Math Insight

Applet: A sphere rotated by a rotating vector field

Applet: A sphere rotated by a rotating vector field

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The sphere embedded in this vector field is rotated by the swirling vector field, while its center is fixed at a point (originally the origin). The rotation of the sphere by the vector field is an indication of the vector field's curl. You can move the point at which the sphere is anchored by dragging it with your mouse.

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. Note that $\curl \dlvf$ is parallel with the axis of rotation of the sphere. Since the curl is constant independent of the point $(x,y,z)$, the sphere spins in the same direction and with the same speed, no matter where it is moved.

You can rotate the graph with your mouse to better visualize the three-dimensional perspective.

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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.