Applet: A rotating sphere indicating the presence of curl
The rotation of the sphere when its center is fixed in an indication of the curl of the vector field. The curl of the vector field is shown by the green arrow. Left panel shows the full vector field; right panel shows its projection in the $xy$-plane. To see the rotation of the sphere, hold your mouse cursor over the graph. (If you double-click, the animation will stop; double-click again to restart the animation.)
This vector field is $\dlvf(x,y,z)= (-y,x,0).$ Its curl is \begin{align*} \curl \dlvf(x,y,z) &= \left(\pdiff{}{y}0-\pdiff{}{z}x, \pdiff{}{z}(-y)- \pdiff{}{x}0, \pdiff{}{x}x - \pdiff{}{y}(-y) \right)\\ &=(0,0,2), \end{align*} which is a constant independent of point $(x,y,z)$ and always points in the positive $z$ direction.
Applet files: curl2movedsphere.m, curl2circle2D.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.
