Divergence

Divergence is a local version of flux. Where flux asks "how much of the field leaves this big closed surface?", divergence asks "how much does it leave a tiny volume around this one point, per unit volume, in the limit of zero size?". The answer at each point is a scalar: positive where the field is diverging (a source), negative where it is converging (a sink), zero where it is just passing through.

For the electric field, divergence has a spectacular property: ∇·E = ρ/ε₀. The divergence of E at any point equals the local charge density divided by the permittivity of free space. This is the differential form of Gauss's law — it says that charges are the sources and sinks of the electric field, and nothing else is. Empty regions of space have zero divergence; regions with positive charge have positive divergence; regions with negative charge have negative divergence.

The divergence theorem (Gauss's theorem) connects the two viewpoints: the flux of F through a closed surface equals the integral of ∇·F over the enclosed volume. This theorem is why "electric field lines begin and end on charges": wherever the divergence is nonzero, field lines must sprout or terminate. Divergence is the calculus that makes Faraday's intuition about lines of force rigorous.