# 5. Divergence-Free Flow Regional conservation uses the flow $\mathbf{S}$. It gives region by region numerical accounting. The shape of the reorganization as a complete process comes next. For that we introduce a flow field $\mathbf{F}$. In empty space, a source-free transport cannot begin or end at an isolated point. If energy leaves one small region, it must pass into another neighboring one. Looked at as a whole, the transport forms closed loops rather than disconnected starts and stops. This is the geometric content of calling the flow divergence-free. For the fundamental flow field, that condition is $$ \nabla \cdot \mathbf{F} = 0. $$ Source-free transport, understood as a complete pattern, has no primitive endpoints. Local gain or loss of stored energy is still tracked by the regional accounting of chapter 4 through $\mathbf{S}$. What is added here is the shape of the process as a whole, described by $\mathbf{F}$. Locally, the picture is circulation. Circulation lines are closed. The next question is how local evolution of $\mathbf{F}$ must be described in order to preserve this source-free structure. Divergence-free language is therefore not the origin of anything. It is the mathematical encoding of a prior physical fact: source-free flow has no primitive beginnings or endings. The connected structure comes first. The vector equation is the language we later use to write it down.