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Curl (3D)

∇×F in three dimensions

Curl in 3D

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curl F = ∇×F =

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

In 2D only the z-component exists. In 3D, curl is a full vector.

Arrows are colored by the selected quantity. Toggle below to see each curl component.

Color by (curl F)_x

Color by (curl F)_y

Color by (curl F)_z

Color by |curl F| (default)

Curl Direction & Axis

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The direction of curl F tells you the axis of rotation (right-hand rule). Enable curl arrows to see the curl vector overlaid on the field.

Show curl direction arrows (orange)

Stokes' Theorem

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∮_∂S F·dr = ∬_S (curl F)·n̂ dS

The circulation of F around a closed curve equals the surface integral of curl F over any surface bounded by that curve. Here we verify with a disk in the z = 0 plane.

Show verification disk

Circle radius 2.0

Key Properties

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curl(∇f) = 0

The curl of any gradient field is zero (Schwarz's theorem). This means gradient fields are irrotational.

div(curl F) = 0

The divergence of any curl field is zero. This means curl fields are always solenoidal (divergence-free).

Display

Arrows show F direction & magnitude. Color = curl (blue→red).
Rotate the scene to see rotation axes in 3D space.

Flow Particles

Field Arrows

Field Lines (Streamlines)

Grid & Axes

Arrow Settings

Density (per axis)6

Arrow Scale2.0

Inspect Point

Evaluates at z = 0 (base plane). Hover on scene to inspect.

Enter coordinates or hover on the scene

Actions

Controls

Left-drag Orbit camera
Right-drag Pan
Scroll Zoom
R Reset camera

F = (z², x², y²)

Arrows = |curl F|

Neg ← 0 → Pos

Hover for coords | F = (–, –, –) | |F| = – | |curl F| = – | – FPS