Divergencefree

Divergence-free describes a property of a vector field in which its divergence is identically zero throughout a region. If F is a vector field defined on a domain in R3, it is divergence-free when ∇·F = 0 everywhere in that domain. It is also called solenoidal.

The divergence of a vector field measures the net outflow of the field per unit volume around a point. A divergence-free field has no local sources or sinks, so the total flux through any closed surface within the domain is zero, by Gauss's divergence theorem.

Common examples include a constant vector field, whose divergence is zero because its components do not vary with position, and the rotational field F = (−y, x, 0), which has zero divergence but nonzero curl. A field like F = (x, y, z) has divergence equal to 3 and is not divergence-free.

In mathematical physics, divergence-free fields are often called solenoidal. A fundamental result is that, on a simply connected region, a divergence-free field F can be expressed as the curl of another vector field, F = ∇×A, up to a gradient potential, as in the Helmholtz decomposition. This reflects the idea that divergence-free fields carry rotational content rather than net sources.

Physical contexts frequently impose divergence-free conditions. In incompressible fluid dynamics, the velocity field is divergence-free, reflecting mass conservation. In electromagnetism, the magnetic field B is divergence-free (Gauss's law for magnetism), reflecting the absence of magnetic monopoles.

In numerical simulations, preserving divergence-free conditions is important for accuracy and stability; methods such as projection or staggered grids are used to maintain ∇·F ≈ 0.