Divergence and Curl Visualizer

What you are seeing: a parameterized 2D vector field drawn as a grid of arrows. The two numerical readouts are the divergence and curl evaluated at the center of the plot (or at any chosen point - here the origin). Four field families are available: radial source (uniform divergence, zero curl), uniform rotation (zero divergence, uniform curl), shear (zero divergence, constant negative curl), and saddle (both zero everywhere).

The scalar parameter $a$ scales each family uniformly. For the source family, $\nabla \cdot \mathbf{F} = 2a$ is independent of position; for the rotation family, $\nabla \times \mathbf{F} = 2a$ is similarly uniform. Watching the arrows reorganize as you slide $a$ makes the differential operators visceral.

Figure 1. Divergence and curl of a parameterized 2D vector field. Method: analytic derivatives; finite-difference cross-check in the invariants test.

family rotation

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WHAT TO TRY

Switch families: a source sprays the tracers outward (div > 0), a sink pulls them in (div < 0), a vortex spins them with zero divergence, and a shear or saddle mixes the two.
Read the inset: flux and circulation through the dashed loop both grow as r squared, the area scaling of the divergence and Green theorems. Flux tracks div, circulation tracks curl.
A pure rotation has div = 0 but curl = 2a: the arrows never point outward, yet the loop still has net circulation. Divergence and curl are independent.