### Applet: Circling sphere in a vector field with zero curl

#### Applet loading

#### Applet loading

The sphere is circling around the $z$-axis due to the macroscopic circulation of the vector field. However, away from the $z$-axis, the vector field has no microscopic circulation, i.e., its curl is zero. This example illustrates that one cannot infer curl from the macroscopic circulation of the vector field. Macroscopic and microscopic circulation can be very different. First panel shows the full vector field; second panel shows its projection in the $xy$-plane. You can rotate the first panel with the mouse to better visualize the three-dimensional perspective.

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 links

This applet is found in the pages

#### General information about three.js applets

The applet was made using three.js and requires Javascript as well as a browser that supports WebGL. For most three.js applets, you can drag with the mouse to rotate the view, drag with the right button to pan, and zoom in/out with the mouse wheel. Many applets contain points that you can drag to change values of variables.