Applet: A nonrotating sphere indicating absence of curl
The sphere does not rotate when its centered is fixed, indicating that, away from the $z$-axis, the vector field has no microscopic circulation, i.e., its curl is zero. On the other hand, the macroscopic circulation of the vector field around the $z$-axis is evident from the graph. This example illustrates that one cannot infer curl from the macroscopic circulation of the vector field. Macroscopic and microscopic circulation can be very different. Left panel shows the full vector field; right panel shows its projection in the $xy$-plane.
This vector field is $$\dlvf(x,y,z) = \frac{(-y,x,0)}{x^2+y^2}$$ for $(x,y) \ne (0,0)$. Since for $(x,y) \ne (0,0)$, $$\pdiff{\dlvfc_2}{x} =\pdiff{\dlvfc_1}{y}= \frac{y^2-x^2}{(x^2+y^2)^2}$$ one can compute that away from the $z$-axis, $\curl \dlvf(x,y,z) = (0,0,0)$.
Applet files: curl3movedsphere.m, curl3circle2D.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.
