Editing Continuity equation (section)

==={{anchor|Integral form|integral form}} Integral form===

The integral form of the continuity equation states that:
* The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward through the surface of the region, and decreases when it flows outward;
* The amount of {{math|''q''}} in a region increases when new {{math|''q''}} is created inside the region, and decreases when {{math|''q''}} is destroyed;
* Apart from these two processes, there is ''no other way'' for the amount of {{math|''q''}} in a region to change.

Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is:
{{Equation box 1
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|equation= <math>\frac{\partial q}{\partial t} + \oint_{S}\mathbf{j} \cdot d\mathbf{S} = \Sigma</math>
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|border colour = #0073CF
|background colour=#F5FFFA
}}

[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any [[closed surface]] that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]]
where
* {{math|''S''}} is any imaginary [[closed surface]], that encloses a volume {{math|''V''}},
* <math>\oint_{S} d\mathbf{S}</math> denotes a [[surface integral]] over that closed surface,
* {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}},
* {{math|'''j'''}} is the flux of {{math|''q''}},
* {{math|''t''}} is time,
* {{math|Σ}} is the net rate that {{math|''q''}} is being generated inside the volume {{math|''V''}} per unit time. When {{math|''q''}} is being generated (i.e., when   <math>\tfrac{\partial q}{\partial t}>0</math> ), the region is called a ''source'' of {{math|''q''}}, and it makes {{math|Σ}} more positive. When {{math|''q''}} is being destroyed (i.e., when   <math>\tfrac{\partial q}{\partial t}<0</math>), the region is called a ''sink'' of {{math|''q''}}, and it makes {{math|Σ}} more negative. The term {{math|Σ}} is sometimes written as <math>dq/dt|_\text{gen}</math> or the total change of {{math|''q''}} from its generation or destruction inside the control volume.

In a simple example, {{math|''V''}} could be a building, and {{math|''q''}} could be the number of living people in the building. The surface {{math|''S''}} would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of living people in the building (1) increases when living people enter the building (i.e., when there is an inward flux through the surface), (2) decreases when living people exit the building (i.e., when there is an outward flux through the surface), (3) increases when someone in the building gives birth to new life (i.e.,  when there is a positive time rate of change within the volume), and (4) decreases when someone in the building no longer lives (i.e.,  when there is a negative time rate of change within the volume). In conclusion, in this example there are four distinct ways that the net rate {{math|Σ}} may be altered.