# Finding max flow in a undirected graph

According to this we can do so by replacing every edge in the undirected graph with two edges backwards and forwards with the same capacity. But I'm having a hard time seeing how this prevents something like $$f((u,v))=c$$ and $$f((v,u))=c$$ (where c is the capacity in the original undirected edge) but then that means the original graph has flow $$2c$$. Could someone prove or explain why the capacity restrictions are still satisfied?

What you describe can happen in a feasible flow. However the net flow across the (undirected) edge $$\{u,v\}$$ would be $$0$$. Also, you say that the original graph has flow $$2c$$, this is false since the amount of flow from a vertex $$s$$ to a vertex $$t$$ in a graph is not the sum of the flows across all the edges, but rather the amount of flow leaving $$s$$ or, equivalently, entering in $$t$$.