Applet: Non-differentiable function with partial derivatives
The partial derivatives of this function $f(x,y)$ exist at the origin, $\pdiff{f}{x}(0,0)=0$ and $\pdiff{f}{y}(0,0)=0$, as the function is constant along the $x$ and $y$ axis, $f(x,0)=f(0,y)=0$. However, the slopes coming into the origin from other directions are non-zero. If there were a tangent plane at the origin, it would have to be the horizontal plane $z=0$, as that is the only plane that would be tangent along the $x$ and $y$ directions. Clearly, the slopes would not match in other directions, so this plane is not tangent. Therefore, there is no tangent plane at $\vc{a}=(0,0)$, and the function is not differentiable there. You can drag the blue point on the slider to remove the folds in the surface, but that does not change the partial derivatives at the origin. The wrinkle at the origin is enough to make the function non-differentiable.
Applet file: dabilityex.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.
