Editing Continuity equation (section)

===Definition of flux===
{{main|Flux}}

A continuity equation is useful when a '''flux''' can be defined. To define flux, first there must be a quantity {{math|''q''}} which can flow or move, such as [[mass]], [[energy]], [[electric charge]], [[momentum]], number of molecules, etc. Let {{math|''ρ''}} be the volume [[density]] of this quantity, that is, the amount of {{math|''q''}} per unit volume.

The way that this quantity {{math|''q''}} is flowing is described by its flux. The flux of {{math|''q''}} is a [[vector field]], which we denote as '''j'''. Here are some examples and properties of flux:
* The dimension of flux is "amount of {{math|''q''}} flowing per unit time, through a unit area". For example, in the mass continuity equation for flowing water, if 1 gram per second of water is flowing through a pipe with cross-sectional area 1&nbsp;cm<sup>2</sup>, then the average mass flux {{math|'''j'''}} inside the pipe is {{nowrap|(1 g/s) / cm<sup>2</sup>}}, and its direction is along the pipe in the direction that the water is flowing. Outside the pipe, where there is no water, the flux is zero.
* If there is a [[velocity field]] {{math|'''u'''}} which describes the relevant flow—in other words, if all of the quantity {{math|''q''}} at a point {{math|'''x'''}} is moving with velocity {{math|'''u'''('''x''')}}—then the flux is by definition equal to the density times the velocity field:
: <math display="block">\mathbf{j} = \rho \mathbf{u}</math>
: For example, if in the mass continuity equation for flowing water, {{math|'''u'''}} is the water's velocity at each point, and {{math|''ρ''}} is the water's density at each point, then {{math|'''j'''}} would be the mass flux, also known as the material [[Discharge (hydrology)|discharge]].
* In a well-known example, the flux of [[electric charge]] is the [[electric current density]].
[[File:Continuity eqn open surface.svg|390px|right|thumb|Illustration of how the fluxes, or flux densities, {{math|'''j'''<sub>1</sub>}} and {{math|'''j'''<sub>2</sub>}} of a quantity {{math|''q''}} pass through open surfaces {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}}. (vectors {{math|'''S'''<sub>1</sub>}} and {{math|'''S'''<sub>2</sub>}} represent [[vector area]]s that can be differentiated into infinitesimal area elements).]]
* If there is an imaginary surface {{math|''S''}}, then the [[surface integral]] of flux over {{math|''S''}} is equal to the amount of {{math|''q''}} that is passing through the surface {{math|''S''}} per unit time:
{{Equation box 1
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|equation = <math> (\text{Rate that }q\text{ is flowing through the imaginary surface }S) = \iint_S \mathbf{j} \cdot d\mathbf{S}</math>
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|border colour = #50C878
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: in which <math display="inline">\iint_S d\mathbf{S}</math> is a [[surface integral]].

(Note that the concept that is here called "flux" is alternatively termed '''flux density''' in some literature, in which context "flux" denotes the surface integral of flux density. See the main article on [[Flux]] for details.)