= Solution
For any feasible <circulation with vertex demands>, sum its defining <flow balance> over all <vertices>:
$$
\sum_{v\in V}d_v
=\sum_{v\in V}\sum_{u:uv\in E}f_{uv}
-\sum_{v\in V}\sum_{w:vw\in E}f_{vw}.
$$
Every directed <edge> $ab\in E$ contributes $+f_{ab}$ exactly once to the first sum, at its head $b$, and $-f_{ab}$ exactly once to the second, at its tail $a$. The two sums therefore contain exactly the same terms. They cancel, giving
$$
\boxed{\sum_{v\in V}d_v=0.}
$$
This is global <flow conservation>: internal transport cannot create or destroy the total amount. <Lower bounds on network flow> and <capacity constraints> restrict which <flows> are feasible but do not affect the cancellation. The condition is necessary, not sufficient: insufficient <flow network edge capacities> can still prevent satisfying balanced demands.
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