Today we discuss Ford Fulkerson method to solve maximum flow problem. A maximum flow problem is one where we can interpret a directed graph as a flow network. Each directed edge has a stated capacity.In the maximum flow problem, we  wish to capture the greatest rate at which material can be perceived to flow through the graph without violating any capacity constraints.
The Ford Fulkerson method is defined as follows:
Ford-Fulkerson-method(G,s,t)
initialize flow f to 0
while there exists an augmenting path p
   do augment flow f along p
return f
As we can see, this method is iterative. At each iteration, we increase the flow by an augmenting path which is simply a path along which we can send more flow and then augmenting the flow along this path. We repeat this process until no augmenting path can be found. There is a way to show that this yield maximum flow which is the max flow min cut theorem. In the Ford Fulkerson method, there can be different implementations to attain the maximum flow and hence named as a method instead of an algorithm.
In the graph with a flow network and a flow, we will have some edges that can admit more flow. This additional flow that we can push through is called residual capacity. The Ford Fulkerson method repeatedly augments the flow along augmenting paths until a maximum flow can be found. The max-flow min-cut theorem  tells us that a flow is maximum if and only if its residual network contains no augmenting path. To prove this theorem, a technique called a cut of flow is used.
A cut (S, T) of flow network is a partition of V into S and T = V - S such that s belongs to S and t belongs to T. If f is the flow, then the net flow across the cut is f(S,T). The capacity of the cut is C(S,T). A minimum cut of a network is a cut whose capacity is minimum over all cuts of the network.
The max-flow min-cut theorem states that if f is a flow in a flow network G = (V, E) with source s and sink t, then the following conditions are equivalent:
1. f is a maximum flow in G
2. The residual network Gf contains no augmenting paths
3. |f| = c(S,T) for some cut (S,T) of G
The first condition implies the second condition. This we can show by proof of contradiction. If f is  the maximum flow and there is an augmenting path then the flow sum f + fp has a flow in G with a strictly greater than |f| contradicting the assumption we just made.
The third condition implies the first because the value of any flow in network is bounded from above by the capacity of any cut of G. The condition |f|  = c(S,T) thus implies that f is a maximum flow.
In a basic Ford Fulkerson algorithm, in each iteration, we find some augmenting path and increase the flow f on each edge of p by the residual capacity cf(p). In this implementation, we make use of the formula that the net flow across the cut (S,T) is f(S,T) = |f| which lets us calculate the residual capacity. Given that edges have capacity and flow and no edge implies no flow and no capacity, we update the flow f[u,v] between each pair of vertices that are connected by an edge by calculating the residual capacity in a temporary variable and augmenting the flow with this capacity.
The residual capacity is the minimum of the capacities. We update the flow in each step until there is none.