# How are vertex capacities defined in a flow network?

I'm new to network flows and I'm reading this topic from Cormen's Algorithms book (3rd edition) from 26 chapter. I came across this problem from the 26.1 section

Suppose that, in addition to edge capacities, a flow network has vertex capacities. That is each vertex has a limit $$l$$ on how much flow can pass though . Show how to transform a flow network $$G = (V, E)$$ with vertex capacities into an equivalent flow network $$G' = (V', E')$$ without vertex capacities, such that a maximum flow in $$G'$$ has the same value as a maximum flow in $$G$$.

I want to understand how the author has defined vertex capacities here? Because from my understanding, the most general definition would be

$$l(v) = \sum\limits_{e \text{ into } v} f(v) - \sum\limits_{e \text{ out of } v} f(v)$$

but if this is so, then for all $$v$$ not in the set of source and sink nodes,

$$l(v)= 0$$, so we are never really setting any limit. Please help me understand the vertex capacity term.

If a node $$v$$ has capacity $$C_v$$, then any flow with $$\sum_{e \text{ into } v} f(e)>C_v$$ is considered invalid (it violates the node capacity rule).