All the examples of continuity equations below express the same idea. Continuity equations are the (stronger) local form of conservation laws.
Any continuity equation has a "differential form" (in terms of the divergence operator) and an "integral form" (in terms of a flux integral). In this article, only the "differential form" versions will be given; see the article divergence theorem for how to express any of these laws in "integral form".
The general form for a continuity equation is
where is some quantity, ƒ is a function describing the flux of , and s describes the generation (or removal) rate of . This equation may be derived by considering the fluxes into an infinitesimal box. This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation.
In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing (local) charge conservation, or can be derived as a consequence of two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
Another one of Maxwell's equations, Gauss's law, states that
Substitute this into equation (1) to obtain
which is the continuity equation.
In fluid dynamics, the continuity equation is a mathematical statement that, in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff's Current Law in electric circuits.
The differential form of the continuity equation is:
where is fluid density, t is time, and u is fluid velocity. If density () is a constant, as in the case of incompressible flow, the mass continuity equation simplifies to a volume continuity equation:
which means that the divergence of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.
Further, the Navier-Stokes equations form a vector continuity equation describing the conservation of linear momentum.
where J is probability flux.
so that since