Divergence

Divergence is a differential operator that measures the rate at which a vector field spreads out from or converges to a point. For a vector field F = (F1, F2, F3) defined on a region of R^3, the divergence is div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z. In n-dimensional space, div F is the sum of the partial derivatives ∂Fi/∂xi. Informally, it represents the net outward flux per unit volume at a point.

Interpretation: positive divergence indicates a source of the field at that point; negative divergence indicates a sink. A field with zero divergence is called divergence-free or solenoidal, and it locally preserves volume (in fluid contexts).

Divergence is connected to the divergence theorem: the flux of F across a closed surface ∂V equals the integral of div F over the volume V: ∮∂V F · n dS = ∭V div F dV. This links local behavior to global flux.

Common applications: in physics, Maxwell's equations include div E = ρ/ε0 and div B = 0; in fluid dynamics, the continuity equation uses divergence to express conservation of mass. In heat conduction and other diffusion problems, divergence appears in constitutive relations and governing equations. Computationally, divergence is used to analyze vector fields and to test for incompressibility.

On manifolds, divergence generalizes via the Levi-Civita connection and the metric, yielding a coordinate-free definition that agrees with the usual Cartesian form in Euclidean space.