Divergence (3D)

∇·F in three dimensions

Divergence in 3D

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div F = ∇ · F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

In 2D only the first two terms exist. The third term is what 3D adds.

Arrows are colored by the selected quantity. Toggle below to see each partial derivative's contribution.

Color by ∂F₁/∂x

Color by ∂F₂/∂y

Color by ∂F₃/∂z

Color by div F = sum (default)

Volume Sources & Sinks

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In 3D, sources and sinks fill volumes, not just areas. Red arrows = positive divergence (source). Blue = negative (sink). White = zero (incompressible).

Divergence Theorem (3D)

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∯_∂V F·n̂ dS = ∭_V div F dV

The total outward flux through a closed surface equals the volume integral of divergence inside. This is the 3D generalization of the 2D divergence theorem (circle → sphere).

Show verification sphere

Sphere radius 2.0

Gauss's Law

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∯ E·dA = Q_enc / ε₀

The divergence theorem applied to Coulomb's law gives Gauss's law: the electric flux through any closed surface depends only on the enclosed charge, not on the surface's shape or size.

The 3D Coulomb field E = r̂/r² has div E = 0 everywhere except the origin. Yet the flux through any sphere enclosing the origin is always 4π.

Drag the radius slider to verify: the flux stays constant as the sphere grows or shrinks.

Sphere radius 1.5

Display

Arrows show F direction & magnitude. Color = divergence (blue→red).
Rotate the scene to see sources/sinks in 3D space.

Flow Particles

Field Arrows

Field Lines (Streamlines)

Grid & Axes

Arrow Settings

Density (per axis)6

Arrow Scale1.0

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 = (x, y, z)

Arrows = div F

Sink ← 0 → Source

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