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 article: continuity equation
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, 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).[7]
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.[7]
History
Joseph-Louis Lagrange introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his Mécanique Analytique. Lagrange employed surface integrals in his work on fluid mechanics.[8] He discovered the divergence theorem in 1762.[9]
Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem.[10][8] He proved additional special cases in 1833 and 1839.[11] But it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow.[12] Special cases were proven by George Green in 1828 in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism,[13][11] Siméon Denis Poisson in 1824 in a paper on elasticity, and Frédéric Sarrus in 1828 in his work on floating bodies.
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