Water flows through rivers, gas through pipes, electric current through wires, traffic through cities, people through buildings and cash flows through our hands. Indeed, charge can flow and form an electric current, but charge cannot just disappear: it is conserved. There is always a distribution of charge over space and time, called its density. The continuity equation is the precise mathematical relation between a flow or current and a change in density, and is called a conservation law. This type of equation is called a partial differential equation because it involves time and spatial derivatives of functions of time and space.

The continuity equation is a flow equation that expresses a conservation law that is local because it applies to any given volume, on all scales. If in an office there are a certain number of people, that number can only change if people enter or leave (through the doors). This is true for any one room, but also for a floor, a whole building or a city.

The continuity equation basically states that the amount ρ (of charge, or gas, or money) in a given volume element can only change if there is a corresponding current or flow j going through the surface bounding the volume element: charge may flow, but it cannot disappear.

For a stationary flow• we have ∂r/∂t=0. This condition says that the ‘fluid’ is incompressible, because the local density ρ of the fluid does not change. For a stationary flow the equation says that for any volume, the same amount of fluid has to flow in and out. That's why a river streams faster through a narrow and shallow passage. This condition typically applies to a real liquid, but not to a gas or a stream of people. Many people may enter a room while nobody leaves, in which case the average ρ in the room would increase.