The science
Vector field. F(x,y,z) assigns an arrow to every point. Arrows show direction and (by length and colour) magnitude; streamlines follow dx/ds = F/|F|; the tracer dots are integrated with a 4th-order Runge-Kutta step.
Divergence ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z is the net outflow per unit volume. Gauss' theorem: ∯ F·n dA = ∭ ∇·F dV. The page evaluates the left side with 2000 points on the sphere (Fibonacci spiral, equal-area weights) and the right side with 2000 equal-volume points inside, using a central finite difference for the divergence, so both sides are computed independently.
Curl ∇×F measures local rotation. Stokes' theorem: ∮ F·dl = ∬ (∇×F)·n dA. The yellow loop is placed perpendicular to the curl vector, where the circulation is largest. A small paddle wheel placed in a flow spins at ω = ½ |∇×F| about the curl axis. A wind-shear field has perfectly straight streamlines and still has curl: the wheel turns because one side is pushed harder.
Point source. For F = r̂ / r² the divergence is zero everywhere except the origin, yet the flux through any sphere around the origin is 4π. The theorem needs a smooth field, so the volume integral misses the singular source, a classic trap that the page flags.
Simplifications. Fields are static and analytic (no fluid dynamics, no viscosity). The paddle wheel is a kinematic probe, not a force model. Arrow lengths are rescaled so the strongest arrows fit the grid. Tracer speed is a visual multiplier. The finite-difference step is 1e-4.