Curl · ∇ × F
probe position (drag the wheel)
position(1.00, 0.50)
derivation at probe
P =
Q =
∂Q/∂x—
∂P/∂y—
curl F = ∂Q/∂x − ∂P/∂y—
Stokes's theorem (locally): the line integral of F around a
tiny loop equals curl·area. The wheel's angular velocity is
curl/2.
What this means: curl measures local rotation. Fluid
vorticity, magnetic curl ∇×B = μ₀J (electromagnetism), and
a leaf spinning in a stream are all the same idea.
The wheel spins counterclockwise here.
positive curl — counterclockwise spin
negative curl — clockwise spin
arrows — field direction (gray)
paddle-wheel probe — drag me!
Pure rotation — same spin everywhere.