🌀 Vector Field, Divergence & Curl 3D

Type any 3D vector field F = (P, Q, R) and see it as a forest of glowing arrows with flowing particles. Calculates divergence and curl at any point, both numerically and visually, and integrates the flux through a probe sphere to demonstrate Gauss's divergence theorem.

drag to orbit · scroll / pinch to zoom · double-click resets

P(x, y, z) — x component

Q(x, y, z) — y component

R(x, y, z) — z component

Arrow grid 5³

Particles 250

Colour arrows by

🎯 Probe point & Gauss sphere

x₀ 1

y₀ 0

z₀ 0

Sphere radius 0.6

F at probe—

Divergence ∇·F—

Curl ∇×F—

Flux out of sphere ∯F·dA—

∭ ∇·F dV inside—

Gauss check—

How this works — reading divergence and curl

A vector field attaches an arrow to every point — wind velocity, fluid flow, an electric or gravitational field. Two derivatives summarise its local behaviour:

Divergence ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z measures how much the field spreads out of a point (source, positive — shown warm) or converges into it (sink, negative — shown cool). Incompressible flow has zero divergence.
Curl ∇×F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y) is a vector along the axis a tiny paddle-wheel would spin about, with length equal to twice its angular speed. The purple arrow at the probe shows it.
Gauss's divergence theorem: the net flux out of a closed surface equals the total divergence inside, ∯ F·n dA = ∭ ∇·F dV. The tool integrates both sides independently (a 48×96 surface grid versus a 24³ spherical volume grid) so you can watch them agree.

Derivatives are central differences with h = 10⁻⁴; the particles are advected with 2nd-order Runge–Kutta. Worked example: F = (x, y, z) has divergence 3 everywhere, so the flux through a sphere of radius r is 3 · (4/3)πr³ = 4πr³.