Applet: A rotating sphere indicating the presence of curl
The rotation of the sphere, with its center fixed at the origin, indicates the presence of nonzero curl at the origin. The curl at the origin is represented by the green arrow. The length of the curl vector indicates the rotation speed. The direction of the curl vector is determined by the rotation direction using the “right hand rule”: curl finger of right hand in direction of rotation and your thumb will indicate the direction of the curl vector. 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 $\dlvf(x,y,z)= (-y,x-z,y).$ Its curl is \begin{align*} \curl \dlvf(x,y,z) &= \left(\pdiff{}{y}y-\pdiff{}{z}(x-z), \pdiff{}{z}(-y)- \pdiff{}{x}y, \pdiff{}{x}(x-z) - \pdiff{}{y}(-y) \right)\\ &=(2,0,2), \end{align*} which is a constant independent of point $(x,y,z)$ and always points at an angle halfway between the positive $x$ and $z$ axes.
Applet file: curlvecsphere.m
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