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All the examples of continuity equations below express the same idea; they are all really examples of the same concept. Continuity equations are the (stronger) local form of conservation laws. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...
Electromagnetic theory
In electromagnetic theory, the continuity equation is derived from 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, Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...
Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...
In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ...
Charge density is the amount of electric charge per unit volume. ...
Derivation One of Maxwell's equations states that  Taking the divergence of both sides results in - ,
but the divergence of a curl is zero, so that  Another one of Maxwell's equations states that  Substitute this into equation (1) to obtain  which is the continuity equation.
Interpretation Current density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.
Fluid dynamics In fluid dynamics, a continuity equation is an equation of conservation of mass. Its differential form is Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ...
of conservation of mass/matter ( The Lomonosov-Lavoisier law ) states that the mass of a system of substances is constant, regardless of the processes acting inside the system. ...
 where ρ is density, t is time, and u is fluid velocity.
Quantum mechanics In quantum mechanics, the conservation of probability also yields a continuity equation. Let P(x, t) be a probability density and write A simple introduction to this subject is provided in Basics of quantum mechanics. ...
In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ...
 where J is probability flux. In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. ...
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