Divergence-free field

A divergence-free or solenoidal field is a differentiable vector field $u$ satisfying $\operatorname{div}u=0$ . A nonsmooth field may satisfy the same condition in the distributional sense, which must be specified when pointwise derivatives are unavailable.

Constructions and cutoffs

Every curl of a $C^2$ vector potential is divergence-free. A scalar cutoff does not generally preserve the condition: $\operatorname{div}(\chi u)=\nabla\chi\cdot u$ for divergence-free $u$ . Cutting off a potential and then taking its curl preserves zero divergence.

Fluid constraint

The kinematic condition for incompressible flow is spatial divergence freedom at every time. The Navier–Stokes equations couple this constraint to a momentum equation.