasses of one kind or another (red-cyan/red-blue,
red-green, or
amber-blue), but there are two 3D viewing options that do not require 3D
glasses, but still can give a truer 3D experience. These options are
Stereo Pair and Cross-eyed. Both of these require some practice, but
give a full color 3D view. See the 3D View
Help for more details on all of these options.