= Divergence of a flow
{title2=$\operatorname{div}\theta$}
= Divergences of a flow
{synonym}
The <divergence of a flow> at a <vertex> $v$ is net outward <flow>. For original nonnegative <edge> amounts and their signed net record, respectively,
$$
\operatorname{div}f(v)
=\sum_{w:(v,w)\in E}f_{vw}-\sum_{u:(u,v)\in E}f_{uv}
=\sum_{w\in V}\theta_{vw}.
$$
Positive <divergence of a flow> means net supply, negative <divergence of a flow> means net demand, and zero is <flow conservation>. This is the same sign convention as <flow balance>.
In the running example, the <divergences of a flow> at $(s,a,b,t)$ are $(5,0,0,-5)$. Adding the four <divergences of a flow> gives zero: every <edge> contributes positively at one endpoint and negatively at the other. In a finite <flow network>, this cancellation explains why net supply at $s$ equals net demand at $t$.
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