Flow conservation

Flow conservation requires equal total inflow and outflow at every vertex other than the source and sink:Equivalently, the divergence of a flow is zero there.^[ref]

This is a steady accounting condition: an intermediate vertex neither creates the routed resource nor accumulates it. Splitting is allowed; equality between individual incoming and outgoing edges is not required. For the running flow, the equation at $a$ is $f_{sa}=f_{ab}+f_{at}$, or $3=1+2$, and at $b$ it is $f_{sb}+f_{ab}=f_{bt}$, or $2+1=3$.

Capacity constraints alone do not ensure flow conservation. For instance, changing only $f_{ab}$ from $1$ to $0$ keeps every edge amount within its flow network edge capacity, but leaves an excess inflow of $1$ at $a$ and an excess outflow of $1$ at $b$. It is therefore not a feasible flow. Changing all affected edge amounts together to $(f_{sa},f_{sb},f_{ab},f_{at},f_{bt})=(2,2,0,2,2)$ restores both equations and gives flow value $4$.