Curl and circulation

Curl measures local circulation

Curl detects the tendency of a small paddle wheel to spin when placed in a vector field. Uniform translation has zero curl; shear can have nonzero curl even when the flow does not look like a circular path globally.

Compare circulation around a tiny loop

Circulation is ∮u·dl. For a small rectangle in the xy plane, the leading flux of curl through its area is (∂u_y/∂x − ∂u_x/∂y)ΔA. Dividing by area and taking the limit gives the z-component of ∇×u. The full vector collects the three oriented loop results.

For u = (−Ωy, Ωx, 0), curl = (0,0,2Ω). A rigid body rotating at angular velocity Ω has vorticity twice its angular velocity, showing the factor of two is geometric.

Zero curl is local, not automatically global

In a multiply connected domain, a field can have zero curl away from a singularity but nonzero circulation around the excluded region. Boundary topology matters when using potential-flow shortcuts.

Work it: Find ∇×u for u = (y², xz, xy). Evaluate at (1,2,3) and identify the axis about which a small paddle wheel tends to rotate.