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Divergence theorem
(section)
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==Applications== ===Differential and integral forms of physical laws=== As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are [[Gauss's law]] (in [[electrostatics]]), [[Gauss's law for magnetism]], and [[Gauss's law for gravity]]. ====Continuity equations==== {{main|continuity equation}} [[Continuity equation]]s 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 equation]]s that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of ''sources'' or ''sinks'' of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).<ref name="C.B. Parker 1994">{{cite book| author=C.B. Parker| edition=2nd| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| year=1994| isbn=978-0-07-051400-3| url=https://archive.org/details/mcgrawhillencycl1993park}}</ref> ===Inverse-square laws=== Any ''[[inverse-square law]]'' can instead be written in a ''Gauss's law''-type form (with a differential and integral form, as described above). Two examples are [[Gauss's law]] (in electrostatics), which follows from the inverse-square [[Coulomb's law]], and [[Gauss's law for gravity]], which follows from the inverse-square [[Newton's law of universal gravitation]]. The derivation of the Gauss's law-type equation from the inverse-square formulation or vice versa is exactly the same in both cases; see either of those articles for details.<ref name="C.B. Parker 1994"/>
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