Applet: A rotating vector field
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This vector field illustrates rotation around an axis. The axis is halfway between the $x$ and $z$ axes. Along the axis of rotation, the vector field is zero. The zero vectors along this axis are displayed as dots. The rotation of the vector field gives it curl, which points parallel to the axis of rotation.
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, parallel with the axis of rotation.
You can rotate the graph with your mouse to better visualize the three-dimensional perspective.
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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.