Math

Multivariable Calculus · Curl

Curl

The curl of a plane vector field, ∂Q/∂x − ∂P/∂y, is twice the rate at which an ideal small paddle wheel placed in the flow spins. Drop a wheel into a straight flow and watch it turn; let it drift round a vortex and see it never turn at all.

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Curl in this visualization

The curl of a plane vector field F=(P,Q)\mathbf F = (P, Q) is the number below, also called the scalar curl; it is the third component of the three-dimensional ∇×F\nabla\times\mathbf F. It is the circulation density: the circulation round a small loop about a point, counterclockwise, divided by the area it encloses, as the loop shrinks to the point. Both hold where F is continuously differentiable.

Curl as a derivative and as a limit
curl⁡F=∂Q∂x−∂P∂y=lim⁡r→01πr2∮F⋅T ds\operatorname{curl}\mathbf F = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \lim_{r\to 0}\frac{1}{\pi r^2}\oint \mathbf F\cdot\mathbf T\,ds

The yellow ticks are the flow's component along the rim, F·T; their integral Γ is the rim's circulation, and Γ/(2πr²) the rim's average angular velocity. It tends to half the curl as r → 0, and equals it at every r when the curl is a linear function. The wheel itself is the ideal small wheel of a smooth flow: it turns at ω=12curl⁡F(P)\omega = \tfrac12\operatorname{curl}\mathbf F(P), the average of the rates at which two perpendicular line elements through P turn at that instant — the level one at ∂Q/∂x, the upright one at −∂P/∂y (the cross). Either one alone need not turn at ω: in the shear flow (y, 0) the upright one turns at −1 and the level one not at all.

Drift lets the wheel go with the flow. At each point it passes it turns at half the curl there, so its total turn is θ=∫12curl⁡F dt\theta = \int \tfrac12\operatorname{curl}\mathbf F\,dt along the way; a mark every half unit of time records its orientation. In the vortex (−y, x)/(x² + y²) the curl is 0 everywhere except at the origin, where F is undefined, and a wheel circling the origin comes back with θ = 0: the flow bending round would turn it counterclockwise, the faster water on its inner side clockwise, and the two cancel.

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Things to notice

  • Green’s theorem (circulation form): ∮∂DF⋅T ds\oint_{\partial D}\mathbf F\cdot\mathbf T\,ds round the boundary of a region D, walked counterclockwise, equals ∬Dcurl⁡F dA\iint_D \operatorname{curl}\mathbf F\,dA, when F is continuously differentiable on an open set containing D. In the loop, the circulation along a shared inner edge contributes once to each of the two cells, with opposite orientations, so the cells’ circulations add up to the outer one, and each is about curl × area. The loop adds the curl at each cell’s centre times its area: a midpoint sum, exact when the curl is linear and closing in on Γ as the cells shrink otherwise.
  • Straight flow lines can have curl (the shear flow, the channel) and round flow lines can have none (the vortex). A loop round the vortex’s centre has circulation 2π although the curl is 0 everywhere else inside: there the theorem does not apply, because F is undefined at the centre (the spiral staircase of the line integral page). The ticks show only the component along the rim; the component across it belongs to the divergence. In three dimensions the curl is a vector and the theorem becomes Stokes’ theorem.
  • The curl comes from symbolic derivatives, and “=” means an exact value. Circulation is in closed form for the presets and by numerical integration with an error estimate for typed fields; the drift is in closed form for the presets except the cubic, and integrated numerically otherwise. The marks are samples every half unit of time: a wheel turning by a whole turn between two of them looks still in the marks, while θ counts every turn. The background particles only show where the flow goes; their crowding means nothing.
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Controls

  • Drag the wheel’s centre P (it catches on the arrows’ lattice) or its rim (it catches on ½, 1, 2 …), or type them: P from −100 to 100, r from 0.001 to 100. Drag elsewhere to pan; zoom with the view buttons. Choose Wheel or Square loop to observe the field. The loop has half-side r and an adjustable number of cells. Drift starts, pauses or resumes the same run in Wheel mode. The wheel keeps moving outside the picture. After pausing, moving its centre sets a new departure point and keeps it still until you press Drift. Return to start restores the departure position and orientation and clears the run’s time, turn and trail. Type P and Q with + − × ÷ ^, sin, exp, ln, sqrt …
  • Presentation mode opens a full-screen view for projectors: Space drifts or pauses the wheel, ← and → double or halve the rim, H hides how the wheel turns. Share copies a link that reopens the field, the wheel and the view.
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Related

Further reading: Wikipedia: Curl (mathematics); OpenStax, Calculus Volume 3, 6.5 Divergence and Curl; Wikipedia: Green's theorem.