# 5. Divergence-Free Flow Regional conservation uses the accounting field $\mathbf{S}$. It gives region-by-region bookkeeping between ordered registrations. The shape of the continuous transport joining those registrations comes next. For that we use the flow field $\mathbf{F}$. This is not an imposed drift carrying a pattern through space. It is the shape of the redistribution of energy as a whole. Source-free transport cannot begin or end at an isolated point, since such points would by definition be a source or a sink of flow [^chosen-boundaries]. If energy leaves one small region, it must pass into another neighboring one. Looked at as a whole, the transport has no primitive starts or stops. It may therefore close on itself, cross the chosen boundaries of neighboring regions, or form other connected recurrent structure rather than disconnected beginnings and endings. [^chosen-boundaries]: The boundaries here are chosen bookkeeping surfaces inside $u$, not physical edges where flow begins or ends. The same continuous flow crosses them from one region into the next. 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 same process as a continuous whole, described by $\mathbf{F}$. Locally, the picture is circulation. Circulatory structure is natural in the source-free case, even though not every individual flow line need be a closed loop. 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.