Applet: Gradient and directional derivative on a mountain shown as level curves
The height of a mountain ranged described by a function $f(x,y)$ is shown as a level curve plot. A point $\vc{a}$ (in dark red) can be moved with the mouse. The height $f(\vc{a})$ is shown on the bottom cyan slider labeled by “f”. The direction of steepest increase of $f$ is given by the gradient vector $\nabla f(\vc{a})$ (the dark blue vector is ten times longer than the actual gradient). The actual length of the gradient $\| \nabla f(\vc{a})\|$ is shown by the dark blue line on the middle (light green) slider. The light green line on that slider indicates the value of the directional derivative $D_{\vc{u}}f(\vc{a})$, where $\vc{u}$ is represented by the light green vector coming out of $\vc{a}$. The direction of $\vc{u}$ is controlled by $\theta$ (changed via top slider), where $\theta$ is the angle between $\nabla f(\vc{a})$ and $\vc{u}$.
Applet file: mountcondirectgrad.m
General information about LiveGraphics3D applets
This applet was created using LiveGraphics3D. To manipulate it, you can
- click and drag to rotate in any direction,
- shift + click and drag up/down to zoom out/in,
- shift + click and drag left/right to rotate around an axis coming out of the screen, and
- press Home to reset to the original view.
Most of the LiveGraphics3D applets have parameters that you can change. In general, you can click and drag points or balls around to change the parameters.
Since these applets use Java, you must have Java installed and properly configured in your browser for the them to display. You can get Java here. If you have trouble getting them to display in your browser, you can visit the LiveGraphics3D troubleshooting page for information on how to get the Java applets to work.
On occasion, even when Java is configured correctly, the Java applets don't load the first time and one instead sees an error message. You can wait a few moments to see if the applet will eventually load itself anyway. You can also reload the page to see if the applet will properly load. Sometimes, one needs to do this multiple times.
