Curl

∇×F — Rotation & Circulation

What is Curl?

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curl F = ∇ × F

= ( ∂F₃/∂y − ∂F₂/∂z , ∂F₁/∂z − ∂F₃/∂x , ∂F₂/∂x − ∂F₁/∂y )

In 2D: (curl F)_z = ∂F₂/∂x − ∂F₁/∂y

The curl measures the local rotation of the field at each point. Positive = counter-clockwise. Negative = clockwise. Zero = irrotational.

Show ∂F₂/∂x only

Show ∂F₁/∂y only

Show curl F = difference (default)

The Paddle Wheel Test

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Imagine placing a tiny paddle wheel in the flow. If the field pushes harder on one side, the wheel spins. The curl measures this tendency to rotate.

Green's Theorem (Circulation Form)

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∮_∂R F·dr = ∬_R curl F dA

The total circulation around a closed curve equals the integral of curl over the enclosed region. This is the circulation form of Green's theorem — the 2D version of Stokes' theorem.

Show verification circle

Circle radius 2.0

Key Identities

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curl(∇f) = 0

The rotation of any gradient field is always zero. This is because ∂²f/∂x∂y = ∂²f/∂y∂x (Schwarz's theorem), so the mixed partials cancel. Geometrically: a gradient field has no circulation around any loop.

Verify on example

Conservative ⇔ Irrotational

A field F is conservative (F = ∇f for some f) if and only if curl F = 0 on a simply connected domain. This means: curl F = 0 ⇔ path independence ⇔ the field is a gradient.

Test a field

div(curl F) = 0

The divergence of the rotation is always zero (in 3D). In 2D, curl F is a scalar (the z-component), so this identity is trivially satisfied. Geometrically: pure rotation creates no net flux.

Verify numerically

Display

Arrows show the field direction & magnitude (color = speed: blue→red).
Particles trace out flow lines — their trails reveal trajectories.
Heatmap shows curl: red = CCW, blue = CW.

Flow Particles

Field Arrows

Curl Heatmap

Grid & Axes

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Enter coordinates or click on canvas

Actions

Controls

Scroll Zoom in/out
Drag Pan the view
R Reset view
Click Pin inspect point
Dbl-click Clear pin

F = (y, −x)

Heatmap = curl F

CW <0 ← 0 → CCW >0

x: 0.00 y: 0.00 | F = (0.00, 0.00) | curl F = 0.00 | – FPS