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A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
The continuity equation for the conserved current is a statement of a conservation law. Examples of canonical conjugate quantities are: Time and energy - the continuous translational symmetry of time implies the conservation of energy; Space and momentum - the continuous translational symmetry of space implies the conservation of momentum
This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology. [ 19 ] [ 20 ] In category theory , a functor F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} between two categories is called continuous if it commutes with small limits .
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is = where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time.
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamics , electromagnetism , quantum mechanics , relativity theory , and a number of other fields, there are continuity equations that describe the conservation of mass, momentum, energy ...
This is called the charge density continuity equation + = The term on the left is the rate of change of the charge density ρ at a point. The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a ...
The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is usually expressed using the continuity equation, given in differential form as + =, where is the density (mass per unit volume), is the time, is the divergence, and is the flow velocity field.
In the analysis of a flow, it is often desirable to reduce the number of equations and/or the number of variables. The incompressible Navier–Stokes equation with mass continuity (four equations in four unknowns) can be reduced to a single equation with a single dependent variable in 2D, or one vector equation in 3D.